{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# TD2: Générateurs pseudo-aléatoires (RICM4, 2013)\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "L'objectif de cette fiche est de présenter les méthodes principales de génération de nombres pseudo-aléatoires, de comprendre leurs limitations, d'apprendre à s'en méfier et de voir comment éventuellement les corriger...\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Générateurs à base de congruence" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "congruent_random <- function(n = 100, a=4, b=2, m=9, x1=1) {\n", " res = c(x1)\n", " for(i in 2:n) {\n", " res[i] = (a * res[i-1] + b) %% m\n", " }\n", " res\n", "}" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Regardons les premières valeurs" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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H291HhJQnQspH291HhJQn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\nAR4QEuABIQEeEBLgwf8Hy5pzyk2C9jAAAAAASUVORK5CYII=", "text/plain": [ "Plot with title “Histogram of X”" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "graphical_check <- function(X) {\n", " par(mfrow=c(1,2))\n", " hist(X)\n", " plot(X[1:length(X)-1],X[2:length(X)])\n", "}\n", "graphical_check(congruent_random(100))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Évidemment, l'espace d'état est trop petit. Essayons avec d'autres paramètres." ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "image/png": 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oFl+qfd+fj/KpWFVX3PQKQCIJIMdZL/jKOKA8e3oQZnVQLLtEjSDc1z3MWp9Fu8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"text/plain": [ "Plot with title “Histogram of X”" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "graphical_check(congruent_random(n=100,a=11,b=1,m=71))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Cela parait mieux mais en s'intéressant à la corrélation et à la covariance entre $X_n$ et $X_{n+1}$, le fait que nos nombres ne sont pas indépendants les uns des autres est plus évident." ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "data": { "text/html": [ "0.0457750907128348" ], "text/latex": [ "0.0457750907128348" ], "text/markdown": [ "0.0457750907128348" ], "text/plain": [ "[1] 0.04577509" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "19.0913087436134" ], "text/latex": [ "19.0913087436134" ], "text/markdown": [ "19.0913087436134" ], "text/plain": [ "[1] 19.09131" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "X <- congruent_random(n=1000,a=11,b=1,m=71)\n", "cor(X[1:length(X)-1],X[2:length(X)])\n", "cov(X[1:length(X)-1],X[2:length(X)])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Il faut donc être suspicieux avec les générateurs de nombres alétaoires et leur faire subir plusieurs tests." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Générateurs à décalage de registres" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[1] 4\n" ] }, { "data": { "text/html": [ "
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