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Bergerak-rata-rata-berdasarkan-estimator-of-integrated-varians

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EViews 9.5 Daftar Fitur EViews menawarkan beragam fitur canggih untuk penanganan data, statistik dan analisis ekonometrik, peramalan dan simulasi, penyajian data, dan pemrograman. Meskipun kami tidak dapat mencantumkan semuanya, daftar berikut ini menawarkan sekilas fitur EView yang penting: Penanganan Data Dasar Label angka numerik, alfanumerik (string), dan label tanggal. Ekstensif perpustakaan operator dan statistik, matematika, tanggal dan fungsi string. Bahasa yang kuat untuk penanganan ekspresi dan transformasi data yang ada menggunakan operator dan fungsi. Sampel dan objek sampel memudahkan pemrosesan pada himpunan bagian data. Dukungan untuk struktur data yang kompleks termasuk data tanggal reguler, data tanggal tidak teratur, data cross-section dengan pengenal observasi, tanggal, dan panel data yang tidak bertanggal. Multi-halaman workfiles. EViews asli, berbasis disk database menyediakan fitur query yang kuat dan integrasi dengan EViews workfiles. Mengkonversi data antara EViews dan berbagai format spreadsheet, statistik, dan database, termasuk (namun tidak terbatas pada): File Microsoft Access dan Excel (termasuk .XSLX dan .XLSM), file Dataset Gauss, file SAS Transport, file asli dan portabel SPSS, File stata, file teks atau biner ASCII yang diformat mentah, database HTML, atau ODBC dan kueri (dukungan ODBC disediakan hanya di Edisi Enterprise). Dukungan OLE untuk menghubungkan keluaran EViews, termasuk tabel dan grafik, ke paket lain, termasuk Microsoft Excel, Word dan Powerpoint. Dukungan OLEDB untuk membaca workfiles dan database EViews menggunakan klien atau program kustom OLEDB. Dukungan untuk database FRED (Data Data Federal Reserve). Edisi Enterprise mendukung basis data Global Insight DRIPro dan DRIBase, Haver Analytics DLX, FAME, EcoWin, Bloomberg, EIA, CEIC, Datastream, FactSet, dan Moodys. EViews Microsoft Excel Add-in memungkinkan Anda untuk menghubungkan atau mengimpor data dari workfiles dan database EViews dari dalam Excel. Dukungan drag and drop untuk membaca data cukup memasukkan file ke dalam EView untuk konversi otomatis dan menghubungkan data asing ke format workfile EViews. Alat yang ampuh untuk membuat halaman kerja baru dari nilai dan tanggal dalam rangkaian yang ada. Cocokkan penggabung, gabung, tambahkan, subset, ubah ukuran, urut, dan buat ulang (tumpukan dan hapuskan) file kerja. Konversi frekuensi otomatis yang mudah digunakan saat menyalin atau menghubungkan data antar halaman dengan frekuensi yang berbeda. Konversi frekuensi dan penggabungan kecocokan mendukung pemutakhiran dinamis kapan pun perubahan data menjadi mendasar. Auto-update seri formula yang secara otomatis menghitung ulang setiap kali perubahan data mendasar. Konversi frekuensi yang mudah digunakan: cukup salin atau tautkan data antar halaman dengan frekuensi yang berbeda. Alat untuk resampling dan random number generation untuk simulasi. Pembangkitan bilangan acak untuk 18 fungsi distribusi berbeda menggunakan tiga generator bilangan acak yang berbeda. Dukungan untuk akses cloud drive, memungkinkan Anda membuka dan menyimpan file langsung ke akun Dropbox, OneDrive, Google Drive dan Box. Penanganan Data Seri Waktu Dukungan terpadu untuk menangani tanggal dan data deret waktu (reguler dan tidak teratur). Dukungan untuk data frekuensi reguler reguler (Tahunan, Semi-tahunan, Triwulanan, Bulanan, Dua Belas, Minggu, Sepuluh hari, mingguan, harian - 5 hari seminggu, harian - 7 hari seminggu). Dukungan untuk data frekuensi tinggi (intraday), memungkinkan frekuensi berjam-jam, menit, dan detik. Selain itu, ada sejumlah frekuensi reguler yang jarang dijumpai, termasuk Multi-year, Bimonthly, Fortnight, Ten-Day, dan Daily dengan rentang waktu yang sewenang-wenang dalam sehari. Fungsi dan operator seri waktu khusus: tertinggal, perbedaan, perbedaan log, moving averages, dll. Konversi frekuensi: berbagai metode high-to-low dan low-to-high. Perataan eksponensial: single, double, Holt-Winters, dan ETS smoothing. Alat built-in untuk memutihkan regresi. Penyaringan Hodrick-Prescott. Filter band-pass (frekuensi): Baxter-King, Christiano-Fitzgerald tetap panjang dan filter asimetris sampel penuh. Penyesuaian musiman: Sensus X-13, X-12-ARIMA, TramoSeats, moving average. Interpolasi untuk mengisi nilai yang hilang dalam rangkaian: Linear, Log-Linear, Catmull-Rom Spline, Cardinal Spline. Statistik Ringkasan data dasar oleh-kelompok ringkasan. Pengujian kesetaraan: uji-t, ANOVA (seimbang dan tidak seimbang, dengan atau tanpa varian heteroskedastis), Wilcoxon, Mann-Whitney, Median Chi-square, Kruskal-Wallis, van der Waerden, uji-F, Siegel-Tukey, Bartlett , Levene, Brown-Forsythe. Tabulasi tabulasi satu arah dengan ukuran asosiasi (Koefisien Phi, Cramers V, Koefisien Kontingensi) dan uji kemandirian (Pearson Chi-Square, Likelihood Ratio G2). Kovarian dan analisis korelasi termasuk Pearson, rank rank Spearman, Kendalls tau-a dan tau-b dan analisis parsial. Analisis komponen utama meliputi plot scree, biplots dan plot pemuatan, dan perhitungan nilai komponen tertimbang. Analisis faktor yang memungkinkan perhitungan ukuran asosiasi (termasuk kovariansi dan korelasi), taksiran keunikan, estimasi pemuatan faktor dan nilai faktor, serta melakukan estimasi diagnostik dan rotasi faktor dengan menggunakan satu dari lebih 30 metode ortogonal dan miring yang berbeda. Fungsi Distribusi Empiris (EDF) Pengujian untuk distribusi Normal, Eksponensial, Ekstrim, Logistik, Chi-kuadrat, Weibull, atau Gamma (Kolmogorov-Smirnov, Lilliefors, Cramer-von Mises, Anderson-Darling, Watson). Histogram, Frekuensi Poligon, Poligon Frekuensi Tepi, Histogram Bergeser Rata-rata, CDF-survivor-quantile, Quantile-Quantile, kepadatan kernel, distribusi teoretis yang sesuai, kotak petak. Scatterplots dengan garis regresi parametrik dan non-parametrik (LOWESS, polynomial lokal), regresi kernel (Nadaraya-Watson, lokal linier, polinomial lokal). Atau elips kepercayaan. Autokorelasi Seri Waktu, autokorelasi parsial, korelasi silang, statistik-Q. Tes kausalitas Granger, termasuk kausalitas kausal Granger. Tes akar unit: Augmented Dickey-Fuller, GLS mentransformasikan Dickey-Fuller, Phillips-Perron, KPSS, Eliot-Richardson-Stock Point Optimal, Ng-Perron, serta tes untuk akar unit dengan breakpoints. Tes kointegrasi: Johansen, Engle-Granger, Phillips-Ouliaris, Park menambahkan variabel, dan stabilitas Hansen. Tes kemandirian: Tes rasio Ragam Brock, Dechert, Scheinkman dan LeBaron: Lo dan MacKinlay, bootstrap Kim liar, peringkat Wrights, skor rangking dan tanda tangan. Wald dan beberapa rasio varians rasio perbandingan (Richardson dan Smith, Chow dan Denning). Perhitungan varians dan kovarians jangka panjang: kovarian jangka panjang simetris atau atau satu sisi menggunakan kernel nonparametrik (Newey-West 1987, Andrews 1991), parametrik VARHAC (Den Haan dan Levin 1997), dan kernel yang diprakarsai (Andrews dan Monahan 1992) Metode. Selain itu, EViews mendukung metode pemilihan bandwidth otomatis Marshall (1991) dan Newey-West (1994) untuk estimator kernel, dan kriteria informasi berdasarkan metode pemilihan panjang lag untuk perkiraan VARHAC dan prewhitening. Statistik berdasarkan panel dan kolam renang dan berdasarkan statistik. Tes akar unit: Levin-Lin-Chu, Breitung, Im-Pesaran-Shin, Fisher, Hadri. Tes kointegrasi: Pedroni, Kao, Maddala dan Wu. Panel dalam kovarian seri dan komponen utama. Uji kausalitas Dumitrescu-Hurlin (2012). Uji ketergantungan lintas bagian. Estimasi Regresi Linear dan nonlinear ordinary least squares (multiple regression). Regresi linier dengan PDL pada sejumlah variabel independen. Regresi kuat Turunan analitik untuk estimasi nonlinier. Kotak terkecil tertimbang Kesalahan standar putih dan Newey-West kuat. Kesalahan standar HAC dapat dihitung dengan menggunakan kernel nonparametrik, parametrik VARHAC, dan metode kernel yang telah digunakan sebelumnya, dan memungkinkan metode seleksi bandwidth Andrews dan Newey-West untuk penduga kernel, dan kriteria berdasarkan metode pemilihan panjang jeda untuk estimasi VARHAC dan prewhitening. Regresi kuantum linier dan penyimpangan absolut (LAD), termasuk perhitungan kortikosterase Hubers Sandwich dan bootstrap. Regresi bertahap dengan tujuh prosedur seleksi berbeda. Ambang batas regresi termasuk TAR dan SETAR. ARMA dan ARMAX Model linier dengan moving average autoregresif, autoregresif musiman, dan kesalahan rata-rata pergerakan musiman. Model nonlinier dengan spesifikasi AR dan SAR. Estimasi menggunakan metode backcasting Box dan Jenkins, conditional least squares, ML atau GLS. Model ARFIMA yang terintegrasi secara acak. Variabel Instrumental dan GMM Linear dan nonlinear two-stage least squaresinstrumental variables (2SLSIV) dan estimasi Generalized Method of Moments (GMM). Estimasi linear dan nonlinear 2SLSIV dengan kesalahan AR dan SAR. Informasi Terbatas Maximum Likelihood (LIML) dan estimasi K-class. Berbagai spesifikasi matriks bobot GMM (White, HAC, User-provided) dengan kontrol terhadap iterasi matriks bobot. Pilihan estimasi GMM termasuk terus memperbarui estimasi (CUE), dan sejumlah opsi kesalahan standar baru, termasuk kesalahan standar Windmeijer. Diagnostik khusus IVGMM meliputi Uji Instrumen Orthogonalitas, Uji Endogeneitas Regresor, Uji Instrumen Lemah, dan tes breakpoint spesifik GMM. ARCHGARCH GARCH (p, q), EGARCH, TARCH, Komponen GARCH, Power ARCH, GARCH Terintegrasi. Persamaan mean linier atau nonlinier dapat mencakup istilah ARCH dan ARMA, baik persamaan mean dan varians yang memungkinkan variabel eksogen. Normal, Student t, dan Generalized Error Distributions. Bollerslev-Wooldridge kesalahan standar yang kuat. In-dan out-of sample prakiraan varians bersyarat dan mean, dan komponen permanen. Model Variabel Ketergantungan Terbatas Binary Logit, Probit, dan Gompit (Nilai Ekstrim). Memerintahkan Logit, Probit, dan Gompit (Nilai Ekstrim). Model disensor dan terpotong dengan kesalahan nilai normal, logistik, dan ekstrim (Tobit, dll.). Hitunglah model dengan spesifikasi Poisson, negative binomial, dan quasi-maximum likelihood (QML). Model Seleksi Heckman. Kesalahan standar kuat HuberWhite. Model hitungan mendukung model linier umum atau kesalahan standar QML. Hosmer-Lemeshow dan Andrews Goodness-of-Fit untuk pengujian model biner. Mudah menyimpan hasil (termasuk residu dan gradien umum) ke objek EView baru untuk analisis lebih lanjut. Mesin estimasi GLM umum dapat digunakan untuk memperkirakan beberapa model ini, dengan opsi untuk memasukkan kovariansi yang kuat. Panel DataPooled Time Series, Cross-Sectional Data Linear dan estimasi nonlinier dengan penampang melintang dan efek tetap atau acak. Pilihan estimator kuadratik tidak bias (QUEs) untuk varians komponen dalam model efek acak: Swamy-Arora, Wallace-Hussain, Wansbeek-Kapteyn. Estimasi 2SLSIV dengan penampang melintang dan efek tetap atau acak. Estimasi dengan kesalahan AR menggunakan kuadrat terkecil nonlinear pada spesifikasi yang ditransformasikan. Kuadrat terkecil yang umum, estimasi 2SLSIV umum, estimasi GMM yang memungkinkan spesifikasi cross-section atau period heteroskedastic dan berkorelasi. Estimasi data panel linier dinamis menggunakan perbedaan pertama atau penyimpangan ortogonal dengan instrumen spesifik periode tertentu (Arellano-Bond). Tes korelasi serial panel (Arellano-Bond). Perhitungan kesalahan standar yang kuat mencakup tujuh jenis kesalahan standar White and Panel-corrected standard (PCSE) yang kuat. Pengujian batasan koefisien, variabel yang dihilangkan dan berlebihan, Hausman menguji efek acak berkorelasi. Unit uji akar unit: Levin-Lin-Chu, Breitung, Im-Pesaran-Shin, uji tipe Fisher menggunakan tes ADF dan PP (Maddala-Wu, Choi), Hadri. Perkiraan kointegrasi panel: OLS yang dimodifikasi sepenuhnya (FMOLS, Pedroni 2000) atau Kotak Terkenal Biasa Dinamis (DOLS, Kao dan Chaing 2000, Mark dan Sul 2003). Estimasi kelompok kolam renang (PMG). Model Linear Generalized Normal, Poisson, Binomial, Binomial Negatif, Gamma, Inverse Gaussian, Mena Eksponensial, Mean Daya, Keluarga Squate Binomial. Identitas, log, log-complement, logit, probit, log-log, log-log gratis, invers, power, rasio odds daya, Box-Cox, Rasio Box-Cox odds ratio. Perbedaan varians dan frekuensi sebelumnya. Spesifikasi dispersi fixed, Pearson Chi-Sq, penyimpangan, dan spesifikasi dispersi yang ditentukan pengguna. Dukungan untuk estimasi QML dan pengujian. Quadratic Hill Climbing, Newton-Raphson, IRLS - Fisher Scoring, dan algoritma estimasi BHHH. Kovarian koefisien biasa dihitung dengan menggunakan Hessian yang diharapkan atau diamati atau produk luar gradien. Perkiraan kovariansi yang kuat menggunakan metode GLM, HAC, atau HuberWhite. Single Equation Cointegrating Regression Support untuk tiga metode estimasi yang sangat efisien, OLS yang dimodifikasi sepenuhnya (Phillips dan Hansen 1992), Canonical Cointegrating Regression (Park 1992), dan Dynamic OLS (Saikkonen 1992, Stock dan Watson 1993 Engle dan Granger (1987) dan Phillips dan Ouliaris (1990) uji berbasis residual, uji ketidakstabilan Hansens (1992b), dan Parks (1992) menambahkan uji variabel. Spesifikasi fleksibel dari trend dan deterministik regresor dalam persamaan dan spesifikasi regresi kointegrasi. Estimasi fitur lengkap varians jangka panjang untuk FMOLS dan CCR Pemilihan jeda otomatis atau tetap untuk kelambatan DOLS dan lead dan untuk regresi pemutihan varians jangka panjang. Rescaled OLS dan perhitungan error standar yang kuat untuk DOLS. Kemungkinan Maksimum yang Ditentukan Pengguna Gunakan ekspresi seri EViews standar untuk menjelaskan kemungkinan log kontribusi. Contoh untuk logit multinomial dan kondisional, model transformasi Box-Cox, model peralihan disekuilibrium, model probit S dengan kesalahan heteroskedastis, nested logit, pemilihan sampel Heckman, dan model bahaya Weibull. Sistem Persamaan Estimasi linier dan nonlinear. Kuadrat terkecil, 2SLS, estimasi bobot rata-rata, Regresi yang Tidak Terkait, dan Kuantitas Tiga Tahap Paling Sedikit. GMM dengan matriks pembobotan White dan HAC. Estimasi AR menggunakan kuadrat terkecil nonlinear pada spesifikasi yang ditransformasikan. Informasi Lengkap Kemungkinan Maksimum (FIML). Perkirakan faktor struktural dalam VARs dengan menerapkan batasan jangka pendek atau jangka panjang. Bayesian VARs. Fungsi respon impuls dalam berbagai format tabel dan grafik dengan kesalahan standar dihitung secara analitis atau dengan metode Monte Carlo. Guncangan respon impuls dihitung dari faktorisasi Cholesky, residu deviasi satu unit atau satu standar (mengabaikan korelasi), impuls umum, faktorisasi struktural, atau bentuk vectormatrix yang ditentukan pengguna. Menerapkan dan menguji batasan linier pada hubungan kointegrasi dan dan koefisien penyesuaian dalam model VEC. Melihat atau menghasilkan hubungan kointegrasi dari model VEC yang diperkirakan. Diagnostik ekstensif termasuk: uji kausalitas Granger, uji pengecualian lag bersama, evaluasi kriteria panjang lag, uji korelasi, autokorelasi, normalitas dan heteroskedastisitas, pengujian kointegrasi, diagnostik multivariat lainnya. Korelasi Konstruktif Bersyarat Multivariat (p, q), Diagonal VECH (p, q), diagonal BEKK (p, q), dengan istilah asimetris. Pilihan parameterisasi yang ekstensif untuk matriks koefisien VEKS Diagonal. Variabel eksogen yang diizinkan dalam mean dan varians persamaan nonlinier dan persyaratan AR diperbolehkan dalam persamaan rata-rata. Bollerslev-Wooldridge kesalahan standar yang kuat. Normal atau Student t multivariate error distribution Pilihan derivatif numerik analitik atau (cepat atau lambat). (Analytics derivatif tidak tersedia untuk beberapa model yang kompleks.) Menghasilkan kovarians, varians, atau korelasi dalam berbagai format tabel dan grafik dari model ARCH yang diperkirakan. Algoritma State Space Kalman untuk memperkirakan model struktural single-and multiequation yang ditentukan pengguna. Variabel eksogen dalam persamaan negara dan spesifikasi varians sepenuhnya parameter. Buat satu langkah di depan, filter, atau penghalusan sinyal, keadaan, dan kesalahan. Contohnya meliputi parameter time-varying, ARMA multivariat, dan model volatilitas stokastik quasilikelihood. Testing and Evaluation Actual, pas, residual plots. Uji Wald untuk pembedaan koefisien linier dan nonlinier elastisitas kepercayaan menunjukkan wilayah kepercayaan bersama dari dua fungsi parameter yang diperkirakan. Diagnostik koefisien lainnya: koefisien standar dan elastisitas koefisien, interval kepercayaan, faktor inflasi varian, dekomposisi varians koefisien. Variabel LR terlewati dan berlebihan, korelasi residual residual dan kuadrat dan statistik Q, korelasi serial residual dan uji ARCH LM. Tes heteroskedastisitas White, Breusch-Pagan, Godfrey, Harvey dan Glejser. Diagnostik stabilitas: uji breakpoint dan perkiraan Chow, uji breakpoint Quandt-Andrews yang tidak diketahui, uji breakpoint Bai-Perron, tes RESET Ramsey, estimasi rekursif OLS, statistik pengaruh, plot leverage. Diagnosa persamaan ARMA: grafik atau tabel akar invers dari polinomial AR andor MA, membandingkan pola autokorelasi teoritis (perkiraan) dengan pola korelasi aktual untuk residu struktural, menampilkan respon impuls ARMA terhadap kejutan inovasi dan frekuensi ARMA spektrum. Mudah menyimpan hasil (koefisien, koefisien matriks kovariansi, residu, gradien, dll.) Ke objek EView untuk analisis lebih lanjut. Lihat juga Estimasi dan Sistem Persamaan untuk prosedur pengujian khusus tambahan. Peramalan dan Simulasi Peramalan statis atau dinamis di luar perkiraan dari perkiraan objek persamaan dengan perhitungan kesalahan standar ramalan. Grafik perkiraan dan evaluasi perkiraan sampel: RMSE, MAE, MAPE, Theil Koefisien Ketidakseimbangan dan proporsi alat bangunan model state-of-the-art untuk peramalan beberapa persamaan dan simulasi multivariat. Model persamaan dapat dimasukkan dalam teks atau sebagai link untuk memperbarui otomatis pada re-estimasi. Tampilkan struktur ketergantungan atau variabel endogen dan eksogen dari persamaan Anda. Gauss-Seidel, pemecah model Broyden dan Newton untuk simulasi stokastik dan stochastic. Solusi forward non-stokastik memecahkan harapan model yang konsisten. Simulasi Stochasitc dapat menggunakan residu bootstrap. Selesaikan masalah kontrol sehingga variabel endogen mencapai target yang ditentukan pengguna. Normalisasi persamaan yang canggih, menambahkan faktor dan mengesampingkan dukungan. Mengelola dan membandingkan beberapa skenario solusi yang melibatkan berbagai rangkaian asumsi. Tampilan dan tampilan model built-in menampilkan hasil simulasi dalam bentuk grafis atau tabular. Grafik dan Tabel Garis, titik petak, area, bar, spike, musiman, pie, xy-line, scatterplots, boxplots, error bar, high-low-open-close, dan area band. Grafik kategoris dan ringkasan yang kuat dan mudah digunakan. Memperbarui otomatis grafik yang diperbarui sebagai perubahan data mendasar. Pengamatan info dan tampilan nilai saat Anda mengarahkan kursor ke titik pada grafik. Histogram, sejarah bergeser rata-rata, poligon frekuensi, poligon frekuensi tepi, kotak petak, kerapatan kernel, distribusi teoretis yang sesuai, kotak peti, CDF, survivor, quantile, quantile-quantile. Scatterplots dengan kombinasi parametrik dan nonparametrik kernel (Nadaraya-Watson, lokal linier, polinomial lokal) dan tetangga terdekat (LOWESS) garis regresi, atau elips kepercayaan. Customization point-and-click atau command-based. Ekstensif menyesuaikan latar belakang grafik, bingkai, legenda, sumbu, penskalaan, garis, simbol, teks, bayangan, pudar, dengan fitur grafik grafik yang disempurnakan. Tabel kustomisasi dengan kontrol atas sel font wajah, ukuran, dan warna, warna latar belakang sel dan perbatasan, penggabungan, dan anotasi. Salin dan tempel grafik ke aplikasi Windows lainnya, atau simpan grafik sebagai metafiles biasa atau metafile Windows, encapsulated file PostScript, bitmap, GIF, PNGs atau JPGs. Salin dan tempel tabel ke aplikasi lain atau simpan ke file RTF, HTML, atau teks. Mengelola grafik dan tabel bersama dalam objek spul yang memungkinkan Anda menampilkan banyak hasil dan analisis dalam satu objek Perintah dan Pemrograman Bahasa perintah berorientasi objek menyediakan akses ke item menu. Batch eksekusi perintah dalam file program. Looping dan kondisi percabangan, subrutin, dan pengolahan makro. Objek vektor string dan string untuk pemrosesan string. Ekstensif perpustakaan string dan daftar string fungsi. Dukungan matriks yang luas: manipulasi matriks, multiplikasi, inversi, produk Kronecker, solusi eigenvalue, dan dekomposisi nilai tunggal. Antarmuka Eksternal dan Add-Ins EViews Dukungan server otomasi COM sehingga program eksternal atau skrip dapat meluncurkan atau mengendalikan EViews, mentransfer data, dan menjalankan perintah EViews. EViews menawarkan aplikasi dukungan klien COM Automation untuk server MATLAB dan R sehingga EViews dapat digunakan untuk meluncurkan atau mengendalikan aplikasi, mentransfer data, atau menjalankan perintah. EViews Microsoft Excel Add-in menawarkan antarmuka sederhana untuk mengambil dan menghubungkan dari dalam Microsoft Excel (2000 dan yang lebih baru) ke objek seri dan matriks yang tersimpan dalam workfiles dan database EViews. Infrastruktur Add-in EViews menawarkan akses tanpa batas ke program yang ditentukan pengguna menggunakan perintah, menu EViews standar, dan antarmuka objek. Download dan instal add-in standar dari situs web EViews. Home AboutContact Untuk informasi penjualan silahkan email saleseviews Untuk dukungan teknis silahkan email supporteviews Mohon sertakan nomor seri anda dengan semua korespondensi email. Untuk informasi kontak tambahan, lihat halaman Tentang kami. Meskipun manajemen portofolio tidak banyak berubah selama 40 tahun setelah karya mani Markowitz dan Sharpe, pengembangan teknik penganggaran berisiko menandai tonggak penting dalam memperdalam hubungan antara manajemen risiko dan aset. . Paritas risiko kemudian menjadi model investasi investasi yang populer setelah krisis keuangan global di tahun 2008. Saat ini, dana pensiun dan investor institusi menggunakan pendekatan ini dalam pengembangan pengindeksan cerdas dan redefinisi kebijakan investasi jangka panjang. Pengantar Risk Parity and Budgeting memberikan perawatan mutakhir metode alternatif ini untuk optimasi Markowitz. Ini menghasilkan eksposur keuangan terhadap ekuitas dan komoditas, mempertimbangkan risiko kredit dalam pengelolaan portofolio obligasi, dan merancang kebijakan investasi jangka panjang. Bagian pertama buku ini memberikan penjelasan teoritis tentang optimasi portofolio dan paritas risiko. Penulis membahas teori portofolio modern dan menawarkan panduan komprehensif untuk penganggaran risiko. Setiap bab di bagian kedua menyajikan penerapan paritas risiko ke kelas aset tertentu. Teks tersebut mencakup indeksasi ekuitas berbasis risiko (juga disebut beta pintar) dan menunjukkan bagaimana menggunakan teknik penganggaran risiko untuk mengelola portofolio obligasi. Ini juga mengeksplorasi investasi alternatif, seperti komoditas dan hedge fund, dan menerapkan teknik paritas risiko ke kelas multi-aset. Buku-buku lampiran pertama menyediakan materi teknis mengenai masalah optimasi, fungsi copula, dan alokasi aset dinamis. Lampiran kedua berisi 30 latihan tutorial. Solusi untuk latihan, slide untuk instruktur, dan program komputer Gauss untuk mereproduksi contoh, tabel, dan angka buku tersedia di situs web buku. Chapman HallCRC Financial Mathematics Series, 410 halaman Pergi ke situs web La Gestion dActifs Kuantitatif La konvergensi de la gestion traditionnelle et de la gestion alternative, gundul bagian, lmergence de la gestion kuantitatif, bagian dautre, refltent la profonde mutation de la gestion dactifs. Ce livre mengusulkan daborder ces diffrents thmes, tous bass sur le contrle risque et les modles dallocation dactifs. Cet ouvrage offre un panorama des diffrentes capitalits de la gestion kuantitatif, allant de la gestion indicielle la gestion hedge fund en passant par les gestions structure, diversifie, profil ou de performance absolue. Louvrage prsente galement les diffrentes stratgies quantitatives, que sont les stratgies de rplication, dallocation, doptions, de volatilit, darbitrage ou encore les stratgies de suivi de tendance et de retour la moyenne. Il montre en particulier comment loptimisation de portefeuille, lconomtrie financire et les stratgies de gestion semboitent pour ex une stratgie quantitative. Gambaran umum tentang unsur-unsur nombreus dan contohnya menggambarkan kelas-kelas doping (tindakan, taux dintret, perubahan et matires premires). Ce livre sadresse aux tudiants de master, qui veulent des quants et travailler dans la financial quantitative, et aux professionnels qui cherchent mieux comprendre les modles mathmatiques et statistiques utiliss dans la gestion dactifs. Edisi Ekonomi, Koleksi Keuangan, 680 halaman Tlcharger la table des matires. Les extraits du livre Lannexe sur les exercices. La mengoreksi des exercices. (Voir aussi les programs koresponden Gauss) et les applications numriques du livre La Gestion des Risques Pemodal (deuxime dition) Cette nouvelle dition pada loccasion de revoir lagi le texte, supprimer un nombre de dveloppement qui ne sont plus dactualit et dapporter un clairage Nouveau par rapport la crise actuelle. Elle contient aussi de nouvelles ilustrasi et de nouvelles aplikasi di dalam konsep kontur serangga mieux qui peuvent apparatre complexes. Plusieurs sujets ont fait lobjet de nouveaux dveloppements comme par exemple le risque de liquidit, les produits exotiques, le risque de contrepartie sur opration de march, des produits structurs, la gestion du risque de maret dans la gestion dactifs, la dpendance dans les drives de Kontribusi crdit et la en risque. Enfin, cette deuxime dition est accompagne dexercices permettant de vrifier ses connaissances. Edisi Ekonomi, Koleksi Keuangan, 560 halaman Tlcharger les programs Gauss. Les errata du livre. La table des matires et la correction des exercices du livre La Gestion des Risques Pemodal La gestion des risques financiers est en pleine volution sous la pression de la rglementation prudentielle et du dveloppement des outils pour mieux les matriser. Le Comit de Ble a publi le Nouvel Accord sur le rasio internasional de solvabilit (Ble II) le 26 juin 2004, ete Commission Europenne a dj mengadopsi proposisi les diffrentes de cet accord. Cet sesuai dengan t accueilli favorablement par la profession bancaire et les tablissements pemodal ont maintenant deux ans et demi pour mener bien cette rforme afin den bnficier pleinement. Les banques nont cependant pas attendu le Nouvel Accord tuangkan modernizer leur gestion des risques. Depuis dix ans, dibantu oleh teknik un dveloppement du manajemen risiko dan peningkatan pensiunan les plus sophistiqus. Le Nouvel Accord participe dailleurs cette volution, puisquil vise dfinir un modal rglementaire plus proche du capital conomique obtenu avec les modles internes. Le prsent ouvrage sinscrit dans ces deux lignes directrices. Rglementasi du risque et modlisation du risque. Il sadresse aussi bien des tudiants de troisime cycle, qui dsirent acqurir une culture financire du risque et de sa gestion, qu des professionnels qui cherchent mieux comprendre les fondements de la modlisation mathmatique du risque. Edisi Ekonomi, Koleksi Gestion, 455 halaman Program Tlcharger les Gauss et les applications numriques du livre TSM (Seri Waktu dan Wavelets untuk Keuangan) TSM adalah perpustakaan GAUSS untuk pemodelan time series baik dalam domain domain dan domain waktu. Hal ini terutama dirancang untuk analisis dan estimasi ARMA, proses VARX, model ruang negara, proses fraksional dan model struktural. Untuk mempelajari model ini, alat khusus telah dikembangkan seperti prosedur simulasi, analisis spektral, matriks Hankel, dll. Estimasi didasarkan pada prinsip Maximum Likelihood atau Metode Momen Gnereralized dan batasan linier dapat dengan mudah dipaksakan. Ini juga berisi beberapa metode penyaringan (Kalman Filter, FLS dan GFLS) dan beberapa prosedur untuk analisis Time-Frequency dari sinyal 1-D (analisis wavelet dan analisis paket wavelet). Source code: 300 Ko, Contoh kode: 390 Ko, Manual: 230 halaman. Deskripsi TSM di halaman web Gauss Aptech Systems Unduh contoh TSM Pendahuluan la programmation sous Gauss 1995 Desain Global, 660 halaman T. Roncalli dan G. Weisang Sebagai regulator di seluruh dunia maju menuju reformasi kehati-hatian dalam sistem keuangan global untuk mengatasi masalah risiko sistemik, Lingkup tugas yang menyapu menyentuh area dan aktor pasar keuangan yang biasanya tidak dipandang penting secara sistemik sebelumnya. Gagasan bahwa industri pengelolaan aset dapat berkontribusi terhadap risiko sistemik adalah baru, dan menjamin pemeriksaan terperinci untuk membentuk kebijakan yang memadai. Dalam tulisan ini, setelah meninjau definisi risiko sistemik dan bagaimana bank dan asuransi yang penting secara sistemik ditetapkan, kami meninjau kembali kegiatan industri pengelolaan aset dan cara-cara penyampaiannya terhadap transmisi risiko sistemik. Kami kemudian melihat secara rinci pada proposal Maret 2015 oleh FSB-IOSCO untuk metodologi penilaian untuk identifikasi lembaga keuangan non-bank yang secara sistemik penting. Kami membandingkan dan mendiskusikan dengan data empiris bagaimana pameran metodologi menentang literatur dan pasca krisis 2007-2008 mengungkapkan peran industri manajemen aset dalam memberikan kontribusi terhadap risiko sistemik. Kami menemukan bahwa usulan saat ini sebagian gagal untuk secara memadai mengidentifikasi kandidat alami untuk penentuan yang secara sistemik penting dan mungkin membingungkan institusi besar dengan institusi strategis sistemik yang memberikan kerugian kekayaan terlalu penting mengenai potensi gangguan ekonomi dan dislokasi pasar yang nyata. Akhirnya, kami meminta pendekatan yang lebih kuat dan sensitif terhadap risiko untuk mengidentifikasi lembaga keuangan yang penting secara sistemik. Risiko sistemik, SIFI, manajer aset, pemilik aset, keterkaitan, risiko likuiditas, risiko reputasi, risiko bisnis, risiko kredit counterpart, risiko pasar, periode likuidasi, dana indeks, dana pasar uang, dana yang diperdagangkan, hedge fund. Download file PDF J-C. Richard dan T. Roncalli Pada artikel ini, kami mempertimbangkan kerangka kerja baru untuk memahami portofolio berbasis risiko (GMV, EW, ERC dan MDP). Kerangka ini mirip dengan model varians terbatas dari Jurczenko et al. (2013), namun dengan definisi batasan diversifikasi lainnya. Masalah optimasi yang sesuai kemudian dapat dipecahkan dengan menggunakan algoritma CCD. Hal ini memungkinkan kita untuk memperluas hasil Cazalet dkk. (2014) dan untuk lebih memahami hubungan trade-off antara penurunan volatilitas, pelacakan kesalahan dan diversifikasi risiko. Secara khusus, kami menunjukkan bahwa portofolio beta cerdas berbeda karena secara implisit menargetkan tingkat penurunan volatilitas yang berbeda. Kami juga mengembangkan strategi beta beta yang baru dengan mengelola tingkat penurunan volatilitas dan menunjukkan bahwa mereka menyajikan properti yang menarik dibandingkan dengan portofolio berbasis risiko tradisional. Smart beta, alokasi berbasis risiko, portofolio varian minimum, GMV, EW, ERC, MDP, optimasi portofolio, algoritma CCD. Download the PDF file Z. Cazalet and T. Roncalli The capital asset pricing model (CAPM) developed by Sharpe (1964) is the starting point for the arbitrage pricing theory (APT). It uses a single risk factor to model the risk premium of an asset class. However, the CAPM has been the subject of important research, which has highlighted numerous empirical contradictions. Based on the APT theory proposed by Ross (1976), Fama and French (1992) and Carhart (1997) introduce other common factors models to capture new risk premia. For instance, they consequently define equity risk factors, such as market, value, size and momentum. In recent years, a new framework based on this literature has emerged to define strategic asset allocation. Similarly, index providers and asset managers now offer the opportunity to invest in these risk factors through factor indexes and mutual funds. These two approaches led to a new paradigm called factor investing (Ang, 2014). Factor investing seems to solve some of the portfolio management issues that emerged in the past, in particular for long-term investors. However, some questions arise, especially with the number of risk factors growing over the last few years (Cochrane, 2011). What is a risk factor Are all risk factors well-rewarded What is their level of stability and robustness How should we allocate between them The main purpose of this paper is to understand and analyze the factor investing approach in order to answer these questions. Factor investing, risk premium, CAPM, risk factor model, anomaly, size, value, momentum, volatility, idiosyncratic risk, liquidity, carry, quality, mutual funds, hedge funds, alternative beta, strategic asset allocation. Download the PDF file Risk parity is an allocation method used to build diversified portfolios that does not rely on any assumptions of expected returns, thus placing risk management at the heart of the strategy. This explains why risk parity became a popular investment model after the global financial crisis in 2008. However, risk parity has also been criticized because it focuses on managing risk concentration rather than portfolio performance, and is therefore seen as being closer to passive management than active management. In this article, we show how to introduce assumptions of expected returns into risk parity portfolios. To do this, we consider a generalized risk measure that takes into account both the portfolio return and volatility. However, the trade-off between performance and volatility contributions creates some difficulty, while the risk budgeting problem must be clearly defined. After deriving the theoretical properties of such risk budgeting portfolios, we apply this new model to asset allocation. First, we consider long-term investment policy and the determination of strategic asset allocation. We then consider dynamic allocation and show how to build risk parity funds that depend on expected returns. Risk parity, risk budgeting, expected returns, ERC portfolio, value-at-risk, expected shortfall, active management, tactical asset allocation, strategic asset allocation. Download the PDF file T. Roncalli and B. Zheng The liquidity of exchange traded funds is of utmost importance for regulators, investors and providers. However, the study of liquidity is still in its infancy. In this work, we show some stylised facts of liquidity statistics (dailyintraday spread, trading volume, etc.). We also propose a new liquidity measure combining these statistics. In this case, liquidity is a power function of the spread where the parameters are determined by actual trading volumes. We also study the relationship between the liquidity of ETFs and the liquidity of the underlying index. We show that they are correlated on a daily basis, but not in terms of intraday frequency. We also define a measure of liquidity improvement and apply it to the EURO STOXX 50 index. Exchange traded fund, liquidity, spread, trading volume, order book, liquidity improvement. Download the PDF file T. Roncalli and G. Weisang T. Griveau-Billion, J-C. Richard and T. Roncalli In this paper we propose a cyclical coordinate descent (CCD) algorithm for solving high dimensional risk parity problems. We show that this algorithm converges and is very fast even with large covariance matrices (n 500). Comparison with existing algorithms also shows that it is one of the most efficient algorithms. Risk parity, risk budgeting, ERC portfolio, cyclical coordinate descent algorithm, SQP algorithm, Jacobi algorithm, Newton algorithm, Nesterov algorithm. Download the PDF file Z. Cazalet, P. Grison and T. Roncalli In this article, we consider smart beta indexing, which is an alternative to capitalization-weighted (CW) indexing. In particular, we focus on risk-based (RB) indexing, the aim of which is to capture the equity risk premium more effectively. To achieve this, portfolios are built which are more diversified and less volatile than CW portfolios. However, RB portfolios are less liquid than CW portfolios by construction. Moreover, they also present two risks in terms of passive management: tracking difference risk and tracking error risk. Smart beta investors then have to a puzzle out the trade-off between diversification, volatility, liquidity and tracking error. This article examines the trade-off relationships. It also defines the return components of smart beta indexes. Smart beta, risk-based indexing, minimum variance portfolio, risk parity, equally weighted portfolio, equal risk contribution portfolio, diversification, low beta anomaly, low volatility anomaly, tracking error, liquidity. Download the PDF file B. Bruder, N. Gaussel, J-C. Richard and T. Roncalli The mean-variance optimization (MVO) theory of Markowitz (1952) for portfolio selection is one of the most important methods used in quantitative finance. This portfolio allocation needs two input parameters, the vector of expected returns and the covariance matrix of asset returns. This process leads to estimation errors, which may have a large impact on portfolio weights. In this paper we review different methods which aim to stabilize the mean-variance allocation. In particular, we consider recent results from machine learning theory to obtain more robust allocation. Portfolio optimization, active management, estimation error, shrinkage estimator, resampling methods, eigendecomposition, norm constraints, Lasso regression, ridge regression, information matrix, hedging portfolio, sparsity. Download the PDF file M. Hassine and T. Roncalli Fund selection is an important issue for investors. This topic has spawned abundant academic literature. Nonetheless, most of the time, these works concern only active management, whereas many investors, such as institutional investors, prefer to invest in index funds. The tools developed in the case of active management are also not suitable for evaluating the performance of these index funds. This explains why information ratios are usually used to compare the performance of passive funds. However, we show that this measure is not pertinent, especially when the tracking error volatility of the index fund is small. The objective of an exchange traded fund (ETF) is precisely to offer an investment vehicle that presents a very low tracking error compared to its benchmark. In this paper, we propose a performance measure based on the value-at-risk framework, which is perfectly adapted to passive management and ETFs. Depending on three parameters (performance difference, tracking error volatility and liquidity spread), this efficiency measure is easy to compute and may help investors in their fund selection process. We provide some examples, and show how liquidity is more of an issue for institutional investors than retail investors. Passive management, index fund, ETF, information ratio, tracking error, liquidity, spread, value-at-risk. Download the PDF file T. Roncalli and G. Weisang T. Roncalli and G. Weisang Portfolio construction and risk budgeting are the focus of many studies by academics and practitioners. In particular, diversification has spawn much interest and has been defined very differently. In this paper, we analyze a method to achieve portfolio diversification based on the decomposition of the portfolios risk into risk factor contributions. First, we expose the relationship between risk factor and asset contributions. Secondly, we formulate the diversification problem in terms of risk factors as an optimization program. Finally, we illustrate our methodology with some real life examples and backtests, which are: budgeting the risk of Fama-French equity factors, maximizing the diversification of an hedge fund portfolio and building a strategic asset allocation based on economic factors. Risk parity, risk budgeting, factor model, ERC portfolio, diversification, concentration, Fama-French model, hedge fund allocation, strategic asset allocation. Download the PDF file B. Bruder, L. Culerier and T. Roncalli Several years ago, the concept of target-date funds emerged to complement traditional balanced funds in defined-contribution pension plans. The main idea is to delegate the dynamic allocation with respect to the retirement date of individuals to the portfolio manager. Owing to its long-term horizon, a target-date fund is unique and cannot be compared to a mutual fund. Moreover, the objective of the individual is to contribute throughout their working life by investing a part of their income in order to maximise their pension benefits. The main purpose of this article is to analyse and understand dynamic allocation in a target-date fund framework. We show that the optimal exposure in the risky portfolio varies over time and is very sensitive to the parameters of both the market and the investors. We then deduce some practical guidelines to better design target-date funds for the asset management industry. Target-date fund, lifecycle fund, retirement system, dynamic asset allocation, stochastic optimal control, market portfolio, risk aversion, stockbond asset mix policy. Download the PDF file Basle II, credit risk measurement, credit portfolio management, time-inconsistency problems. Download the PDF file N. Baud, A. Frachot and T. Roncalli December 01, 2002 Intense reflections are being conducted at the moment regarding the way to pool heteregenous data coming from both banks internal systems and industry-pooled databases. We propose here a sound methodology. As it relies on maximum likelihood principle, it is thus statistically rigorous and should be accepted by supervisors. We believe that it solves the most part of data heterogeneity and scaling issues. Operational risk, capital charge, threshold, conditional distribution, maximum likelihood. Download the PDF file N. Baud, A. Frachot and T. Roncalli It is widely recognized that calibration on internal data may not suffice for computing an accurate capital charge against operational risk. However, pooling external and internal data lead to unacceptable capital charges as external data are generally skewed toward large losses. In a previous paper, we have developped a statistical methodology to ensure that merging both internal and external data leads to unbiased estimates of the loss distribution. This paper shows that this methodology is applicable in real-life risk management and that it permits to pool internal and external data together in an appropriate way. The paper is organized as follows. We first discuss how external databases are designed and how their design may result in statistical flaws. Then we develop a model for the data generating process which underlies external data. In this model, the bias comes simply from the fact that external data are truncated above a specific threshold while this threshold may be either constant but known, or constant but unknown, or finally stochastic. We describe the rationale behind these three cases and we provide for each of them a methodology to circumvent the related bias. In each case, numerical simulations and practical evidences are given. Operational risk, internal data, external data, consortium data, threshold. Download the PDF file N. Baud, A. Frachot and T. Roncalli Slides of the conference Seminarios de Matematica Financiera, Instituto MEFF - Risklab. Madrid. Operational risk, LDA, internal data, external data, implied threshold. Download the PDF file J-F. Jouanin, G. Riboulet and T. Roncalli January 31, 2002 Non-technical version of the paper Modelling dependence for credit derivatives with copulas. Copulas, intensity models, Moodys diversity score. Download the PDF file A. Frachot and T. Roncalli Januray 29, 2002 The Loss Distribution Approach has many appealing features since it is expected to be much more risk-sensitive than any other methods taken into consideration by the last proposals by the Basel Committee. Thus this approach is expected to provide significantly lower capital charges for banks whose track record is particularly good relatively to their exposures and compared with industry-wide benchmarks. Unfortunately LDA when calibrated only on internal data is far from being satisfactory from a regu- latory perspective as it could likely underestimate the necessary capital charge. This happens for two reasons. First if a bank has experienced a lower-than-average number of events, it will benefit from a lower-than-average capital charge even though its good track record happened by chance and does not result from better-than-average risk management practices. As a consequence, LDA is acceptable as long as internal frequency data are tempered by industry-wide references. As such, it immediately raises the issue of how to cope with both internal frequency data and external benchmarks. This paper proposes a solution based on credibility theory which is widely used in the insurance industry to tackle analogous problems. As a result, we show how to make the statistical adjustment to temper the information conveyed by internal frequency data with the use of external references. Similarly if the calibration of severity parameters ignores external data, then the severity distribution will likely be biased towards low-severity losses since internal losses are typically lower than those recorded in industry-wide databases. Again from a regulatory perspective LDA cannot be accepted unless both internal and external data are merged and the merged database is used in the calibration process. Here again it raises the issue regarding the best way to merge these data. Obviously it cannot be done without any care since if internal databases are directly fuelled with external data, severity distributions will be strongly biased towards high-severity losses. This paper proposes also a statistical adjustment to make internal and external databases comparable with one another in order to permit a safe and unbiased merging. Operational risk, LDA, internal data, external data, credibility theory. Download the PDF file October 26, 2001 Slides of the conference Sminaire de Mathmatiques et Finance Louis Bachelier, Institut Henri Poincar. Copulas, credit derivatives, multi-asset options. Download the PDF file S. Coutant, V. Durrleman, G. Rapuch and T. Roncalli September 5, 2001 In this paper, we use copulas to define multivariate risk-neutral distributions. We can then derive general pricing formulas for multi-asset options and best possible bounds with given volatility smiles. Finally, we then apply the copula framework to define forward-looking indicators of the dependence function between asset returns. Copulas, risk-neutral distribution, change of numraire, option pricing, implied multivariate RND. Download the PDF file J-F. Jouanin, G. Rapuch, G. Riboulet and T. Roncalli In this paper, we address the problem of incorporating default dependency in intensity-based credit risk models. Following the works of Li 2000, Giesecke 2001 and Schonbucher and Schubert 2001, we use copulas to model the joint distribution of the default times. Two approaches are considered. The first one consists in modelling the joint survival function directly with survival copulas of default times, whereas in the second approach, copulas are used to correlate the threshold exponential random variables. We compare these two approaches and give some results about their relationships. Then we try some simulations of simple products, such as first-to-defaults. Finally, we discuss the calibration issue according to Moodys diversity score. Copulas, intensity models, Cox processes, Bessel processes, Moodys diversity score. Download the PDF file G. Rapuch and T. Roncalli In this short note, we consider some problems of two-asset options pricing. In particular, we investigate the relationship between options prices and the correlation parameter in the Black-Scholes model. Then, we consider the general case in the framework of the copula construction of risk-neutral distributions. This extension involves results on the supermodular order applied to the Feynman-Kac representation. We show that it could be viewed as a generalization of a maximum principle for parabolic PDE. Copulas, two-asset options (Spread, Basket, Min, Max, BestOf, WorstOf), supermodular order, concordance order, Frchet bounds, Feynman-Kac representation, maximum principle, parabolic PDE. Download the PDF file Slides of the conference Statistics 2001, Concordia University, Montral, Canada. Copulas, gaussian assumption, operational risk, risk-neutral copula, Heston model. Download the PDF file P. Georges, A-G. Lamy, E. Nicolas, G. Quibel and T. Roncalli In this paper, we review the use of copulas for multivariate survival modelling. In particular, we study properties of survival copulas and discuss the dependence measures associated to this construction. Then, we consider the problem of competing risks. We derive the distribution of the failure time and order statistics. After having presented statistical inference, we finally provide financial applications which concern the life time value (attrition models), the link between default, prepayment and credit life, the measure of risk for a credit portfolio and the pricing of credit derivatives. Survival copula, frailty model, ageing concepts, competing risks, failure time, order statistics, prepayment, credit risk measure, default mode, correlated defaults, risk-bucket capital charge, default digital put, credit default swap, first-to-default. Download the PDF file Slides of the seminar Stochastic Models in Finance, Ecole Polytechnique, Paris, 23042001. Copulas, quantile regression, markov copulas, credit risk, uniform convergence, operations on distribution functions. Download the PDF file A. Frachot, P. Georges and T. Roncalli In this paper, we explore the Loss Distribution Approach (LDA) for computing the capital charge of a bank for operational risk where LDA refers to statisticalactuarial methods for modelling the loss distribution. In this framework, the capital charge is calculated using a Value-at-Risk measure. In the first part of the paper, we give a detailed description of the LDA implementation and we explain how it could be used for economic capital allocation. In particular, we show how to compute the aggregate loss distribution by compounding the loss severity distribution and the loss frequency distribution, how to compute the total Capital-at-Risk using copulas, how to control the upper tail of the loss severity distribution with order statistics. In the second part of the paper, we compare LDA with the Internal Measurement Approach (IMA) proposed by the Basel Committee on Banking Supervision to calculate regulatory capital for operational risk. LDA and IMA are bottom-up internal measurement models which are apparently different. Nevertheless, we could map LDA into IMA and give then some justifications about the choice done by regulators to define IMA. Finally, we provide alternative ways of mapping both methods together. Operational risk, aggregated loss, compound distribution, loss severity, loss frequency, Panjer algorithm, Capital-at-Risk, economic capital allocation, order statistics, LDA, IMA, RPI, copulas. Download the PDF file V. Durrleman, A.Nikeghbali and T. Roncalli Slides for the International Finance Conference, Hammam-Sousse, Tunisia, 03172001. Copulas, risky dependence function, singular copulas, extreme points, quantile aggregation, spread option. Download the PDF file January 26, 2001 Slides of the seminar Statistical Methods in Integrated Risk Management organized by Frontiers in Finance. Copulas, 2D option pricing, markov processes, credit risk, CreditMetrics, CreditRisk, first-to-default. Download the PDF file J. Bodeau, G. Riboulet and T. Roncalli December 15, 2000 In this paper, we consider non-uniform grids to solve PDE. We derive the theta-scheme algorithm based on finite difference methods and show its consistency. We then apply it to different option pricing problems. Theta-scheme, non-uniform grids, temporal grids, cubic spline interpolation, european option, american option, barrier option. Download the PDF file Download the corresponding GAUSS library November 16, 2000 Slides of the seminar Financial Applications of Copulas. Copulas, financial applications, risk management, statistical modelling, probabilistic metric spaces, markov operators, quasi-copulas. Download the PDF file V. Durrleman, A.Nikeghbali and T. Roncalli November 23, 2000 In this paper, we consider the open question on Spearmans rho and Kendalls tau of Nelsen 1991. Using a technical hypothesis, we can answer in the positive. One question remains open: how can we understand the technical hypothesis Because this hypothesis is not right in general, we could find some pathological cases which contradict Nelsens conjecture. Spearmans rho, Kendalls tau, cubic copula. Download the PDF file E. Bouy, V. Durrleman, A. Nikeghbali G. Riboulet and T. Roncalli March 23, 2001 (First version: November 10, 2000) In this paper, we show that copulas are a very powerful tool for risk management since it fulfills one of its main goals: the modelling of dependence between the individual risks. That is why this approach is an open field for risk. Copulas, market risk, credit risk, operational risk. Download the PDF file A. Costinot, T.Roncalli and J. Teiumlletche October 24, 2000 We consider the problem of modelling the dependence between financial markets. In financial economics, the classical tool is the Pearson (or linear correlation) coefficient to compare the dependence structure. We show that this coefficient does not give a precise information on the dependence structure. Instead, we propose a conceptual framework based on copulas. Two applications are proposed. The first one concerns the study of extreme dependence between international equity markets. The second one concerns the analysis of the East Asian crisis. Linear correlation, extreme value theory, quantile regression, concordance order, Deheuvels copula, contagion, Asian crisis. Download the PDF file A. Costinot, G. Riboulet et T. Roncalli September 15, 2000 Les banques ont aujourdhui la possibilit de mettre en place un modle interne de risque de march. Lune des composantes indispensables de ce modle est la cration dun programme de stress testing. Cet article prsente un outil potentiel pour la construction dun tel programme. la thorie des valeurs extrmes. Aprs avoir rappel la rglementation propre au stress testing et les principaux rsultats de cette thorie, nous montrons comment les utiliser pour construire des scnarios unidimensionnels, multidimensionnels et enfin pour quantifier des scnarios de crise labors partir de mthodologies diffrentes. Aux considrations mthodologiques sont adjoints les rsultats des simulations que nous avons ralises sur diffrentes sries financires. Copules, fonction de dpendance de queue stable, thorie des valeurs extrmes, stress testing. Tlcharger le fichier PDF V. Durrleman, A. Nikeghbali and T. Roncalli September 10, 2000 In this paper, we consider the problem of bounds for distribution convolutions and we present some applications to risk management. We show that the upper Frchet bound is not always the more risky dependence structure. It is in contradiction with the belief in finance that maximal risk corresponds to the case where the random variables are comonotonic. Triangle functions, dependency bounds, infimal, supremal and sigma-convolutions, Makarov inequalities, Value-at-Risk, square root rule, Dallaglio problem, Kantorovich distance. Download the PDF file V. Durrleman, A.Nikeghbali and T. Roncalli In this paper, we study the approximation procedures introduced by Li, Mikusinski, Sherwood and Taylor 1997. We show that there exists a bijection between the set of the discretized copulas and the set of the doubly stochastic matrices. For the Bernstein and checkerboard approximations, we then provide analytical formulas for the Kendalls tau and Spearmans rho concordance measures. Moreover, we demonstrate that these approximations do not exhibit tail dependences. Finally, we consider the general case of approximations induced by partitions of unity. Moreover, we show that the set of copulas induced by partition of unity is a Markov sub-algebra with respect to the -product of Darsow, Nguyen and Olsen 1992. Doubly stochastic matrices, Bernstein polynomials approximation, checkerboard copula, partitions of unity, Markov algebras, product of copulas. Download the PDF file V. Durrleman, A.Nikeghbali and T. Roncalli In this paper, we give a few methods for the choice of copulas in financial modelling. Maximum likelihood method, inference for margins, CML method, point estimator, non parametric estimation, Deheuvels copula, copula approximation, discrete L norm. Download the PDF file V. Durrleman, A.Nikeghbali and T. Roncalli We study how copulas properties are modified after some suitable transformations. In particular, we show that using appropriate transformations permits to fit the dependence structure in a better way. gamma-transformation, Kendalls tau, Spearmans rho, upper tail dependence. Download the PDF file V. Durrleman, A.Kurpiel, G. Riboulet and T. Roncalli In this paper, we consider 2D option pricing. Most of the problems come from the fact that only few closed-form formulas are available. Numerical algorithms are also necessary to compute option prices. This paper examines some topics on this subject. Numerical integration methods, Gauss quadratures, Monte Carlo, Quasi Monte Carlo, Sobol sequences, Faure sequences, two-dimensional PDE, Hopscotch, LOD, ADI, MOL, Stochastic volatility model, Malliavin calculus. Paper presented at the 17th International Conference in Finance organized by the French Finance Association, Paris (June 28, 2000). Download the PDF file E. Bouy, V. Durrleman, A. Nikeghbali, G. Riboulet and T. Roncalli Copulas are a general tool to construct multivariate distributions and to investigate dependence structure between random variables. However, the concept of copula is not popular in Finance. In this paper, we show that copulas can be extensively used to solve many financial problems. Multivariate distribution, dependence structure, concordance measures, scoring, Markov processes, risk management, extreme value theory, stress testing, operational risk, market risk, credit risk. Paper presented at the 17th International Conference in Finance organized by the French Finance Association, Paris (June 27, 2000) and at First World Congress of the Bachelier Finance Society (June 29, 2000). Download the PDF file N. Baud, P. Demey, D. Jacomy, G. Riboulet et T. Roncalli Comme son nom lindique, le Plan Epargne Logement est un produit dpargne qui permet dacqurir des droits prts pour financer un ventuel achat immobilier. Pour que les tablissements financiers et les particuliers y trouvent un intrt commun, le lgislateur a mis en place un systme de prime pendant la phase dpargne. Celui-ci est peru comme un systme incitatif pour le particulier et doit permettre dassurer la rentabilit du produit pour la banque. Une note rdige par le Trsor en 1996 conclut la rentabilit du PEL pour les banques. Largument repose sur le fait que les pertes (ventuelles) supportes par la banque pendant la phase demprunt sont largement compenses par les revenus de la phase dpargne. En rponse cette note lAFB sest attache montrer le contraire en incluant les coucircts lis aux risques de taux (Note de lAFB du 16121996). Il nest donc pas du tout certain que le systme mis en place soit rentable pour ltablissement financier. Dautant plus que le Plan Epargne Logement est un produit financier relativement complexe et que celui-ci contient diffrentes options caches. Le calcul de sa rentabilit est donc beaucoup plus difficile que ceux prsents par le Trsor ou lAFB. Cest pourquoi le GRO a tent de modliser les options caches du PEL, de les valoriser et de calculer la rentabilit finale de ce produit. Plan dpargne logement, option cache de conversion, option amricaine, problme de contrle optimal. Tlcharger le fichier PDF N. Baud, A. Frachot, P. Igigabel, P. Martineu and T. Roncalli December 1, 1999 Capital allocation within a bank is getting more important as the regulatory requirements are moving towards economic-based measures of risk. Banks are urged to build sound internal measures of credit and market risks for all their activities. Internal models for credit, market and operational risks are fundamental for bank capital allocation in a bottom-up approach. But this approach has to be completed by a top-down approach in order to give to bank managers a more comprehensive (but less detailed) vision of the allocation efficiency. From a top-down viewpoint, we are considering the different business lines of a bank as assets. Then the capital has to be allocated in order to balance a portfolio in an optimal way. In this respect, a bank has to evaluate not only the expected return and the risk of every business line, but also the correlation matrix of these business lines returns. If a bank usually has a good knowledge of its expected returns and risks, the problem is more complex in the case of the correlation matrix: to cope with the lack of internal data and information, we develop an approach based on a Market Factor Model and estimate an implied correlation matrix using the returns of a panel of banks. The allocation problem is not exactly the problem a bank is confronted to. It more precisely deals with capital reallocation. Moving from an allocation to a new one generates costs that have to be taken into account to ensure that the new allocation is better than the former one. That is why reallocation signals are more interesting: they do not point out the optimal allocation but they allow the implementation of a dynamic policy that leads to an optimal situation. Capital allocation, top-down, bottom-up, factor model, optimisation problem, Lagrange multipliers. Paper presented at Les petits djeuners de la Finance, Paris (January 27, 2000). Download the PDF file January 13, 1999 In this paper, we consider the use of interest rate contingent claims as indicators for the monetary policy. We analyze two approches: one based on the term structure of zero bonds and another based on interest-rate option derivatives. We show how traditional tools based on the Black framework could be biased to build indicators for monetary policy. In fact, the second approach could not be viewed as an alternative approach, but as a complementary approach of the term structure approach. Yield curve, Hull-White trinomial model, monetary policy. Download the PDF file A. Kurpiel and T. Roncalli December 8, 1998 The purpose of this paper is to analyse different implications of the stochastic behavior of asset prices volatilities for option hedging purposes. We present a simple stochastic volatility model for option pricing and illustrate its consistency with financial stylized facts. Then, assuming a stochastic volatility environment, we study the accuracy of Black and Scholes implied volatility-based hedging. More precisely, we analyse the hedging ratios biases and investigate different hedging schemes in a dynamic setting. option hedging, stochastic volatility, Heston model, delta, gamma, vega. Download the PDF file A. Kurpiel and T. Roncalli November 17, 1998 In this paper, we consider Hopscotch methods for solving two-state financial models. We first derive a solution algorithm for two-dimensional partial differential equations with mixed boundary conditions. We then consider a number of financial applications including stochastic volatility option pricing, term structure modelling with two states and elliptic irreversible investment problems. Two-dimensional PDE, Hopscotch method, parabolic financial models, elliptic problems. Download the PDF file Download the corresponding GAUSS library Thse de lUniversit de Montesqieu-Bordeaux IV. Structure par terme, taux zro, taux forward, mthode de Nelson-Siegel, modles factoriels, processus de diffusion, modle de Black-Derman-Toy, modle de Hull-White. Tlcharger le fichier PDF Tlcharger la bibliothque Gauss J-S. Pentecte, T. Roncalli et M-A. Sngas 1998, LARE, Universit de Bordeaux IV J-S. Pentecte and T. Roncalli 1997, LARE, University of Bordeaux IVModeling and Simulation Systems Simulation: The Shortest Route to Applications This site features information about discrete event system modeling and simulation. It includes discussions on descriptive simulation modeling, programming commands, techniques for sensitivity estimation, optimization and goal-seeking by simulation, and what-if analysis. Advancements in computing power, availability of PC-based modeling and simulation, and efficient computational methodology are allowing leading-edge of prescriptive simulation modeling such as optimization to pursue investigations in systems analysis, design, and control processes that were previously beyond reach of the modelers and decision makers. To search the site . try E dit F ind in page Ctrl f. Enter a word or phrase in the dialogue box, e.g. quot optimizationquot or quot sensitivityquot If the first appearance of the wordphrase is not what you are looking for, try F ind Next. Statistics and Probability for Simulation Topics in Descriptive Simulation Modeling Techniques for Sensitivity Estimation Simulation-based Optimization Techniques Metamodeling and the Goal seeking Problems What-if Analysis Techniques Introduction Summary Computer system users, administrators, and designers usually have a goal of highest performance at lowest cost. Modeling and simulation of system design trade off is good preparation for design and engineering decisions in real world jobs. In this Web site we study computer systems modeling and simulation. We need a proper knowledge of both the techniques of simulation modeling and the simulated systems themselves. The scenario described above is but one situation where computer simulation can be effectively used. In addition to its use as a tool to better understand and optimize performance andor reliability of systems, simulation is also extensively used to verify the correctness of designs. Most if not all digital integrated circuits manufactured today are first extensively simulated before they are manufactured to identify and correct design errors. Simulation early in the design cycle is important because the cost to repair mistakes increases dramatically the later in the product life cycle that the error is detected. Another important application of simulation is in developing virtual environments. misalnya for training. Analogous to the holodeck in the popular science-fiction television program Star Trek, simulations generate dynamic environments with which users can interact as if they were really there. Such simulations are used extensively today to train military personnel for battlefield situations, at a fraction of the cost of running exercises involving real tanks, aircraft, etc. Dynamic modeling in organizations is the collective ability to understand the implications of change over time. This skill lies at the heart of successful strategic decision process. The availability of effective visual modeling and simulation enables the analyst and the decision-maker to boost their dynamic decision by rehearsing strategy to avoid hidden pitfalls. System Simulation is the mimicking of the operation of a real system, such as the day-to-day operation of a bank, or the value of a stock portfolio over a time period, or the running of an assembly line in a factory, or the staff assignment of a hospital or a security company, in a computer. Instead of building extensive mathematical models by experts, the readily available simulation software has made it possible to model and analyze the operation of a real system by non-experts, who are managers but not programmers. A simulation is the execution of a model, represented by a computer program that gives information about the system being investigated. The simulation approach of analyzing a model is opposed to the analytical approach, where the method of analyzing the system is purely theoretical. As this approach is more reliable, the simulation approach gives more flexibility and convenience. The activities of the model consist of events, which are activated at certain points in time and in this way affect the overall state of the system. The points in time that an event is activated are randomized, so no input from outside the system is required. Events exist autonomously and they are discrete so between the execution of two events nothing happens. The SIMSCRIPT provides a process-based approach of writing a simulation program. With this approach, the components of the program consist of entities, which combine several related events into one process. In the field of simulation, the concept of principle of computational equivalence has beneficial implications for the decision-maker. Simulated experimentation accelerates and replaces effectively the wait and see anxieties in discovering new insight and explanations of future behavior of the real system. Consider the following scenario. You are the designer of a new switch for asynchronous transfer mode (ATM) networks, a new switching technology that has appeared on the marketplace in recent years. In order to help ensure the success of your product in this is a highly competitive field, it is important that you design the switch to yield the highest possible performance while maintaining a reasonable manufacturing cost. How much memory should be built into the switch Should the memory be associated with incoming communication links to buffer messages as they arrive, or should it be associated with outgoing links to hold messages competing to use the same link Moreover, what is the best organization of hardware components within the switch These are but a few of the questions that you must answer in coming up with a design. With the integration of artificial intelligence, agents and other modeling techniques, simulation has become an effective and appropriate decision support for the managers. By combining the emerging science of complexity with newly popularized simulation technology, the PricewaterhouseCoopers, Emergent Solutions Group builds a software that allows senior management to safely play out what if scenarios in artificial worlds. For example, in a consumer retail environment it can be used to find out how the roles of consumers and employees can be simulated to achieve peak performance. Statistics for Correlated Data We concern ourselves with n realizations that are related to time, that is having n correlated observations the estimate of the mean is given by mean S X i n, where the sum is over i 1 to n. where the sum is over j 1 to m, then the estimated variance is: 1 43 2A S 2 n Where S 2 the usual variance estimate r j,x the jth coefficient of autocorrelation m the maximum time lag for which autocorrelations are computed, such that j 1, 2, 3. m As a good rule of thumb, the maximum lag for which autocorrelations are computed should be approximately 2 of the number of n realizations, although each r j,x could be tested to determine if it is significantly different from zero. Sample Size Determination: We can calculate the minimum sample size required by n 1 43 2A S 2 t 2 ( d 2 mean 2 ) Application: A pilot run was made of a model, observations numbered 150, the mean was 205.74 minutes and the variance S 2 101, 921.54, estimate of the lag coefficients were computed as: r 1,x 0.3301 r 2,x 0.2993, and r 3,x 0.1987. Calculate the minimum sample size to assure the estimate lies within 43 d 10 of the true mean with a 0.05. n (1.96) 2 (101,921.54) 1 43 2 (1-14) 0.3301 43 (1 - 24) 0.2993 43 (1- 34) 0.1987 (0.1) 2 (205.74) 2 What Is Central Limit Theorem For practical purposes, the main idea of the central limit theorem (CLT) is that the average of a sample of observations drawn from some population with any shape-distribution is approximately distributed as a normal distribution if certain conditions are met. In theoretical statistics there are several versions of the central limit theorem depending on how these conditions are specified. These are concerned with the types of assumptions made about the distribution of the parent population (population from which the sample is drawn) and the actual sampling procedure. One of the simplest versions of the theorem says that if is a random sample of size n (say, n larger than 30) from an infinite population, finite standard deviation. then the standardized sample mean converges to a standard normal distribution or, equivalently, the sample mean approaches a normal distribution with mean equal to the population mean and standard deviation equal to standard deviation of the population divided by the square root of sample size n. In applications of the central limit theorem to practical problems in statistical inference, however, statisticians are more interested in how closely the approximate distribution of the sample mean follows a normal distribution for finite sample sizes, than the limiting distribution itself. Sufficiently close agreement with a normal distribution allows statisticians to use normal theory for making inferences about population parameters (such as the mean ) using the sample mean, irrespective of the actual form of the parent population. It is well known that whatever the parent population is, the standardized variable will have a distribution with a mean 0 and standard deviation 1 under random sampling. Moreover, if the parent population is normal, then it is distributed exactly as a standard normal variable for any positive integer n. The central limit theorem states the remarkable result that, even when the parent population is non-normal, the standardized variable is approximately normal if the sample size is large enough (say gt 30). It is generally not possible to state conditions under which the approximation given by the central limit theorem works and what sample sizes are needed before the approximation becomes good enough. As a general guideline, statisticians have used the prescription that if the parent distribution is symmetric and relatively short-tailed, then the sample mean reaches approximate normality for smaller samples than if the parent population is skewed or long-tailed. In this lesson, we will study the behavior of the mean of samples of different sizes drawn from a variety of parent populations. Examining sampling distributions of sample means computed from samples of different sizes drawn from a variety of distributions, allow us to gain some insight into the behavior of the sample mean under those specific conditions as well as examine the validity of the guidelines mentioned above for using the central limit theorem in practice. Under certain conditions, in large samples, the sampling distribution of the sample mean can be approximated by a normal distribution. The sample size needed for the approximation to be adequate depends strongly on the shape of the parent distribution. Symmetry (or lack thereof) is particularly important. For a symmetric parent distribution, even if very different from the shape of a normal distribution, an adequate approximation can be obtained with small samples (e.g. 10 or 12 for the uniform distribution). For symmetric short-tailed parent distributions, the sample mean reaches approximate normality for smaller samples than if the parent population is skewed and long-tailed. In some extreme cases (e.g. binomial) samples sizes far exceeding the typical guidelines (e.g. 30) are needed for an adequate approximation. For some distributions without first and second moments (e.g. Cauchy), the central limit theorem does not hold. What Is a Least Squares Model Many problems in analyzing data involve describing how variables are related. The simplest of all models describing the relationship between two variables is a linear, or straight-line, model. The simplest method of fitting a linear model is to eye-ball a line through the data on a plot. A more elegant, and conventional method is that of least squares, which finds the line minimizing the sum of distances between observed points and the fitted line. Realize that fitting the best line by eye is difficult, especially when there is a lot of residual variability in the data. Know that there is a simple connection between the numerical coefficients in the regression equation and the slope and intercept of regression line. Know that a single summary statistic like a correlation coefficient does not tell the whole story. A scatter plot is an essential complement to examining the relationship between the two variables. ANOVA: Analysis of Variance The tests we have learned up to this point allow us to test hypotheses that examine the difference between only two means. Analysis of Variance or ANOVA will allow us to test the difference between 2 or more means. ANOVA does this by examining the ratio of variability between two conditions and variability within each condition. For example, say we give a drug that we believe will improve memory to a group of people and give a placebo to another group of people. We might measure memory performance by the number of words recalled from a list we ask everyone to memorize. A t-test would compare the likelihood of observing the difference in the mean number of words recalled for each group. An ANOVA test, on the other hand, would compare the variability that we observe between the two conditions to the variability observed within each condition. Recall that we measure variability as the sum of the difference of each score from the mean. When we actually calculate an ANOVA we will use a short-cut formula Thus, when the variability that we predict (between the two groups) is much greater than the variability we dont predict (within each group) then we will conclude that our treatments produce different results. Exponential Density Function An important class of decision problems under uncertainty concerns the chance between events. For example, the chance of the length of time to next breakdown of a machine not exceeding a certain time, such as the copying machine in your office not to break during this week. Exponential distribution gives distribution of time between independent events occurring at a constant rate. Its density function is: where l is the average number of events per unit of time, which is a positive number. The mean and the variance of the random variable t (time between events) are 1 l. and 1 l 2. respectively. Applications include probabilistic assessment of the time between arrival of patients to the emergency room of a hospital, and arrival of ships to a particular port. Comments: Special case of both Weibull and gamma distributions. You may like using Exponential Applet to perform your computations. You may like using the following Lilliefors Test for Exponentially to perform the goodness-of-fit test. Poisson Process An important class of decision problems under uncertainty is characterized by the small chance of the occurrence of a particular event, such as an accident. Gives probability of exactly x independent occurrences during a given period of time if events take place independently and at a constant rate. May also represent number of occurrences over constant areas or volumes. The following statements describe the Poisson Process . The occurrences of the events are independent. The occurrence of events from a set of assumptions in an interval of space or time has no effect on the probability of a second occurrence of the event in the same, or any other, interval. Theoretically, an infinite number of occurrences of the event must be possible in the interval. The probability of the single occurrence of the event in a given interval is proportional to the length of the interval. In any infinitesimally small portion of the interval, the probability of more than one occurrence of the event is negligible. Poisson process are often used, for example in quality control, reliability, insurance claim, incoming number of telephone calls, and queuing theory. An Application: One of the most useful applications of the Poisson Process is in the field of queuing theory. In many situations where queues occur it has been shown that the number of people joining the queue in a given time period follows the Poisson model. For example, if the rate of arrivals to an emergency room is l per unit of time period (say 1 hr), then: P ( n arrivals) l n e - l n The mean and variance of random variable n are both l. However if the mean and variance of a random variable having equal numerical values, then it is not necessary that its distribution is a Poisson. P ( 0 arrival) e - l P ( 1 arrival) l e - l 1 P ( 2 arrival) l 2 e - l 2 and so on. In general: P ( n1 arrivals ) l Pr ( n arrivals ) n. You may like using Poisson Applet to perform your computations. Goodness-of-Fit for Poisson Replace the numerical example data with your up-to-14 pairs of Observed values their frequencies . and then click the Calculate button. Blank boxes are not included in the calculations. Dalam memasukkan data Anda untuk berpindah dari sel ke sel di matriks data gunakan tombol Tab bukan panah atau masukkan kunci. Uniform Density Function Application: Gives probability that observation will occur within a particular interval when probability of occurrence within that interval is directly proportional to interval length. Example: Used to generate random numbers in sampling and Monte Carlo simulation. Comments: Special case of beta distribution. The mass function of geometric mean of n independent uniforms 0,1 is: P(X x) n x (n - 1) (Log1x n ) (n -1) (n - 1). z L U L -(1-U) L L is said to have Tukeys symmetrical l -distribution. You may like using Uniform Applet to perform your computations. Some Useful SPSS Commands For more SPSS programs useful to simulation inputoutput analysis, visit Data Analysis Routines. Random Number Generators Classical uniform random number generators have some major defects, such as, short period length and lack of higher dimension uniformity. However, nowadays there are a class of rather complex generators which is as efficient as the classical generators while enjoy the property of a much longer period and of a higher dimension uniformity. Computer programs that generate random numbers use an algorithm. That means if you know the algorithm and the seedvalues you can predict what numbers will result. Because you can predict the numbers they are not truly random - they are pseudorandom. For statistical purposes good pseudorandom numbers generators are good enough. The Random Number Generator RANECU A FORTRAN code for a generator of uniform random numbers on 0,1. RANECU is multiplicative linear congruential generator suitable for a 16-bit platform. It combines three simple generators, and has a period exceeding 81012. It is constructed for more efficient use by providing for a sequence of such numbers, LEN in total, to be returned in a single call. A set of three non-zero integer seeds can be supplied, failing which a default set is employed. If supplied, these three seeds, in order, should lie in the ranges 1,32362, 1,31726 and 1,31656 respectively. The Shuffling Routine in Visual Basic The Square Histogram Method We are given a histogram, with vertical bars having heights proportional to the probability with which we want to produce a value indicated by the label at the base. A simple such histogram, layed flat, might be: The idea is to cut the bars into pieces then reassemble them into a square histogram, all heights equal, with each final bar having a lower part, as well as an upper part indicating where it came from. A single uniform random variable U can then be used to choose one of the final bars and to indicate whether to use the lower or upper part. There are many ways to do this cutting and reassembling the simplest seems to be the Robin Hood Algorithm: Take from richest to bring the poorest up to average. STEP 1: The original (horizontal) histogram, average height 20: Take 17 from strip a to bring strip e up to average. Record donor and use old poor level to mark lower part of donee: Then bring d up to average with donor b. Record donor and use old poor level to mark lower part of donee: Then bring a up to average with donor c. Record donor and use old poor level to mark lower part of donee: Finally, bring b up to average with donor c. Record donor and use old poor level to mark lower part of donee: We now have a squared histogram, i.e. a rectangle with 4 strips of equal area, each strip with two regions. A single uniform variate U can be used to generate a,b,c,d,e with the required probabilities. 32. 27. 26. 12 .06. Setup: Make tables, Let j be the integer part of 15U, with U uniform in (0,1). If U lt Tj return Vj, else return VKj. In many applications no V table is necessary: Vii and the generating procedure becomes If U lt Tj return j, else return Kj. References Further Readings: Aiello W. S. Rajagopalan, and R. Venkatesan, Design of practical and provably good random number generators, Journal of Algorithms . 29, 358-389, 1998. Dagpunar J. Principles of Random Variate Generation . Clarendon, 1988. Fishman G. Monte Carlo . Springer, 1996. James, Fortran version of LEcuyer generator, Comput. Fisik. Comm. . 60, 329-344, 1990. Knuth D. The Art of Computer Programming, Vol. 2. Addison-Wesley, 1998. LEcuyer P. Efficient and portable combined random number generators, Comm. ACM, 31, 742-749, 774, 1988. LEcuyer P. Uniform random number generation, Ann. Op. Res . 53, 77-120, 1994. LEcuyer P. Random number generation. In Handbook on Simulation . J. Banks (ed.), Wiley, 1998. Maurer U. A universal statistical test for random bit generators, J. Cryptology . 5, 89-105, 1992. Sobol I. and Y. Levitan, A pseudo-random number generator for personal computers, Computers Mathematics with Applications . 37(4), 33-40, 1999. Tsang W-W. A decision tree algorithm for squaring the histogram in random number generation, Ars Combinatoria . 23A, 291-301, 1987 Test for Randomness We need to test for both randomness as well as uniformity. The tests can be classified in 2 categories: Empirical or statistical tests, and theoretical tests. Theoretical tests deal with the properties of the generator used to create the realization with desired distribution, and do not look at the number generated at all. For example, we would not use a generator with poor qualities to generate random numbers. Statistical tests are based solely on the random observations produced. Test for Randomness: A. Test for independence: Plot the x i realization vs x i1 . If there is independence, the graph will not show any distinctive patterns at all, but will be perfectly scattered. B. Runs tests.(run-ups, run-downs): This is a direct test of the independence assumption. There are two test statistics to consider: one based on a normal approximation and another using numerical approximations. Test based on Normal approximation: Suppose you have N random realizations. Let a be the total number of runs in a sequence. If the number of positive and negative runs are greater than say 20, the distribution of a is reasonably approximated by a Normal distribution with mean (2N - 1) 3 and (16N - 29) 90. Reject the hypothesis of independence or existence of runs if Zo gt Z(1-alpha2) where Zo is the Z score. C. Correlation tests: Do the random numbers exhibit discernible correlation Compute the sample Autcorrelation Function. Frequency or Uniform Distribution Test: Use Kolmogorov-Smirimov test to determine if the realizations follow a U(0,1) References Further Readings: Headrick T. Fast fifth-order polynomial transforms for generating univariate and multivariate nonnormal distributions, Computational Statistics and Data Analysis . 40 (4), 685-711, 2002. Karian Z. and E. Dudewicz, Modern Statistical Systems and GPSS Simulation . CRC Press, 1998. Kleijnen J. and W. van Groenendaal, Simulation: A Statistical Perspective . Wiley, Chichester, 1992 Korn G. Real statistical experiments can use simulation-package software, Simulation Modelling Practice and Theory . 13(1), 39-54, 2005. Lewis P. and E. Orav, Simulation Methodology for Statisticians, Operations Analysts, and Engineers . Wadsworth Inc. 1989 Madu Ch. and Ch-H. Kuei, Experimental Statistical Designs and Analysis in Simulation Modeling . Greenwood Publishing Group, 1993. Pang K. Z. Yang, S. Hou, and P. Leung, Non-uniform random variate generation by the vertical strip method, European Journal of Operational Research . 142(3), 595-609, 2002. Robert C. and G. Casella, Monte Carlo Statistical Methods . Springer, 1999. Modeling Simulation Simulation in general is to pretend that one deals with a real thing while really working with an imitation. In operations research the imitation is a computer model of the simulated reality. A flight simulator on a PC is also a computer model of some aspects of the flight: it shows on the screen the controls and what the pilot (the youngster who operates it) is supposed to see from the cockpit (his armchair). Why to use models To fly a simulator is safer and cheaper than the real airplane. For precisely this reason, models are used in industry commerce and military: it is very costly, dangerous and often impossible to make experiments with real systems. Provided that models are adequate descriptions of reality (they are valid), experimenting with them can save money, suffering and even time. When to use simulations Systems that change with time, such as a gas station where cars come and go (called dynamic systems) and involve randomness. Nobody can guess at exactly which time the next car should arrive at the station, are good candidates for simulation. Modeling complex dynamic systems theoretically need too many simplifications and the emerging models may not be therefore valid. Simulation does not require that many simplifying assumptions, making it the only tool even in absence of randomness. How to simulate Suppose we are interested in a gas station. We may describe the behavior of this system graphically by plotting the number of cars in the station the state of the system. Every time a car arrives the graph increases by one unit while a departing car causes the graph to drop one unit. This graph (called sample path), could be obtained from observation of a real station, but could also be artificially constructed. Such artificial construction and the analysis of the resulting sample path (or more sample paths in more complex cases) consists of the simulation. Types of simulations: Discrete event. The above sample path consisted of only horizontal and vertical lines, as car arrivals and departures occurred at distinct points of time, what we refer to as events. Between two consecutive events, nothing happens - the graph is horizontal. When the number of events are finite, we call the simulation discrete event. In some systems the state changes all the time, not just at the time of some discrete events. For example, the water level in a reservoir with given in and outflows may change all the time. In such cases continuous simulation is more appropriate, although discrete event simulation can serve as an approximation. Further consideration of discrete event simulations. How is simulation performed Simulations may be performed manually. Most often, however, the system model is written either as a computer program (for an example click here) or as some kind of input into simulator software. State: A variable characterizing an attribute in the system such as level of stock in inventory or number of jobs waiting for processing. Event: An occurrence at a point in time which may change the state of the system, such as arrival of a customer or start of work on a job. Entity: An object that passes through the system, such as cars in an intersection or orders in a factory. Often an event (e.g. arrival) is associated with an entity (e.g. customer). Queue: A queue is not only a physical queue of people, it can also be a task list, a buffer of finished goods waiting for transportation or any place where entities are waiting for something to happen for any reason. Creating: Creating is causing an arrival of a new entity to the system at some point in time. Scheduling: Scheduling is the act of assigning a new future event to an existing entity. Random variable: A random variable is a quantity that is uncertain, such as interarrival time between two incoming flights or number of defective parts in a shipment. Random variate: A random variate is an artificially generated random variable. Distribution: A distribution is the mathematical law which governs the probabilistic features of a random variable. A Simple Example: Building a simulation gas station with a single pump served by a single service man. Assume that arrival of cars as well their service times are random. At first identify the: states: number of cars waiting for service and number of cars served at any moment events: arrival of cars, start of service, end of service entities: these are the cars queue: the queue of cars in front of the pump, waiting for service random realizations: interarrival times, service times distributions: we shall assume exponential distributions for both the interarrival time and service time. Next, specify what to do at each event. The above example would look like this: At event of entity arrival: Create next arrival. If the server is free, send entity for start of service. Otherwise it joins the queue. At event of service start: Server becomes occupied. Schedule end of service for this entity. At event of service end: Server becomes free. If any entities waiting in queue: remove first entity from the queue send it for start of service. Some initiation is still required, for example, the creation of the first arrival. Lastly, the above is translated into code. This is easy with an appropriate library which has subroutines for creation, scheduling, proper timing of events, queue manipulations, random variate generation and statistics collection. How to simulate Besides the above, the program records the number of cars in the system before and after every change, together with the length of each event. Development of Systems Simulation Discrete event systems (DES) are dynamic systems which evolve in time by the occurrence of events at possibly irregular time intervals. DES abound in real-world applications. Examples include traffic systems, flexible manufacturing systems, computer-communications systems, production lines, coherent lifetime systems, and flow networks. Most of these systems can be modeled in terms of discrete events whose occurrence causes the system to change from one state to another. In designing, analyzing and operating such complex systems, one is interested not only in performance evaluation but also in sensitivity analysis and optimization. A typical stochastic system has a large number of control parameters that can have a significant impact on the performance of the system. To establish a basic knowledge of the behavior of a system under variation of input parameter values and to estimate the relative importance of the input parameters, sensitivity analysis applies small changes to the nominal values of input parameters. For systems simulation, variations of the input parameter values cannot be made infinitely small. The sensitivity of the performance measure with respect to an input parameter is therefore defined as (partial) derivative. Sensitivity analysis is concerned with evaluating sensitivities (gradients, Hessian, etc.) of performance measures with respect to parameters of interest. It provides guidance for design and operational decisions and plays a pivotal role in identifying the most significant system parameters, as well as bottleneck subsystems. I have carried out research in the fields of sensitivity analysis and stochastic optimization of discrete event systems with an emphasis on computer simulation models. This part of lecture is dedicated to the estimation of an entire response surface of complex discrete event systems (DES) from a single sample path (simulation), such as the expected waiting time of a customer in a queuing network, with respect to the controllable parameters of the system, such as service rates, buffer sizes and routing probabilities. With the response surfaces at hand, we are able to perform sensitivity analysis and optimization of a DES from a single simulation, that is, to find the optimal parameters of the system and their sensitivities (derivatives), with respect to uncontrollable system parameters, such as arrival rates in a queuing network. We identified three distinct processes. Descriptive Analysis includes: Problem Identification Formulation, Data Collection and Analysis, Computer Simulation Model Development, Validation, Verification and Calibration, and finally Performance Evaluation. Prescriptive Analysis: Optimization or Goal Seeking. These are necessary components for Post-prescriptive Analysis: Sensitivity, and What-If Analysis. The prescriptive simulation attempts to use simulation to prescribe decisions required to obtain specified results. It is subdivided into two topics- Goal Seeking and Optimization. Recent developments on single-run algorithms for the needed sensitivities (i.e. gradient, Hessian, etc.) make the prescriptive simulation feasible. Click on the image to enlarge it and THEN print it. Problem Formulation: Identify controllable and uncontrollable inputs. Identify constraints on the decision variables. Define measure of system performance and an objective function. Develop a preliminary model structure to interrelate the inputs and the measure of performance. Click on the image to enlarge it and THEN print it. Data Collection and Analysis: Regardless of the method used to collect the data, the decision of how much to collect is a trade-off between cost and accuracy. Simulation Model Development: Acquiring sufficient understanding of the system to develop an appropriate conceptual, logical and then simulation model is one of the most difficult tasks in simulation analysis. Model Validation, Verification and Calibration: In general, verification focuses on the internal consistency of a model, while validation is concerned with the correspondence between the model and the reality. The term validation is applied to those processes which seek to determine whether or not a simulation is correct with respect to the real system. More prosaically, validation is concerned with the question Are we building the right system. Verification, on the other hand, seeks to answer the question Are we building the system right Verification checks that the implementation of the simulation model (program) corresponds to the model. Validation checks that the model corresponds to reality. Calibration checks that the data generated by the simulation matches real (observed) data. Validation: The process of comparing the models output with the behavior of the phenomenon. In other words: comparing model execution to reality (physical or otherwise) Verification: The process of comparing the computer code with the model to ensure that the code is a correct implementation of the model. Calibration: The process of parameter estimation for a model. Calibration is a tweakingtuning of existing parameters and usually does not involve the introduction of new ones, changing the model structure. In the context of optimization, calibration is an optimization procedure involved in system identification or during experimental design. Input and Output Analysis: Discrete-event simulation models typically have stochastic components that mimic the probabilistic nature of the system under consideration. Successful input modeling requires a close match between the input model and the true underlying probabilistic mechanism associated with the system. The input data analysis is to model an element (e.g. arrival process, service times) in a discrete-event simulation given a data set collected on the element of interest. This stage performs intensive error checking on the input data, including external, policy, random and deterministic variables. System simulation experiment is to learn about its behavior. Careful planning, or designing, of simulation experiments is generally a great help, saving time and effort by providing efficient ways to estimate the effects of changes in the models inputs on its outputs. Statistical experimental-design methods are mostly used in the context of simulation experiments. Performance Evaluation and What-If Analysis: The what-if analysis is at the very heart of simulation models. Sensitivity Estimation: Users must be provided with affordable techniques for sensitivity analysis if they are to understand which relationships are meaningful in complicated models. Optimization: Traditional optimization techniques require gradient estimation. As with sensitivity analysis, the current approach for optimization requires intensive simulation to construct an approximate surface response function. Incorporating gradient estimation techniques into convergent algorithms such as Robbins-Monroe type algorithms for optimization purposes, will be considered. Gradient Estimation Applications: There are a number of applications which measure sensitivity information, (i.e. the gradient, Hessian, etc.), Local information, Structural properties, Response surface generation, Goal-seeking problem, Optimization, What-if Problem, and Meta-modelling Report Generating: Report generation is a critical link in the communication process between the model and the end user. A Classification of Stochastic Processes A stochastic process is a probabilistic model of a system that evolves randomly in time and space. Formally, a stochastic process is a collection of random variables all defined on a common sample (probability) space. The X(t) is the state while (time) t is the index that is a member of set T. Examples are the delay of the ith customer and number of customers in the queue at time t in an MM1 queue. In the first example, we have a discrete- time, continuous state, while in the second example the state is discrete and time in continuous. The following table is a classification of various stochastic processes. The man made systems have mostly discrete state. Monte Carlo simulation deals with discrete time while in discrete even system simulation the time dimension is continuous, which is at the heart of this site. Change in the States of the System A Classification of Stochastic Processes Simulation Output Data and Stochastic Processes To perform statistical analysis of the simulation output we need to establish some conditions, e.g. output data must be a covariance stationary process (e.g. the data collected over n simulation runs). Stationary Process (strictly stationary): A stationary stochastic process is a stochastic process with the property that the joint distribution all vectors of h dimension remain the same for any fixed h. First Order Stationary: A stochastic process is a first order stationary if expected of X(t) remains the same for all t. For example in economic time series, a process is first order stationary when we remove any kinds of trend by some mechanisms such as differencing. Second Order Stationary: A stochastic process is a second order stationary if it is first order stationary and covariance between X(t) and X(s) is function of t-s only. Again, in economic time series, a process is second order stationary when we stabilize also its variance by some kind of transformations such as taking square root. Clearly, a stationary process is a second order stationary, however the reverse may not hold. In simulation output statistical analysis we are satisfied if the output is covariance stationary . Covariance Stationary: A covariance stationary process is a stochastic process having finite second moments, i.e. expected of X(t) 2 be finite. Clearly, any stationary process with finite second moment is covariance stationary. A stationary process may have no finite moment whatsoever. Since a Gaussian process needs a mean and covariance matrix only, it is stationary (strictly) if it is covariance stationary. Two Contrasting Stationary Process: Consider the following two extreme stochastic processes: - A sequence Y 0 . Y 1 . of independent identically distributed, random-value sequence is a stationary process, if its common distribution has a finite variance then the process is covariance stationary. - Let Z be a single random variable with known distribution function, and set Z 0 Z 1 . Z. Note that in a realization of this process, the first element, Z 0, may be random but after that there is no randomness. The process i . i 0, 1, 2. is stationary if Z has a finite variance. Output data in simulation fall between these two type of process. Simulation outputs are identical, and mildly correlated (how mild It depends on e.g. in a queueing system how large is the traffic intensity r ). An example could be the delay process of the customers in a queueing system. Techniques for the Steady State Simulation Unlike in queuing theory where steady state results for some models are easily obtainable, the steady state simulation is not an easy task. The opposite is true for obtaining results for the transient period (i.e. the warm-up period). Gather steady state simulation output requires statistical assurance that the simulation model reached the steady state. The main difficulty is to obtain independent simulation runs with exclusion of the transient period. The two technique commonly used for steady state simulation are the Method of Batch means, and the Independent Replication. None of these two methods is superior to the other in all cases. Their performance depend on the magnitude of the traffic intensity. The other available technique is the Regenerative Method, which is mostly used for its theoretical nice properties, however it is rarely applied in actual simulation for obtaining the steady state output numerical results. Suppose you have a regenerative simulation consisting of m cycles of size n 1 . n 2 ,n m . Masing-masing. The cycle sums is: The overall estimate is: Estimate S y i S n i . the sums are over i1, 2. m The 100(1- a 2) confidence interval using the Z-table (or T-table, for m less than, say 30), is: Estimate 177 Z. S (n. m ) n S n i m, the sum is over i1, 2. m and the variance is: S 2 S (y i - n i . Estimate) 2 (m-1), the sum is over i1, 2. m Method of Batch Means: This method involves only one very long simulation run which is suitably subdivided into an initial transient period and n batches. Each of the batch is then treated as an independent run of the simulation experiment while no observation are made during the transient period which is treated as warm-up interval. Choosing a large batch interval size would effectively lead to independent batches and hence, independent runs of the simulation, however since number of batches are few on cannot invoke the central limit theorem to construct the needed confidence interval. On the other hand, choosing a small batch interval size would effectively lead to significant correlation between successive batches therefore cannot apply the results in constructing an accurate confidence interval. Suppose you have n equal batches of m observations each. The means of each batch is: mean i S x ij m, the sum is over j1, 2. m The overall estimate is: Estimate S mean i n, the sum is over i1, 2. n The 100(1- a 2) confidence interval using the Z-table (or T-table, for n less than, say 30), is: Estimate 177 Z. S where the variance is: S 2 S (mean i - Estimate) 2 (n-1), the sum is over i1, 2. n Method of Independent Replications: This method is the most popularly used for systems with short transient period. This method requires independent runs of the simulation experiment different initial random seeds for the simulators random number generator. For each independent replications of the simulation run it transient period is removed. For the observed intervals after the transient period data is collected and processed for the point estimates of the performance measure and for its subsequent confidence interval. Suppose you have n replications with of m observations each. The means of each replication is: mean i S x ij m, the sum is over j1, 2. m The overall estimate is: Estimate S mean i n, the sum is over i1, 2. n The 100(1- a 2) confidence interval using the Z-table (or T-table, for n less than, say 30), is: Estimate 177 Z. S where the variance is: S 2 S (mean i - Estimate) 2 (n-1), the sum is over i1, 2. n Further Reading: Sherman M. and D. Goldsman, Large-sample normality of the batch-means variance estimator, Operations Research Letters . 30, 319-326, 2002. Whitt W. The efficiency of one long run versus independent replications in steady-state simulation, Management Science . 37(6), 645-666, 1991. Determination of the Warm-up Period To estimate the long-term performance measure of the system, there are several methods such as Batch Means, Independent Replications and Regenerative Method. Batch Means is a method of estimating the steady-state characteristic from a single-run simulation. The single run is partitioned into equal size batches large enough for estimates obtained from different batches to be approximately independent. In the method of Batch Means, it is important to ensure that the bias due to initial conditions is removed to achieve at least a covariance stationary waiting time process. An obvious remedy is to run the simulation for a period large enough to remove the effect of the initial bias. During this warm-up period, no attempt is made to record the output of the simulation. The results are thrown away. At the end of this warm-up period, the waiting time of customers are collected for analysis. The practical question is How long should the warm-up period be. Abate and Whitt provided a relatively simple and nice expression for the time required (t p ) for an MM1 queue system (with traffic intensity r ) starting at the origin (empty) to reach and remain within 100p of the steady- state limit as follows: C( r )2 r ( r 2 4 r ) 4. Some notions of t p ( r ) as a function of r and p, are given in following table: Time ( t p ) required for an MM1 queue to reach and remain with 100p limits of the steady-state value. Although this result is developed for MM1 queues, it has already been established that it can serve as an approximation for more general i.e. GIG1 queues. Further Reading: Abate J. and W. Whitt, Transient behavior of regular Brownian motion, Advance Applied Probability . 19, 560-631, 1987. Chen E. and W. Kelton, Determining simulation run length with the runs test, Simulation Modelling Practice and Theory . 11, 237-250, 2003. Determination of the Desirable Number of Simulation Runs The two widely used methods for experimentation on simulation models are method of bath means, and independent replications. Intuitively one may say the method of independent replication is superior in producing statistically a good estimate for the systems performance measure. In fact, not one method is superior in all cases and it all depends on the traffic intensity r. After deciding what method is more suitable to apply, the main question is determination of number of runs. That is, at the planning stage of a simulation investigation of the question of number of simulation runs (n) is critical. The confidence level of simulation output drawn from a set of simulation runs depends on the size of data set. The larger the number of runs, the higher is the associated confidence. However, more simulation runs also require more effort and resources for large systems. Thus, the main goal must be in finding the smallest number of simulation runs that will provide the desirable confidence. Pilot Studies: When the needed statistics for number of simulation runs calculation is not available from existing database, a pilot simulation is needed. For large pilot simulation runs (n), say over 30, the simplest number of runs determinate is: where d is the desirable margin of error (i.e. the absolute error), which is the half-length of the confidence interval with 100(1- a ) confidence interval. S 2 is the variance obtained from the pilot run. One may use the following sample size determinate for a desirable relative error D in , which requires an estimate of the coefficient of variation (C.V. in ) from a pilot run with n over 30: These sample size determinates could also be used for simulation output estimation of unimodal output populations, with discrete or continuous random variables provided the pilot run size (n) is larger than (say) 30. The aim of applying any one of the above number of runs determinates is at improving your pilot estimates at feasible costs. You may like using the following Applet for determination of number of runs. Further Reading: Daz-Emparanza I, Is a small Monte Carlo analysis a good analysis Checking the size power and consistency of a simulation-based test, Statistical Papers . 43(4), 567-577, 2002. Whitt W. The efficiency of one long run versus independent replications in steady-state simulation, Management Science . 37(6), 645-666, 1991. Determination of Simulation Runs Size At the planning stage of a simulation modeling the question of number of simulation runs (n) is critical. The following Java applets compute the needed Runs Size based on current avialable information ontained from a pilot simulation run, to achieve an acceptable accuracy andor risk. Enter the needed information, and then click the Calculate button. The aim of applying any one of the following number of simulation runs determinates is at improving your pilot estimates at a feasible cost. Notes: The normality condition might be relaxed for number of simulation runs over, say 30. Moreover, determination of number of simulation runs for mean could also be used for other unimodal simulation output distributions including those with discrete random variables, such as proportion, provided the pilot run is sufficiently large (say, over 30). Runs Size with Acceptable Absolute Precision Simulation Software Selection The vast amount of simulation software available can be overwhelming for the new users. The following are only a random sample of software in the market today: ACSL, APROS, ARTIFEX, Arena, AutoMod, CSIM, CSIM, Callim, FluidFlow, GPSS, Gepasi, JavSim, MJX, MedModel, Mesquite, Multiverse, NETWORK, OPNET Modeler, POSES, Simulat8, Powersim, QUEST, REAL, SHIFT, SIMPLE, SIMSCRIPT, SLAM, SMPL, SimBank, SimPlusPlus, TIERRA, Witness, SIMNON, VISSIM, and javasim. There are several things that make an ideal simulation package. Some are properties of the package, such as support, reactivity to bug notification, interface, etc. Some are properties of the user, such as their needs, their level of expertise, etc. For these reasons asking which package is best is a sudden failure of judgment. The first question to ask is for what purpose you need the software Is it for education, teaching, student-projects or research The main question is: What are the important aspects to look for in a package The answer depends on specific applications. However some general criteria are: Input facilities, Processing that allows some programming, Optimization capability, Output facilities, Environment including training and support services, Input-output statistical data analysis capability, and certainly the Cost factor. You must know which features are appropriate for your situation, although, this is not based on a Yes or No judgment. For description of available simulation software, visit Simulation Software Survey. Reference Further Reading: Nikoukaran J. Software selection for simulation in manufacturing: A review, Simulation Practice and Theory . 7(1), 1-14, 1999. Animation in Systems Simulation Animation in systems simulation is a useful tool. Most graphically based software packages have default animation. This is quite useful for model debugging, validation, and verification. This type of animation comes with little or no additional effort and gives the modeler additional insight into how the model. This type of animation comes with little or no additional effort and gives the modeler additional insight into how the model works. However, it augments the modeling tools available. The more realistic animation presents qualities which intend to be useful to the decision-maker in implementing the developed simulation model. There are also, good model management tools. Some tools have been developed which combined a database with simulation to store models, data, results, and animations. However, there is not one product that provides all of those capabilities. SIMSCRIPT II.5 Without computer one cannot perform any realistic dynamic systems simulation. SIMSCRIPT II.5 is a powerful, free-format, English-like simulation language designed to greatly simplify writing programs for simulation modelling. Programs written in SIMSCRIPT II.5 are easily read and maintained. They are accurate, efficient, and generate results which are acceptable to users. Unlike other simulation programming languages, SIMSCRIPT II.5 requires no coding in other languages. SIMSCRIPT II.5 has been fully supported for over 33 years. Contributing to the wide acceptance and success of SIMSCRIPT II.5 modelling are: A powerful worldview, consisting of Entities and Processes, provides a natural conceptual framework with which to relate real objects to the model. SIMSCRIPT II.5 is a modern, free-form language with structured programming constructs and all the built-in facilities needed for model development. Model components can be programmed so they clearly reflect the organization and logic of the modeled system. The amount of program needed to model a system is typically 75 less than its FORTRAN or C counterpart. A well designed package of program debug facilities is provided. The required tools are available to detect errors in a complex computer program without resorting an error. Simulation status information is provided, and control is optionally transferred to a user program for additional analysis and output. This structure allows the model to evolve easily and naturally from simple to detailed formulation as data becomes available. Many modifications, such as the choice of set disciplines and statistics are simply specified in the Preamble. You get a powerful, English-like language supporting a modular implementation. Because each model component is readable and self-contained, the model documentation is the model listing it is never obsolete or inaccurate. For more information contact SIMSCRIPT Guidelines for Running SIMSCRIPT on the VAX System System Dynamics and Discrete Event Simulation The modeling techniques used by system dynamics and discrete event simulations are often different at two levels: The modeler way of representing systems might be different, the underlying simulators algorithms are also different. Each technique is well tuned to the purpose it is intended. However, one may use a discrete event approach to do system dynamics and vice versa. Traditionally, the most important distinction is the purpose of the modeling. The discrete event approach is to find, e.g. how many resources the decision maker needs such as how many trucks, and how to arrange the resources to avoid bottlenecks, i.e. excessive of waiting lines, waiting times, or inventories. While the system dynamics approach is to prescribe for the decision making to, e.g. timely respond to any changes, and how to change the physical structure, e.g. physical shipping delay time, so that inventories, sales, production, etc. System dynamics is the rigorous study of problems in system behavior using the principles of feedback, dynamics and simulation. In more words system dynamics is characterized by: Searching for useful solutions to real problems, especially in social systems (businesses, schools, governments. ) and the environment. Using computer simulation models to understand and improve such systems. Basing the simulation models on mental models, qualitative knowledge and numerical information. Using methods and insights from feedback control engineering and other scientific disciplines to assess and improve the quality of models. Seeking improved ways to translate scientific results into achieved implemented improvement. Systems dynamics approach looks at systems at a very high level so is more suited to strategic analysis. Discrete event approach may look at subsystems for a detailed analysis and is more suited, e.g. to process re-engineering problems. Systems dynamics is indicative, i.e. helps us understand the direction and magnitude of effects (i.e. where in the system do we need to make the changes), whereas discrete event approach is predictive (i.e. how many resources do we need to achieve a certain goal of throughout). Systems dynamics analysis is continuous in time and it uses mostly deterministic analysis, whereas discrete event process deals with analysis in a specific time horizon and uses stochastic analysis. Some interesting and useful areas of system dynamics modeling approach are: Short-term and long term forecasting of agricultural produce with special reference to field crops and perennial fruits such as grapes, which have significant processing sectors of different proportions of total output where both demand and supply side perspectives are being considered. Long term relationship between the financial statements of balance sheet, income statement and cash flow statement balanced against scenarios of the stock markets need to seek a stablegrowing share price combined with a satisfactory dividend and related return on shareholder funds policy. Managerial applications include the development and evaluation of short-term and long-term strategic plans, budget analysis and assessment, business audits and benchmarking. A modeler must consider both as complementary tools to each other. Systems dynamic to look at the high level problem and identify areas which need more detailed analysis. Then, use discrete event modeling tools to analyze (and predict) the specific areas of interest. What Is Social Simulation Social scientists have always constructed models of social phenomena. Simulation is an important method for modeling social and economic processes. In particular, it provides a middle way between the richness of discursive theorizing and rigorous but restrictive mathematical models. There are different types of computer simulation and their application to social scientific problems. Faster hardware and improved software have made building complex simulations easier. Computer simulation methods can be effective for the development of theories as well as for prediction. For example, macro-economic models have been used to simulate future changes in the economy and simulations have been used in psychology to study cognitive mechanisms. The field of social simulation seems to be following an interesting line of inquiry. As a general approach in the field, a world is specified with much computational detail. Then the world is simulated (using computers) to reveal some of the non-trivial implications (or emergent properties) of the world. When these non trivial implications are made known (fed back) in world, apparently it constitutes some added values. Artificial Life is an interdisciplinary study enterprise aimed at understanding life-as-it-is and life-as-it-could-be, and at synthesizing life-like phenomena in chemical, electronic, software, and other artificial media. Artificial Life redefines the concepts of artificial and natural, blurring the borders between traditional disciplines and providing new media and new insights into the origin and principles of life. Simulation allows the social scientist to experiment with artificial societies and explore the implications of theories in ways not otherwise possible. Reference and Further Readings: Gilbert N. and K. Troitzsch, Simulation for the Social Scientist . Open University Press, Buckingham, UK, 1999. Sichman J. R. Conte, and N. Gilbert, (eds,), Multi-Agent Systems and Agent-Based Simulation . Berlin, Springer-Verlag, 1998. What Is Web-based Simulation Web-based simulation is quickly emerging as an area of significant interest for both simulation researchers and simulation practitioners. This interest in web-based simulation is a natural outgrowth of the proliferation of the World-Wide Web and its attendant technologies, e.g. HTML, HTTP, CGI, etc. Also the surging popularity of, and reliance upon, computer simulation as a problem solving and decision support systems tools. The appearance of the network-friendly programming language, Java, and of distributed object technologies like the Common Object Request Broker Architecture (CORBA) and the Object Linking and Embedding Component Object Model (OLECOM) have had particularly acute effects on the state of simulation practice. Currently, the researchers in the field of web-based simulation are interested in dealing with topics such as methodologies for web-based model development, collaborative model development over the Internet, Java-based modeling and simulation, distributed modeling and simulation using web technologies, and new applications. Parallel and Distributed Simulation The increasing size of the systems and designs requires more efficient simulation strategies to accelerate the simulation process. Parallel and distributed simulation approaches seem to be a promising approach in this direction. Current topics under extensive research are: Synchronization, scheduling, memory management, randomized and reactiveadaptive algorithms, partitioning and load balancing. Synchronization in multi-user distributed simulation, virtual reality environments, HLA, and interoperability. System modeling for parallel simulation, specification, re-use of modelscode, and parallelizing existing simulations. Language and implementation issues, models of parallel simulation, execution environments, and libraries. Theoretical and empirical studies, prediction and analysis, cost models, benchmarks, and comparative studies. Computer architectures, VLSI, telecommunication networks, manufacturing, dynamic systems, and biologicalsocial systems. Web based distributed simulation such as multimedia and real time applications, fault tolerance, implementation issues, use of Java, and CORBA. References Further Readings: Bossel H. Modeling Simulation . A. K. Peters Pub. 1994. Delaney W. and E. Vaccari, Dynamic Models and Discrete Event Simulation . Dekker, 1989. Fishman G. Discrete-Event Simulation: Modeling, Programming and Analysis . Springer-Verlag, Berlin, 2001. Fishwick P. Simulation Model Design and Execution: Building Digital Worlds . Prentice-Hall, Englewood Cliffs, 1995. Ghosh S. and T. Lee, Modeling Asynchronous Distributed Simulation: Analyzing Complex Systems . IEEE Publications, 2000. Gimblett R. Integrating Geographic Information Systems and Agent-Based Modeling: Techniques for Simulating Social and Ecological Processes . Oxford University Press, 2002. Harrington J. and K. Tumay, Simulation Modeling Methods: An Interactive Guide to Results-Based Decision . McGraw-Hill, 1998. Haas P. Stochastic Petri Net Models Modeling and Simulation . Springer Verlag, 2002. Hill D. Object-Oriented Analysis and Simulation Modeling . Addison-Wesley, 1996. Kouikoglou V. and Y. Phillis, Hybrid Simulation Models of Production Networks . Kluwer Pub. 2001. Law A. and W. Kelton, Simulation Modeling and Analysis . McGraw-Hill, 2000. Nelson B. Stochastic Modeling: Analysis Simulation . McGraw-Hill, 1995. Oakshott L., Business Modelling and Simulation . Pitman Publishing, London, 1997. Pidd M. Computer Simulation in Management Science . Wiley, 1998. Rubinstein R. and B. Melamed, Modern Simulation and Modeling . Wiley, 1998. Severance F. System Modeling and Simulation: An Introduction . Wiley, 2001. Van den Bosch, P. and A. Van der Klauw, Modeling, Identification Simulation of Dynamical Systems . CRC Press, 1994. Woods R. and K. Lawrence, Modeling and Simulation of Dynamic Systems . Prentice Hall, 1997. Techniques for Sensitivity Estimation Simulation continues to be the primary method by which engineers and managers obtain information about complex stochastic systems, such as telecommunication networks, health service, corporate planning, financial modeling, production assembly lines, and flexible manufacturing systems. These systems are driven by the occurrence of discrete events and complex interactions within these discrete events occur over time. For most discrete event systems (DES) no analytical methods are available, so DES must be studied via simulation. DES are studied to understand their performance, and to determine the best ways to improve their performance. In particular, one is often interested in how system performance depends on the systems parameter v, which could be a vector. DESs system performance is often measured as an expected value. Consider a system with continuous parameter v 206 V 205 R n . where V is an open set. Let be the steady state expected performance measure, where Y is a random vector with known probability density function (pdf), f(y v) depends on v, and Z is the performance measure. In discrete event systems, Monte Carlo simulation is usually needed to estimate J(v) for a given value v v 0 . By the law of large numbers converges to the true value, where y i . i 1, 2. n are independent, identically distributed, random vector realizations of Y from f (y v 0 ), and n is the number of independent replications. We are interested in sensitivities estimation of J(v) with respect to v. Applications of sensitivity information There are a number of areas where sensitivity information (the gradient, Hessian, etc.) of a performance measure J(v) or some estimate of it, is used for the purpose of analysis and control. In what follows, we single out a few such areas and briefly discuss them. Local information: An estimate for dJdv is a good local measure of the effect of on performance. For example, simply knowing the sign of the derivative dJdv at some point v immediately gives us the direction in which v should be changed. The magnitude of dJd also provides useful information in an initial design process: If dJdv is small, we conclude that J is not very sensitive to changes in. and hence focusing concentration on other parameters may improve performance. Structural properties: Often sensitivity analysis provides not only a numerical value for the sample derivative, but also an expression which captures the nature of the dependence of a performance measure on the parameter v. The simplest case arises when dJdv can be seen to be always positive (or always negative) for any sample path we may not be able to tell if the value of J(v) is monotonically increasing (or decreasing) in v. This information in itself is very useful in design and analysis. More generally, the form of dJdv can reveal interesting structural properties of the DES (e.g. monotonicity, convexity). Such properties must be exploited in order to determine optimal operating policies for some systems. Response surface generation: Often our ultimate goal is to obtain the function J(v), i.e. a curve describing how the system responds to different values of v. Since J(v) is unknown, one alternative is to obtain estimates of J(v) for as many values of v as possible. This is clearly a prohibitively difficult task. Derivative information, however may include not only first-order but also higher derivatives which can be used to approximate J(v). If such derivative information can be easily and accurately obtained, the task of response surface generation may be accomplished as well. Goal-seeking and What-if problems: Stochastic models typically depend upon various uncertain parameters that must be estimated from existing data sets. Statistical questions of how input parameter uncertainty propagates through the model into output parameter uncertainty is the so-called what-if analysis. A good answer to this question often requires sensitivity estimates. The ordinary simulation output results are the solution of a direct problem: Given the underlying pdf with a particular parameter value v. we may estimate the output function J(v). Now we pose the goal-seeking problem: given a target output value J 0 of the system and a parameterized pdf family, find an input value for the parameter, which generates such an output. There are strong motivations for both problems. When v is any controllable or uncontrollable parameter the decision maker is, for example, interested in estimating J(v) for a small change in v , the so called what-if problem, which is a direct problem and can be solved by incorporating sensitivity information in the Taylors expansion of J(v) in the neighborhood of v. However, when v is a controllable input, the decision maker may be interested in the goal-seeking problem: what change in the input parameter will achieve a desired change in output value J(v). Another application of goal-seeking arises when we want to adapt a model to satisfy a new equality constraint (condition) for some stochastic function. The solution to the goal-seeking problem is to estimate the derivative of the output function with respect to the input parameter for the nominal system use this estimate in a Taylors expansion of the output function in the neighborhood of the parameter and finally, use Robbins-Monro (R-M) type of stochastic approximation algorithm to estimate the necessary controllable input parameter value within the desired accuracy. Optimization: Discrete-event simulation is the primary analysis tool for designing complex systems. However, simulation must be linked with a mathematical optimization technique to be effectively used for systems design. The sensitivity dJdv can be used in conjunction with various optimization algorithms whose function is to gradually adjust v until a point is reached where J(v) is maximized (or minimized). If no other constraints on v are imposed, we expect dJdv 0 at this point. Click on the image to enlarge it and THEN print it. Finite difference approximation Kiefer and Wolfowitz proposed a finite difference approximation to the derivative. One version of the Kiefer-Wolfwitz technique uses two-sided finite differences. The first fact to notice about the K-W estimate is that it requires 2N simulation runs, where N is the dimension of vector parameter q. If the decision maker is interested in gradient estimation with respect to each of the components of q. then 2N simulations must be run for each component of v. This is inefficient. The second fact is that it may have a very poor variance, and it may result in numerical calculation difficulties. Simultaneous perturbation methods The simultaneous perturbation (SP) algorithm introduced by Dr. J. Spall has attracted considerable attention. There has recently been much interest in recursive optimization algorithms that rely on measurements of only the objective function to be optimized, not requiring direct measurements of the gradient of the objective function. Such algorithms have the advantage of not requiring detailed modeling information describing the relationship between the parameters to be optimized and the objective function. For example, many systems involving complex simulations or human beings are difficult to model, and could potentially benefit from such an optimization approach. The simultaneous perturbation stochastic approximation (SPSA) algorithm operates in the same framework as the above K-W methods, but has the strong advantage of requiring a much lower number of simulation runs to obtain the same quality of result. The essential feature of SPSA, which accounts for its power and relative ease of use in difficult multivariate optimization problems--is the underlying gradient approximation that requires only TWO objective function measurements regardless of the dimension of the optimization problem (one variation of basic SPSA uses only ONE objective function measurement per iteration). The underlying theory for SPSA shows that the N-fold savings in simulation runs per iteration (per gradient approximation) translates directly into an N-fold savings in the number of simulations to achieve a given quality of solution to the optimization problem. In other words, the K-W method and SPSA method take the same number of iterations to converge to the answer despite the N-fold savings in objective function measurements (e.g. simulation runs) per iteration in SPSA. Perturbation analysis Perturbation analysis (PA) computes (roughly) what simulations would have produced, had v been changed by a small amount without actually making this change. The intuitive idea behind PA is that a sample path constructed using v is frequently structurally very similar to the sample path using the perturbed v. There is a large amount of information that is the same for both of them. It is wasteful to throw this information away and to start the simulation from scratch with the perturbed v. In PA, moreover, we can let the change approach zero to get a derivative estimator without numerical problems. We are interested in the affect of a parameter change on the performance measure. However, we would like to realize this change by keeping the order of events exactly the same. The perturbations will be so small that only the duration, not the order, of the states will be affected. This effect should be observed in three successive stages: Step 1: How does a change in the value of a parameter vary the sample duration related to that parameter Step 2: How does the change in an individual sample duration reflect itself as a change in a subsequent particular sample realization Step 3: Finally, what is the relationship between the variation of the sample realization and its expected value Score function methods Using the score function method, the gradient can be estimated simultaneously, at any number of different parameter values, in a single-run simulation. The basic idea is that, the gradient of the performance measure function, J( v ), is expressed as an expectation with respect to the same distribution as the performance measure function itself. Therefore, the sensitivity information can be obtained with little computational (not simulation) cost, while estimating the performance measure. It is well-known that the crude form of the SF estimator suffers from the problem of linear growth in its variance as the simulation run increases. However, in the steady-state simulation the variance can be controlled by run length. Furthermore, information about the variance may be incorporated into the simulation algorithm. A recent flurry of activity has attempted to improve the accuracy of the SF estimates. Under regenerative conditions, the estimator can easily be modified to alleviate this problem, yet the magnitude of the variance may be large for queueing systems with heavy traffic intensity. The heuristic idea is to treat each component of the system (e.g. each queue) separately, which synchronously assumes that individual components have local regenerative cycles. This approach is promising since the estimator remains unbiased and efficient while the global regenerative cycle is very long. Now we look at the general (non-regenerative) case. In this case any simulation will give a biased estimator of the gradient, as simulations are necessarily finite. If n (the length of the simulation) is large enough, this bias is negligible. However, as noted earlier, the variance of the SF sensitivity estimator increases with increase in n so, a crude SF estimator is not even approximately consistent. There are a number of ways to attack this problem. Most of the variations in an estimator comes from the score function. The variation is especially high, when all past inputs contribute to the performance and the scores from all are included. When one uses batch means, the variation is reduced by keeping the length of the batch small. A second way is to reduce the variance of the score to such an extent that we can use simulations long enough to effectively eliminate the bias. This is the most promising approach. The variance may be reduced further by using the standard variance reduction techniques (VRT), such as importance sampling. Finally, we can simply use a large number of iid replications of the simulation. Harmonic analysis Another strategy for estimating the gradient simulation is based on the frequency domain method, which differs from the time domain experiments in that the input parameters are deterministically varied in sinusoidal patterns during the simulation run, as opposed to being kept fixed as in the time domain runs. The range of possible values for each input factor should be identified. Then the values of each input factor within its defined range should be changed during a run. In time series analysis, t is the time index. In simulation, however, t is not necessarily the simulation clock time. Rather, t is a variable of the model, which keeps track of certain statistics during each run. For example, to generate the inter-arrival times in a queueing simulation, t might be the variable that counts customer arrivals. Frequency domain simulation experiments identify the significant terms of the polynomial that approximates the relationship between the simulation output and the inputs. Clearly, the number of simulation runs required to identify the important terms by this approach is much smaller than those of the competing alternatives, and the difference becomes even more conspicuous as the number of parameters increases. Conclusions Further Readings PA and SF (or LR) can be unified. Further comparison of the PA and SF approaches reveals several interesting differences. Both approaches require an interchange of expectation and differentiation. However, the conditions for this interchange in PA depend heavily on the nature of the problem, and must be verified for each application, which is not the case in SF. Therefore, in general, it is easier to satisfy SF unbiased conditions. PA assumes that the order of events in the perturbed path is the same as the order in the nominal path, for a small enough change in v. allowing the computation of the sensitivity of the sample performance for a particular simulation. For example, if the performance measure is the mean number of customer in a busy period, the PA estimate of the gradient with respect to any parameter is zero The number of customers per busy period will not change if the order of events does not change. In terms of ease of implementation, PA estimators may require considerable analytical work on the part of algorithm developer, with some customization for each application, whereas SF has the advantage of remaining a general definable algorithm whenever it can be applied. Perhaps the most important criterion for comparison lies in the question of accuracy of an estimator, typically measured through its variance. If an estimator is strongly consistent, its variance is gradually reduced over time and ultimately approaches to zero. The speed with which this happens may be extremely important. Since in practice, decisions normally have to be made in a limited time, an estimator whose variance decreases fast is highly desirable. In general, when PA does provide unbiased estimators, the variance of these estimators is small. PA fully exploits the structure of DES and their state dynamics by extracting the needed information from the observed sample path, whereas SF requires no knowledge of the system other than the inputs and the outputs. Therefore when using SF methods, variance reduction is necessary. The question is whether or not the variance can be reduced enough to make the SF estimator useful in all situations to which it can be applied. The answer is certainly yes. Using the standard variance reduction techniques can help, but the most dramatic variance reduction occurs using new methods of VR such as conditioning, which is shown numerically to have a mean squared error that is essentially the same as that of PA. References Further Readings: Arsham H. Algorithms for Sensitivity Information in Discrete-Event Systems Simulation, Simulation Practice and Theory . 6(1), 1-22, 1998. Fu M. and J-Q. Hu, Conditional Monte Carlo: Gradient Estimation and Optimization Applications . Kluwer Academic Publishers, 1997. Rubinstein R. and A. Shapiro, Discrete Event Systems: Sensitivity Analysis and Stochastic Optimization by the Score Function Method . John Wiley Sons, 1993. Whitt W. Minimizing delays in the GIG1 queue, Operations Research . 32(1), 41-51, 1984. Simulation-based Optimization Techniques Discrete event simulation is the primary analysis tool for designing complex systems. Simulation, however, must be linked with a optimization techniques to be effectively used for systems design. We present several optimization techniques involving both continuous and discrete controllable input parameters subject to a variety of constraints. The aim is to determine the techniques most promising for a given simulation model. Many man-made systems can be modeled as Discrete Event Systems (DES) examples are computer systems, communication networks, flexible manufacturing systems, production assembly lines, and traffic transportation systems. DES evolve with the occurrence of discrete events, such as the arrival of a job or the completion of a task, in contrast with continuously variable dynamic processes such as aerospace vehicles, which are primarily governed by differential equations. Owing to the complex dynamics resulting from stochastic interactions of such discrete events over time, the performance analysis and optimization of DES can be difficult tasks. At the same time, since such systems are becoming more widespread as a result of modern technological advances, it is important to have tools for analyzing and optimizing the parameters of these systems. Analyzing complex DES often requires computer simulation. In these systems, the objective function may not be expressible as an explicit function of the input parameters rather, it involves some performance measures of the system whose values can be found only by running the simulation model or by observing the actual system. On the other hand, due to the increasingly large size and inherent complexity of most man-made systems, purely analytical means are often insufficient for optimization. In these cases, one must resort to simulation, with its chief advantage being its generality, and its primary disadvantage being its cost in terms of time and money. Even though, in principle, some systems are analytically tractable, the analytical effort required to evaluate the solution may be so formidable that computer simulation becomes attractive. While the price for computing resources continue to dramatically decrease, one nevertheless can still obtain only a statistical estimate as opposed to an exact solution. For practical purposes, this is quite sufficient. These man-made DES are costly, and therefore it is important to operate them as efficiently as possible. The high cost makes it necessary to find more efficient means of conducting simulation and optimizing its output. We consider optimizing an objective function with respect to a set of continuous andor discrete controllable parameters subject to some constraints. Click on the image to enlarge it and THEN print it. The above figure illustrates the feedback loop application. Although the feedback concept is not a simulation but a systemic concept, however, whatever paradigm we use one can always incorporate feedback. For example, consider a discrete event system (DES) model that employs resources to achieve certain tasksprocesses, by only incorporating decision rules regarding how to manage the stocks and thence how the resource will be deployed depending on the stock level, clearly, in the system structure there are feedback loops. Usually when modelers choose a DES approach they often model the system as open loop or nearly open loop system, making the system behave as if there where no superior agent controlling the whole productionservice process. Closing the loops should be an elemental task that simulation modeler should take care of, even if the scope does not involve doing it, there must be awareness of system behavior, particularly if there is known to be that the system if under human decision making processesactivities. In almost all simulation models, an expected value can express the systems performance. Consider a system with continuous parameter v 206 V, where V is the feasible region. Let be the steady state expected performance measure, where Y is a random vector with known probability density function (pdf), f(y v) depends on v, and Z is the performance measure. In discrete event systems, Monte Carlo simulation is usually needed to estimate J(v) for a given value v v 0 . By the law of large numbers converges to the true value, where y i . i 1, 2. n are independent, identically distributed, random vector realizations of Y from f (y v 0 ), and n is the number of independent replications. The aim is to optimize J(v) with respect to v. We shall group the optimization techniques for simulation into seven broad categories namely, Deterministic Search, Pattern Search, Probabilistic Search, Evolutionary Techniques, Stochastic Approximation, Gradient Surface, and some Mixtures of the these techniques Click on the image to enlarge it and THEN print it. Deterministic search techniques A common characteristic of deterministic search techniques is that they are basically borrowed from deterministic optimization techniques. The deterministic objective function value required in the technique is now replaced with an estimate obtained from simulation. By having a reasonably accurate estimate, one hopes that the technique will perform well. Deterministic search techniques include heuristic search, complete enumeration, and random search techniques. Heuristic search technique The heuristic search technique is probably most commonly used in optimizing response surfaces. It is also the least sophisticated scheme mathematically, and it can be thought of as an intuitive and experimental approach. The analyst determines the starting point and stopping rule based on previous experience with the system. After setting the input parameters (factors) to levels that appear reasonable, the analyst makes a simulation run with the factors set at those levels and computes the value of the response function. If it appears to be a maximum (minimum) to the analyst, the experiment is stopped. Otherwise the analyst changes parameter settings and makes another run. This process continues until the analyst believes that the output has been optimized. Suffice it to say that, if the analyst is not intimately familiar with the process being simulated, this procedure can turn into a blind search and can expend an inordinate amount of time and computer resources without producing results commensurate with input. The heuristic search can be ineffective and inefficient in the hand of a novice. Complete enumeration and random techniques The complete enumeration technique is not applicable to continuous cases, but in discrete space v it does yield the optimal value of the response variable. All factors ( v ) must assume a finite number of values for this technique to be applicable. Then, a complete factorial experiment is run. The analyst can attribute some degree of confidence to the determined optimal point when using this procedure. Although the complete enumeration technique yields the optimal point, it has a serious drawback. If the number of factors or levels per factor is large, the number of simulation runs required to find the optimal point can be exceedingly large. For example, suppose that an experiment is conducted with three factors having three, four, and five levels, respectively. Also suppose that five replications are desired to provide the proper degree of confidence. Then 300 runs of the simulator are required to find the optimal point. Hence, this technique should be used only when the number of unique treatment combinations is relatively small or a run takes little time. The random search technique resembles the complete enumeration technique except that one selects a set of inputs at random. The simulated results based on the set that yields the maximum (minimum) value of the response function is taken to be the optimal point. This procedure reduces the number of simulation runs required to yield an optimal result however, there is no guarantee that the point found is actually the optimal point. Of course, the more points selected, the more likely the analyst is to achieve the true optimum. Note that the requirement that each factor assumes only a finite number of values is not a requirement in this scheme. Replications can be made on the treatment combinations selected, to increase the confidence in the optimal point. Which strategy is better, replicating a few points or looking at a single observation on more points, depends on the problem. Response surface search Response surface search attempts to fit a polynomial to J(v). If the design space v is suitably small, the performance function J(v) may be approximated by a response surface, typically a first order, or perhaps quadratic order in v. possibly after transformation, e.g. log ( v ). The response surface method (RSM) requires running the simulation in a first order experimental design to determine the path of steepest descent. Simulation runs made along this path continue, until one notes no improvement in J(v). The analyst then runs a new first order experimental design around the new optimal point reached, and finds a new path of steepest descent. The process continues, until there is a lack of fit in the fitted first order surface. Then, one runs a second order design, and takes the optimum of the fittest second order surface as the estimated optimum. Although it is desirable for search procedures to be efficient over a wide range of response surfaces, no current procedure can effectively overcome non-unimodality (surfaces having more than one local maximum or minimum). An obvious way to find the global optimal would be to evaluate all the local optima. One technique that is used when non-unimodality is known to exist, is called the Las Vegas technique. This search procedure estimates the distribution of the local optima by plotting the estimated J( v ) for each local search against its corresponding search number. Those local searches that produce a response greater than any previous response are then identified and a curve is fitted to the data. This curve is then used to project the estimated incremental response that will be achieved by one more search. The search continues until the value of the estimated improvement in the search is less than the cost of completing one additional search. It should be noted that a well-designed experiment requires a sufficient number of replications so that the average response can be treated as a deterministic number for search comparisons. Otherwise, since replications are expensive, it becomes necessary to effectively utilize the number of simulation runs. Although each simulation is at a different setting of the controllable variables, one can use smoothing techniques such as exponential smoothing to reduce the required number of replications. Pattern search techniques Pattern search techniques assume that any successful set of moves used in searching for an approximated optimum is worth repeating. These techniques start with small steps then, if these are successful, the step size increases. Alternatively, when a sequence of steps fails to improve the objective function, this indicates that shorter steps are appropriate so we may not overlook any promising direction. These techniques start by initially selecting a set of incremental values for each factor. Starting at an initial base point, they check if any incremental changes in the first variable yield an improvement. The resulting improved setting becomes the new intermediate base point. One repeats the process for each of the inputs until one obtains a new setting where the intermediate base points act as the initial base point for the first variable. The technique then moves to the new setting. This procedure is repeated, until further changes cannot be made with the given incremental values. Then, the incremental values are decreased, and the procedure is repeated from the beginning. When the incremental values reach a pre-specified tolerance, the procedure terminates the most recent factor settings are reported as the solution. Conjugate direction search The conjugate direction search requires no derivative estimation, yet it finds the optimum of an N-dimensional quadratic surface after, at most, N-iterations, where the number of iterations is equal to the dimension of the quadratic surface. The procedure redefines the n dimensions so that a single variable search can be used successively. Single variable procedures can be used whenever dimensions can be treated independently. The optimization along each dimension leads to the optimization of the entire surface. Two directions are defined to be conjugate whenever the cross-product terms are all zero. The conjugate direction technique tries to find a set of n dimensions that describes the surface such that each direction is conjugate to all others. Using the above result, the technique attempts to find two search optima and replace the n th dimension of the quadratic surface by the direction specified by the two optimal points. Successively replacing the original dimension yields a new set of n dimensions in which, if the original surface is quadratic, all directions are conjugate to each other and appropriate for n single variable searches. While this search procedure appears to be very simple, we should point out that the selection of appropriate step sizes is most critical. The step size selection is more critical for this search technique because - during axis rotation - the step size does not remain invariant in all dimensions. As the rotation takes place, the best step size changes, and becomes difficult to estimate. Steepest ascent (descent) The steepest ascent (descent) technique uses a fundamental result from calculus ( that the gradient points in the direction of the maximum increase of a function), to determine how the initial settings of the parameters should be changed to yield an optimal value of the response variable. The direction of movement is made proportional to the estimated sensitivity of the performance of each variable. Although quadratic functions are sometimes used, one assumes that performance is linearly related to the change in the controllable variables for small changes. Assume that a good approximation is a linear form. The basis of the linear steepest ascent is that each controllable variable is changed in proportion to the magnitude of its slope. When each controllable variable is changed by a small amount, it is analogous to determining the gradient at a point. For a surface containing N controllable variables, this requires N points around the point of interest. When the problem is not an n-dimensional elliptical surface, the parallel-tangent points are extracted from bitangents and inflection points of occluding contours. Parallel tangent points are points on the occluding contour where the tangent is parallel to a given bitangent or the tangent at an inflection point. Tabu search technique An effective technique to overcome local optimality for discrete optimization is the Tabu Search technique. It explores the search space by moving from a solution to its best neighbor, even if this results in a deterioration of the performance measure value. This approach increases the likelihood of moving out of local optima. To avoid cycling, solutions that were recently examined are declared tabu (Taboo) for a certain number of iterations. Applying intensification procedures can accentuate the search in a promising region of the solution space. In contrast, diversification can be used to broaden the search to a less explored region. Much remains to be discovered about the range of problems for which the tabu search is best suited. Hooke and Jeeves type techniques The Hooke and Jeeves pattern search uses two kinds of moves namely, an exploratory and a pattern move. The exploratory move is accomplished by doing a coordinate search in one pass through all the variables. This gives a new base point from which a pattern move is made. A pattern move is a jump in the pattern direction determined by subtracting the current base point from the previous base point. After the pattern move, another exploratory move is carried out at the point reached. If the estimate of J(v) is improved at the final point after the second exploratory move, it becomes the new base point. If it fails to show improvement, an exploratory move is carried out at the last base point with a smaller step in the coordinate search. The process stops when the step gets small enough. Simplex-based techniques The simplex-based technique performs simulation runs first at the vertices of the initial simplex i.e. a polyhedron in the v -space having N1 vertices. A subsequent simplex (moving towards the optimum) are formed by three operations performed on the current simplex: reflection, contraction, and expansion. At each stage of the search process, the point with the highest J(v) is replaced with a new point foundvia reflection through the centroid of the simplex. Depending on the value of J(v) at this new point, the simplex is either expanded, contracted, or unchanged. The simplex technique starts with a set of N1 factor settings. These N1 points are all the same distance from the current point. Moreover, the distance between any two points of these N1 points is the same. Then, by comparing their response values, the technique eliminates the factor setting with the worst functional value and replaces it with a new factor setting, determined by the centroid of the N remaining factor settings and the eliminated factor setting. The resulting simplex either grows or shrinks, depending on the response value at the new factor settings. One repeats the procedure until no more improvement can be made by eliminating a point, and the resulting final simplex is small. While this technique will generally performance well for unconstrained problems, it may collapse to a point on a boundary of a feasible region, thereby causing the search to come to a premature halt. This technique is effective if the response surface is generally bowl- shaped even with some local optimal points. Probabilistic search techniques All probabilistic search techniques select trial points governed by a scan distribution, which is the main source of randomness. These search techniques include random search, pure adaptive techniques, simulated annealing, and genetic methods. Random search A simple, but very popular approach is the random search, which centers a symmetric probability density function (pdf) e.g. the normal distribution, about the current best location. The standard normal N(0, 1) is a popular choice, although the uniform distribution U-1, 1 is also common. A variation of the random search technique determines the maximum of the objective function by analyzing the distribution of J(v) in the bounded sub-region. In this variation, the random data are fitted to an asymptotic extreme-value distribution, and J is estimated with a confidence statement. Unfortunately, these techniques cannot determine the location of J. which can be as important as the J value itself. Some techniques calculate the mean value and the standard deviation of J(v) from the random data as they are collected. Assuming that J is distributed normally in the feasible region. the first trial, that yields a J-value two standard deviations within the mean value, is taken as a near-optimum solution. Pure adaptive search Various pure adaptive search techniques have been suggested for optimization in simulation. Essentially, these techniques move from the current solution to the next solution that is sampled uniformly from the set of all better feasible solutions. Evolutionary Techniques Nature is a robust optimizer. By analyzing natures optimization mechanism we may find acceptable solution techniques to intractable problems. Two concepts that have most promise are simulated annealing and the genetic techniques. Simulated annealing Simulated annealing (SA) borrows its basic ideas from statistical mechanics. A metal cools, and the electrons align themselves in an optimal pattern for the transfer of energy. In general, a slowly cooling system, left to itself, eventually finds the arrangement of atoms, which has the lowest energy. The is the behavior, which motivates the method of optimization by SA. In SA we construct a model of a system and slowly decrease the temperature of this theoretical system, until the system assumes a minimal energy structure. The problem is how to map our particular problem to such an optimizing scheme. SA as an optimization technique was first introduced to solve problems in discrete optimization, mainly combinatorial optimization. Subsequently, this technique has been successfully applied to solve optimization problems over the space of continuous decision variables. SA is a simulation optimization technique that allows random ascent moves in order to escape the local minima, but a price is paid in terms of a large increase in the computational time required. It can be proven that the technique will find an approximated optimum. The annealing schedule might require a long time to reach a true optimum. Genetic techniques Genetic techniques (GT) are optimizers that use the ideas of evolution to optimize a system that is too difficult for traditional optimization techniques. Organisms are known to optimize themselves to adapt to their environment. GT differ from traditional optimization procedures in that GT work with a coding of the decision parameter set, not the parameters themselves GT search a population of points, not a single point GT use objective function information, not derivatives or other auxiliary knowledge and finally, GT use probabilistic transition rules, not deterministic rules. GT are probabilistic search optimizing techniques that do not require mathematical knowledge of the response surface of the system, which they are optimizing. They borrow the paradigms of genetic evolution, specifically selection, crossover, and mutation. Selection: The current points in the space are ranked in terms of their fitness by their respective response values. A probability is assigned to each point that is proportional to its fitness, and parents (a mating pair) are randomly selected. Crossover: The new point, or offspring, is chosen, based on some combination of the genetics of the two parents. Mutation: The location of offspring is also susceptible to mutation, a process, which occurs with probability p, by which a offspring is replaced randomly by a new offspring location. A generalized GT generates p new offspring at once and kills off all of the parents. This modification is important in the simulation environment. GT are well suited for qualitative or policy decision optimization such as selecting the best queuing disciplines or network topologies. They can be used to help determine the design of the system and its operation. For applications of GT to inventory systems, job-shop, and computer time-sharing problems. GT do not have certain shortcomings of other optimization techniques, and they will usually result in better calculated optima than those found with the traditionally techniques. They can search a response surface with many local optima and find (with a high probability) the approximate global optimum. One may use GT to find an area of potential interest, and then resort to other techniques to find the optimum. Recently, several classical GT principles have been challenged. Differential Evolution. Differential Evolution (DE) is a genetic type of algorithm for solving continuous stochastic function optimization. The basic idea is to use vector differences for perturbing the vector population. DE adds the weighted difference between two population vectors to a third vector. This way, no separate probability distribution has to be used, which makes the scheme completely self-organizing. A short comparison When performing search techniques in general, and simulated annealing or genetic techniques specifically, the question of how to generate the initial solution arises. Should it be based on a heuristic rule or on a randomly generated one Theoretically, it should not matter, but in practice this may depend on the problem. In some cases, a pure random solution systematically produces better final results. On the other hand, a good initial solution may lead to lower overall run times. This can be important, for example, in cases where each iteration takes a relatively long time therefore, one has to use some clever termination rule. Simulation time is a crucial bottleneck in an optimization process. In many cases, a simulation is run several times with different initial solutions. Such a technique is most robust, but it requires the maximum number of replications compared with all other techniques. The pattern search technique applied to small problems with no constraints or qualitative input parameters requires fewer replications than the GT. GT, however, can easily handle constraints, and have lower computational complexity. Finally, simulated annealing can be embedded within the Tabu search to construct a probabilistic technique for global optimization. References Further Readings: Choi D.-H. Cooperative mutation based evolutionary programming for continuous function optimization, Operations Research Letters . 30, 195-201, 2002. Reeves C. and J. Rowe, Genetic Algorithms: Principles and Perspectives . Kluwer, 2002. Saviotti P. (Ed.), Applied Evolutionary Economics: New Empirical Methods and Simulation Techniques . Edward Elgar Pub. 2002. Wilson W. Simulating Ecological and Evolutionary Systems in C . Cambridge University Press, 2000. Stochastic approximation techniques Two related stochastic approximation techniques have been proposed, one by Robbins and Monro and one by Kiefer and Wolfowitz. The first technique was not useful for optimization until an unbiased estimator for the gradient was found. Kiefer and Wolfowitz developed a procedure for optimization using finite differences. Both techniques are useful in the optimization of noisy functions, but they did not receive much attention in the simulation field until recently. Generalization and refinement of stochastic approximation procedures give rise to a weighted average, and stochastic quasi-gradient methods. These deal with constraints, non-differentiable functions, and some classes of non-convex functions, among other things. Kiefer-Wolfowitz type techniques Kiefer and Wolfowitz proposed a finite difference approximation to the derivative. One version of the Kiefer-Wolfwitz technique uses two-sided finite differences. The first fact to notice about the K-W estimate is that it requires 2N simulation runs, where N is the dimension of vector parameter v. If the decision maker is interested in gradient estimation with respect to each of the components of v. then 2N simulations must be run for each component of v. This is inefficient. The second fact is that it may have a very poor variance, and it may result in numerical calculation difficulties. Robbins-Monro type techniques The original Robbins-Monro (R-M) technique is not an optimization scheme, but rather a root finding procedure for functions whose exact values are not known but are observed with noise. Its application to optimization is immediate: use the procedure to find the root of the gradient of the objective function. Interest was renewed in the R-M technique as a means of optimization, with the development of the perturbation analysis, score function (known also as likelihood ratio method), and frequency domain estimates of derivatives. Optimization for simulated systems based on the R-M technique is known as a single-run technique. These procedures optimize a simulation model in a single run simulation with a run length comparable to that required for a single iteration step in the other methods. This is achieved essentially be observing the sample values of the objective function and, based on these observations, updating the values of the controllable parameters while the simulation is running, that is, without restarting the simulation. This observing-updating sequence is done repeatedly, leading to an estimate of the optimum at the end of a single-run simulation. Besides having the potential of large computational savings, this technique can be a powerful tool in real-time optimization and control, where observations are taken as the system is evolving in time. Gradient surface method One may combine the gradient-based techniques with the response surface methods (RSM) for optimization purposes. One constructs a response surface with the aid of n response points and the components of their gradients. The gradient surface method (GSM) combines the virtue of RSM with that of the single- run, gradient estimation techniques such as Perturbation Analysis, and Score Function techniques. A single simulation experiment with little extra work yields N 1 pieces of information i.e. one response point and N components of the gradient. This is in contrast to crude simulation, where only one piece of information, the response value, is obtained per experiment. Thus by taking advantage of the computational efficiency of single-run gradient estimators. In general, N-fold fewer experiments will be needed to fit a global surface compared to the RSM. At each step, instead of using Robbins-Monro techniques to locate the next point locally, we determine a candidate for the next point globally, based on the current global fit to the performance surface. The GSM approach has the following advantages The technique can quickly get to the vicinity of the optimal solution because its orientation is global 23, 39. Thus, it produces satisfying solutions quickly Like RSM, it uses all accumulated information And, in addition, it uses gradient surface fitting, rather than direct performance response-surface fitting via single-run gradient estimators. This significantly reduces the computational efforts compared with RSM. Similar to RSM, GSM is less sensitive to estimation error and local optimality And, finally, it is an on-line technique, the technique may be implemented while the system is running. A typical optimization scheme involves two phases: a Search Phase and an Iteration Phase. Most results in analytic computational complexity assume that good initial approximations are available, and deal with the iteration phase only. If enough time is spent in the initial search phase, we can reduce the time needed in the iteration phase. The literature contains papers giving conditions for the convergence of a process a process has to be more than convergent in order to be computationally interesting. It is essential that we be able to limit the cost of computation. In this sense, GSM can be thought of as helping the search phase and as an aid to limit the cost of computation. One can adopt standard or simple devices for issues such as stopping rules. For on-line optimization, one may use a new design in GSM called single direction design. Since for on-line optimization it may not be advisable or feasible to disturb the system, random design usually is not suitable. Post-solution analysis Stochastic models typically depend upon various uncertain and uncontrollable input parameters that must be estimated from existing data sets. We focus on the statistical question of how input-parameter uncertainty propagates through the model into output- parameter uncertainty. The sequential stages are descriptive, prescriptive and post-prescriptive analysis. Rare Event Simulation Large deviations can be used to estimate the probability of rare events, such as buffer overflow, in queueing networks. It is simple enough to be applied to very general traffic models, and sophisticated enough to give insight into complex behavior. Simulation has numerous advantages over other approaches to performance and dependability evaluation most notably, its modelling power and flexibility. For some models, however, a potential problem is the excessive simulation effort (time) required to achieve the desired accuracy. In particular, simulation of models involving rare events, such as those used for the evaluation of communications and highly-dependable systems, is often not feasible using standard techniques. In recent years, there have been significant theoretical and practical advances towards the development of efficient simulation techniques for the evaluation of these systems. Methodologies include: Techniques based on importance sampling, The restart method, and Hybrid analyticsimulation techniques among newly devised approaches. Conclusions Further Readings With the growing incidence of computer modeling and simulation, the scope of simulation domain must be extended to include much more than traditional optimization techniques. Optimization techniques for simulation must also account specifically for the randomness inherent in estimating the performance measure and satisfying the constraints of stochastic systems. We described the most widely used optimization techniques that can be effectively integrated with a simulation model. We also described techniques for post-solution analysis with the aim of theoretical unification of the existing techniques. All techniques were presented in step-by-step format to facilitate implementation in a variety of operating systems and computers, thus improving portability. General comparisons among different techniques in terms of bias, variance, and computational complexity are not possible. However, a few studies rely on real computer simulations to compare different techniques in terms of accuracy and number of iterations. Total computational effort for reduction in both the bias andvariance of the estimate depends on the computational budget allocated for a simulation optimization. No single technique works effectively andor efficiently in all cases. The simplest technique is the random selection of some points in the search region for estimating the performance measure. In this technique, one usually fixes the number of simulation runs and takes the smallest (or largest) estimated performance measure as the optimum. This technique is useful in combination with other techniques to create a multi-start technique for global optimization. The most effective technique to overcome local optimality for discrete optimization is the Tabu Search technique. In general, the probabilistic search techniques, as a class, offer several advantages over other optimization techniques based on gradients. In the random search technique, the objective function can be non-smooth or even have discontinuities. The search program is simple to implement on a computer, and it often shows good convergence characteristics in noisy environments. More importantly, it can offer the global solution in a multi-modal problem, if the technique is employed in the global sense. Convergence proofs under various conditions are given in. The Hooke-Jeeves search technique works well for unconstrained problems with less than 20 variables pattern search techniques are more effective for constrained problems. Genetic techniques are most robust and can produce near-best solutions for larger problems. The pattern search technique is most suitable for small size problems with no constraint, and it requires fewer iterations than the genetic techniques. The most promising techniques are the stochastic approximation, simultaneous perturbation, and the gradient surface methods. Stochastic approximation techniques using perturbation analysis, score function, or simultaneous perturbation gradient estimators, optimize a simulation model in a single simulation run. They do so by observing the sample values of the objective function, and based on these observations, the stochastic approximation techniques update the values of the controllable parameters while the simulation is running and without restarting the simulation. This observing-updating sequence, done repeatedly, leads to an estimate of the optimum at the end of a single-run simulation. Besides having the potential of large savings in computational effort in the simulation environment, this technique can be a powerful tool in real-time optimization and control, where observations are taken as the system is evolving over time. Response surface methods have a slow convergence rate, which makes them expensive. The gradient surface method combines the advantages of the response surface methods (RSM) and efficiency of the gradient estimation techniques, such as infinitesimal perturbation analysis, score function, simultaneous perturbation analysis, and frequency domain technique. In the gradient surface method (GSM) the gradient is estimated, and the performance gradient surface is estimated from observations at various points, similar to the RSM. Zero points of the successively approximating gradient surface are then taken as the estimates of the optimal solution. GSM is characterized by several attractive features: it is a single run technique and more efficient than RSM at each iteration step, it uses the information from all of the data points rather than just the local gradient it tries to capture the global features of the gradient surface and thereby quickly arrive in the vicinity of the optimal solution, but close to the optimum, they take many iterations to converge to stationary points. Search techniques are therefore more suitable as a second phase. The main interest is to figure out how to allocate the total available computational budget across the successive iterations. For when the decision variable is qualitative, such as finding the best system configuration, a random or permutation test is proposed. This technique starts with the selection of an appropriate test statistic, such as the absolute difference between the mean responses under two scenarios. The test value is computed for the original data set. The data are shuffled (using a different seed) the test statistic is computed for the shuffled data and the value is compared to the value of the test statistic for the original, un-shuffled data. If the statistics for the shuffled data are greater than or equal to the actual statistic for the original data, then a counter c, is incremented by 1. The process is repeated for any desired m number of times. The final step is to compute (c1)(m1), which is the significant level of the test. The null hypothesis is rejected if this significance level is less than or equal to the specified rejection level for the test. There are several important aspects to this nonparametric test. First, it enables the user to select the statistic. Second, assumptions such as normality or equality of variances made for the t-test, ranking-and-selection, and multiple-comparison procedures, are no longer needed. A generalization is the well-known bootstrap technique. What Must Be Done computational studies of techniques for systems with a large number of controllable parameters and constraints. effective combinations of several efficient techniques to achieve the best results under constraints on computational resources.development of parallel and distributed schemesdevelopment of an expert system that incorporates all available techniques. References Further Readings: Arsham H. Techniques for Monte Carlo Optimizing, Monte Carlo Methods and Applications . 4(3), 181-230, 1998. Arsham H. Stochastic Optimization of Discrete Event Systems Simulation, Microelectronics and Reliability . 36(10), 1357-1368, 1996. Fu M. and J-Q. Hu, Conditional Monte Carlo: Gradient Estimation and Optimization Applications . Kluwer Academic Publishers, 1997. Rollans S. and D. McLeish, Estimating the optimum of a stochastic system using simulation, Journal of Statistical Computation and Simulation . 72, 357 - 377, 2002. Rubinstein R. and A. Shapiro, Discrete Event Systems: Sensitivity Analysis and Stochastic Optimization by the Score Function Method . John Wiley Sons, 1993. Metamodeling and the Goal seeking Problems The simulation models although simpler than the real-world system, are still a very complex way of relating input (v) to output J(v). Sometimes a simpler analytic model may be used as an auxiliary to the simulation model. This auxiliary model is often referred to as a metamodel. In many simulation applications such as systems analysis and design applications, the decision maker may not be interested in optimization but wishes to achieve a certain value for J(v), say J 0 . This is the goal-seeking problem. given a target output value J 0 of the performance and a parameterized pdf family, one must find an input value for the parameter, which generates such an output. Metamodeling The simulation models although simpler than the real-world system, are still a very complex way of relating input (v) to output J(v). Sometimes a simpler analytic model may be used as an auxiliary to the simulation model. This auxiliary model is often referred to as a metamodel. There are several techniques available for metamodeling including: design of experiments, response surface methodology, Taguchi methods, neural networks, inductive learning, and kriging. Metamodeling may have different purposes: model simplification and interpretation, optimization, what-if analysis, and generalization to models of the same type. The following polynomial model can be used as an auxiliary model. where d v v-v 0 and the primes denote derivatives. This metamodel approximates J(v) for small d v. To estimate J(v) in the neighborhood of v 0 by a linear function, we need to estimate the nominal J(v) and its first derivative. Traditionally, this derivative is estimated by crude Monte Carlo i.e. finite difference which requires rerunning the simulation model. Methods which yield enhanced efficiency and accuracy in estimating, at little additional computational (Not simulation) cost, are presented in this site. The Score Function method of estimating the first derivative is: where Sf(y v) f(y v)d Lnf(y v) dv is the Score function and differentiations is with respect to v, provided that, f(y v) exist, and f(y v) is positive for all v in V. The Score function approach can be extended in estimating the second and higher order of derivatives. For example, an estimate for the second derivative based on the Score Function method is: Where S and H S S 2 are the score and information functions, respectively, widely used in statistics literature, such as in the construction of Cramer-Rao bounds. By having gradient and Hessian in our disposal, we are able to construct a second order local metamodel using the Taylors series. An Illustrative Numerical Example: For most complex reliability systems, the performance measures such as mean time to failure (MTTF) are not available in analytical form. We resort to Monte Carlo Simulation (MCS) to estimate MTTF function from a family of single-parameter density functions of the components life with specific value for the parameter. The purpose of this section is to solve the inverse problem, which deals with the calculation of the components life parameters (such as MTTF) of a homogeneous subsystem, given a desired target MTTF for the system. A stochastic approximation algorithm is used to estimate the necessary controllable input parameter within a desired range of accuracy. The potential effectiveness is demonstrated by simulating a reliability system with a known analytical solution. Consider the coherent reliability sub-system with four components component 1, and 2 are in series, and component 3 and 4 also in series, however these two series of components are in parallel, as illustrated in the following Figure. All components are working independently and are homogeneous i.e. manufactured by an identical process, components having independent random lifetimes Y1, Y2, Y3, and Y4, which are distributed exponentially with rates v v 0 0.5. The system lifetime is Z (Y1,Y2,Y3,Y4 v 0 ) max min (Y3,Y4), min (Y1,Y2). It is readily can be shown that the theoretical expected lifetime of this sub-system is The underlying pdf for this system is: f(y v) v 4 exp(-v S y i ), the sum is over i 1, 2, 3, 4. Applying the Score function method, we have: S(y) f (y v) f(y v) 4v - S y i . the sum is over i 1, 2, 3, 4. H(y) f (y v) f(y v) v 2 ( S y i ) 2 - 8v ( S y i ) 12 v 2 , the sums are over i 1, 2, 3, 4. The estimated average lifetime and its derivative for the nominal system with v v 0 0.5, are: respectively, where Y i, j is the j th observation for the i th component (i 1, 2, 3, 4). We have performed a Monte Carlo experiment for this system by generating n 10000 independent replications using SIMSCRIPT II.5 random number streams 1 through 4 to generate exponential random variables Y1, Y2, Y3, Y4. respectively, on a VAX system. The estimated performance is J(0.5) 1.5024, with a standard error of 0.0348. The first and second derivatives estimates are -3.0933 and 12.1177 with standard errors of 0.1126 and 1.3321, respectively. The response surface approximation in the neighborhood of v 0.5 is: J(v) 1.5024 (v - 0.5) (-3.0933) (v - 0.5) 2 (12.1177)2 6.0589v 2 - 9.1522v 4.5638 A numerical comparison based on exact and the approximation by this metamodel reveals that the largest absolute error is only 0.33 for any v in the range of 0.40, 0.60. This error could be reduced by either more accurate estimates of the derivatives andor using a higher order Taylor expansion. A comparison of the errors indicates that the errors are smaller and more stable in the direction of increasing v. This behavior is partly due to the fact that lifetimes are exponentially distributed with variance 1v. Therefore, increasing v causes less variance than the nominal system (with v 0.50). Goal seeking problem In many systems modeling and simulation applications, the decision maker may not be interested in optimization but wishes to achieve a certain value for J(v), say J 0 . This is the goal-seeking problem. given a target output value J 0 of the performance and a parameterized pdf family, one must find an input value for the parameter, which generates such an output. When is a controllable input, the decision maker may be interested in the goal-seeking problem: namely, what change of the input parameter will achieve a desired change in the output value. Another application of the goal-seeking problem arises when we want to adapt a model to satisfy a new equality constraint with some stochastic functions. We may apply the search techniques, but the goal-seeking problem can be considered as an interpolation based on a meta-model. In this approach, one generates a response surface function for J(v). Finally, one uses the fitted function to interpolate for the unknown parameter. This approach is tedious, time-consuming, and costly moreover, in a random environment, the fitted model might have unstable coefficients. For a given J(v) the estimated d v, using the first order approximation is: provided that the denominator does not vanish for all v 0 in set V. The Goal-seeker Module: The goal-seeking problem can be solved as a simulation problem. By this approach, we are able to apply variance reduction techniques (VRT) used in the simulation literature. Specifically, the solution to the goal-seeking problem is the unique solution of the stochastic equation J(v) - J 0 0. The problem is to solve this stochastic equation by a suitable experimental design, to ensure convergence. The following is a Robbins- Monro (R-M) type technique. where d j is any divergent sequence of positive numbers. Under this conditions, d v J 0 - J(v j ) converges to approach zero while dampening the effect of the simulation random errors. These conditions are satisfied, for example, by the harmonic sequence d j 1j. With this choice, the rate of reduction of di is very high initially but may reduce to very small steps as we approach the root. Therefore, a better choice is, for example d j 9 (9 j). This technique involves placing experiment i1 according to the outcome of experiment i immediately preceding it, as is depicted in the following Figure: Under these not unreasonable conditions, this algorithm will converge in mean square moreover, it is an almost sure convergence. Finally, as in Newtons root-finding method, it is impossible to assert that the method converges for just any initial v v 0 . even though J(v) may satisfy the Lipschits condition over set V. Indeed, if the initial value v 0 is sufficiently close to the solution, which is usually the case, then this algorithm requires only a few iterations to obtain a solution with very high accuracy. An application of the goal-seeker module arises when we want to adapt a model to satisfy a new equality constraint (condition) for some stochastic function. The proposed technique can also be used to solve integral equations by embedding the Importance Sampling techniques within a Monte Carlo sampling. One may extend the proposed methodology to the inverse problems with two or more unknown parameters design by considering two or more relevant outputs to ensure uniqueness. By this generalization we could construct a linear (or even nonlinear) system of stochastic equations to be solved simultaneously by a multidimensional version of the proposed algorithm. The simulation design is more involved for problems with more than a few parameters. References and Further Readings: Arsham H. The Use of Simulation in Discrete Event Dynamic Systems Design, Journal of Systems Science . 31(5), 563-573, 2000. Arsham H. Input Parameters to Achieve Target Performance in Stochastic Systems: A Simulation-based Approach, Inverse Problems in Engineering . 7(4), 363-384, 1999. Arsham H. Goal Seeking Problem in Discrete Event Systems Simulation, Microelectronics and Reliability . 37(3), 391-395, 1997. Batmaz I. and S. Tunali, Small response surface designs for metamodel estimation, European Journal of Operational Research . 145(3), 455-470, 2003. Ibidapo-Obe O. O. Asaolu, and A. Badiru, A New Method for the Numerical Solution of Simultaneous Nonlinear Equations, Applied Mathematics and Computation . 125(1), 133-140, 2002. Lamb J. and R. Cheng, Optimal allocation of runs in a simulation metamodel with several independent variables, Operations Research Letters . 30(3), 189-194, 2002. Simpson T. J. Poplinski, P. Koch, and J. Allen, Metamodels for Computer-based Engineering Design: Survey and Recommendations, Engineering with Computers . 17(2), 129-150, 2001. Tsai C-Sh. Evaluation and optimisation of integrated manufacturing system operations using Taguchs experiment design in computer simulation, Computers And Industrial Engineering . 43(3), 591-604, 2002. What-if Analysis Techniques Introduction The simulation models are often subject to errors caused by the estimated parameter(s) of underlying input distribution function. What-if analysis is needed to establish confidence with respect to small changes in the parameters of the input distributions. However the direct approach to what-if analysis requires a separate simulation run for each input value. Since this is often inhibited by cost, as an alternative, what people are basically doing in practice is to plot results and use a simple linear interpolationextrapolation. This section presents some simulation-based techniques that utilize the current information for estimating performance function for several scenarios without any additional simulation runs. Simulation continues to be the primary method by which system analysts obtain information about analysis of complex stochastic systems. In almost all simulation models, an expectedvalue can express the systems performance. Consider a system with continuous parameter v 206 V, where V is the feasible region. Let be the steady state expected performance measure, where Y is a random vector with known probability density function (pdf), f(y v) depends on v, and Z is the performance measure. In discrete event systems, Monte Carlo simulation is usually needed to estimate J(v) for a given value v. By the law of large numbers where y i . i 1, 2. n are independent, identically distributed, random vector realizations of Y from f (y v ), and n is the number of independent replications. This is an unbiased estimator for J(v) and converges to J(v) by law of large numbers. There are strong motivations for estimating the expected performance measure J(v) for a small change in v to v d v, that is to solve the so-called what if problem. The simulationist must meet managerial demands to consider model validation and cope with uncertainty in the estimation of v. Adaptation of a model to new environments also requires an adjustment in v. An obvious solution to the what if problem is the Crude Monte Carlo (CMC) method, which estimates J(v d v) for each v separately by rerunning the system for each v d v. Therefore costs in CPU time can be prohibitive The use of simulation as a tool to design complex computer stochastic systems is often inhibited by cost. Extensive simulation is needed to estimate performance measures for changes in the input parameters. As as an alternative, what people are basically doing in practice is to plot results of a few simulation runs and use a simple linear interpolationextrapolation. In this section we consider the What-if analysis problem by extending the information obtained from a single run at the nominal value of parameter v to the closed neighborhood. We also present the use of results from runs at two or more points over the intervening interval. We refer to the former as extrapolation and the latter as interpolation by simulation. The results are obtained by some computational cost as opposed to simulation cost . Therefore, the proposed techniques are for estimating a performance measure at multiple settings from a simulation at a nominal value. Likelihood Ratio (LR) Method A model based on Radon-Nikodym theorem to estimate J(v d v) for stochastic systems in a single run is as follows: where the likelihood ratio W is: W f(y v d v) f(y v) adjusts the sample path, provided f(y v) does not vanish. Notice that by this change of probability space, we are using the common realization as J(v). The generated random vector y is roughly representative of Y, with f(v). Each of these random observations, could also hypothetically came from f(v d v). W weights the observations according to this phenomenon. Therefore, the What-if estimate is: which is based on only one sample path of the system with parameter v and the simulation for the system with v d v is not required. Unfortunately LR produces a larger variance compared with CMC. However, since E(W)1, the following variance reduction techniques (VRT) may improve the estimate. Exponential Tangential in Expectation Method In the statistical literature the efficient score function is defined to be the gradient S(y) d Ln f(y v) dv We consider the exponential (approximation) model for J(v d v) in a first derivative neighborhood of v by: J(v d v) E Z(y). exp d vS(y) Eexp( d S(y)) Now we are able to estimate J(v d v) based on n independent replications as follows: Taylor Expansion of Response Function The following linear Taylor model can be used as an auxiliary model. J(v d v) J(v) d v.J (v) . where the prime denotes derivative. This metamodel approximates J(v d v)) for small d v. For this estimate, we need to estimate the nominal J(v) and its first derivative. Traditionally, this derivative is estimated by crude Monte Carlo i.e. finite difference, which requires rerunning the simulation model. Methods which yield enhanced efficiency and accuracy in estimating, at little additional cost, are of great value. There are few ways to obtain efficiently the derivatives of the output with respect to an input parameter as presented earlier on this site. The most straightforward method is the Score Function (SF). The SF approach is the major method for estimating the performance measure and its derivative, while observing only a single sample path from the underlying system. The basic idea of SF is that the derivative of the performance function, J(v), is expressed as expectation with respect to the same distribution as the performance measure itself. Therefore, for example, using the estimated values of J(v) and its derivative J(v), the estimated J(v d v) is: VarJ(v d v) VarJ(v) ( d v) 2 VarJ(v) 2 d v CovJ(v), J(v). This variation is needed for constructing a confidence interval for the perturbed estimate. Interpolation Techniques Given two points, v1 and v2 (scalars only) sufficiently close, one may simulate at these two points then interpolates for any desired points in between. Assuming the given v1 and v2 are sufficiently close and looks for the best linear interpolation in the sense of minimum error on the interval. Clearly, Similar to the Likelihood Ratio approach, this can be written as: where the likelihood ratios W1 and W2 are W1 f(y v) f(y v1) and W2 f(y v) f(y v2), respectively. One obvious choice is f f(y v1) f(y v1)f(y v2). This method can easily extended to k-point interpolation. For 2-point interpolation, if we let f to be constant within the interval 0, 1, then the linear interpolated what-if estimated value is: where the two estimates on the RHS of are two independent Likelihood Ratio extrapolations using the two end-points. We define f as the f in this convex combination with the minimum error in the estimate. That is, it minimizes By the first order necessary and sufficient conditions, the optimal f is: Thus, the best linear interpolation for any point in interval v1, v2 is: which is the optimal interpolation in the sense of having minimum variance. Conclusions Further Readings Estimating system performance for several scenarios via simulation generally requires a separate simulation run for each scenario. In some very special cases, such as the exponential density f(y v)ve -vy. one could have obtained the perturbed estimate using Perturbation Analysis directly as follow. Clearly, one can generate random variate Y by using the following inverse transformation: where Ln is the natural logarithm and U i is a random number distributed Uniformly 0,1. In the case of perturbed v, the counterpart realization using the same U i is Clearly, this single run approach is limited, since the inverse transformation is not always available in closed form. The following Figure illustrates the Perturbation Analysis Method: Since the Perturbation Analysis Approach has this serious limitation, for this reason, we presented some techniques for estimating performance for several scenarios using a single-sample path, such as the Likelihood Ratio method, which is illustrated in the following Figure. Research Topics: Items for further research include: i) to introduce efficient variance reduction and bias reduction techniques with a view to improving the accuracy of the existing and the proposed methods ii) to incorporate the result of this study in a random search optimization technique. In this approach one can generate a number of points in the feasible region uniformly distributed on the surface of a hyper-sphere each stage the value of the performance measure is with a specified radius centered at a starting point. At estimated at the center (as a nominal value). Perturbation analysis is used to estimate the performance measure at the sequence of points on the hyper-sphere. The best point (depending whether the problem is max or min) is used as the center of a smaller hyper- sphere. Iterating in this fashion one can capture the optimal solution within a hyper-sphere with a specified small enough radius. Clearly, this approach could be considered as a sequential self-adaptive optimization technique. iii) to estimate the sensitivities i.e. the gradient, Hessian, etc. of J(v) can be approximated using finite difference. For example the first derivative can be obtained in a single run using the Likelihood Ratio method as follows: the sums are over all i, i 1, 2, 3. n, where The last two estimators may induce some variance reductions. iv) Other interpolation techniques are also possible. The most promising one is based on Kriging. This technique gives more weight to neighboring realizations, and is widely used in geo-statistics. Other items for further research include some experimentation on large and complex systems such as a large Jacksonian network with routing that includes feedback loops in order to study the efficiency of the presented technique. References Further Readings: Arsham H. Performance Extrapolation in Discrete-event Systems Simulation, Journal of Systems Science . 27(9), 863-869, 1996. Arsham H. A Simulation Technique for Estimation in Perturbed Stochastic Activity Networks, Simulation . 58(8), 258-267, 1992. Arsham H. Perturbation Analysis in Discrete-Event Simulation, Modelling and Simulation . 11(1), 21-28, 1991. Arsham H. What-if Analysis in Computer Simulation Models: A Comparative Survey with Some Extensions, Mathematical and Computer Modelling . 13(1), 101-106, 1990. Arsham H. Feuerverger, A. McLeish, D. Kreimer J. and Rubinstein R. Sensitivity analysis and the what-if problem in simulation analysis, Mathematical and Computer Modelling . 12(1), 193-219, 1989. PDF Version The Copyright Statement: The fair use, according to the 1996 Fair Use Guidelines for Educational Multimedia. of materials presented on this Web site is permitted for non-commercial and classroom purposes only. This site may be mirrored intact (including these notices), on any server with public access. All files are available at home.ubalt.eduntsbarshBusiness-stat for mirroring. Kindly e-mail me your comments, suggestions, and concerns. Terima kasih. This site was launched on 2111995, and its intellectual materials have been thoroughly revised on a yearly basis. The current version is the 9 th Edition. All external links are checked once a month. EOF: 211 1995-2015.
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