{ "cells": [ { "cell_type": "markdown", "id": "c57b2ce5", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "# Sampling From Unknown Distributions\n", "\n", "\n", "Feng Li\n", "\n", "School of Statistics and Mathematics\n", "\n", "Central University of Finance and Economics\n", "\n", "[feng.li@cufe.edu.cn](mailto:feng.li@cufe.edu.cn)\n", "\n", "[https://feng.li/statcomp](https://feng.li/statcomp)\n" ] }, { "cell_type": "code", "execution_count": 1, "id": "69326237", "metadata": { "slideshow": { "slide_type": "skip" } }, "outputs": [], "source": [ "options(jupyter.plot_scale=0.8)" ] }, { "cell_type": "markdown", "id": "47a7f101", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## Direct Methods\n", "\n", "- Only need random numbers from uniform\n", "\n", "- Assume that we have a way to simulate from a uniform distribution between $0$ and $1$, $u\\sim U(0,1)$\n", "\n", "- If this is available, it is possible to simulate many other probability distributions.\n", "\n", "- The most simple method is the **Direct Method**" ] }, { "cell_type": "markdown", "id": "b5ca565a", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### Discrete Case: Example 1\n", "\n", "- Assume that we want to simulate a binary variable $X$ with\n", " $\\mbox{Pr}(X=1)=0.3$ and $\\mbox{Pr}(X=0)=0.7$\n", "\n", "- Let $u\\sim U(0,1)$. Then the following rule can be used\n", " $$x=\\left\\{\\begin{array}{c}\n", " 1 \\quad \\mbox{if u<0.3}\\\\\n", " 0 \\quad \\mbox{if u>0.3}\\\\\n", " \\end{array}\n", " \\right.$$\n", "\n", "- It is expected that if this is repeated many times 30% $X=1$ and 70%\n", " $X=0$" ] }, { "cell_type": "code", "execution_count": 2, "id": "df9d4801", "metadata": { "slideshow": { "slide_type": "slide" } }, "outputs": [ { "data": { "text/html": [ "\n", "
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Then the following rule can be used\n", " $$x=\\left\\{\\begin{array}{l}\n", " 0 \\quad \\mbox{if u<0.3}\\\\\n", " 1 \\quad \\mbox{if 0.30.55}\n", " \\end{array}\n", " \\right.$$\n", "\n", "- It is expected that if this is repeated many times, roughly 30%\n", " $X=0$, 25% $X=1$ and 45% $X=2$" ] }, { "cell_type": "markdown", "id": "e6617d2b", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "![image](figures/d2p3.png) " ] }, { "cell_type": "markdown", "id": "88814415", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### Continuous Case\n", "\n", "- How do we extend this idea to the continuous case?\n", "\n", "- What was the step function in our discrete example?\n", "\n", "- It is the **cumulative distribution function (CDF)**\n", "\n", "- Can we replace the discrete cdf with a continuous cdf?\n", "\n", "- Yes!" ] }, { "cell_type": "markdown", "id": "27dfe1db", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "![image](figures/cp1.png) " ] }, { "cell_type": "markdown", "id": "f704e110", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "![image](figures/cp3.png) " ] }, { "cell_type": "markdown", "id": "11721518", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "\n", "### Continuous Case\n", "\n", "- The cdf, $F(X)$ takes values of $X$ and gives a value between $0$\n", " and $1$\n", "\n", "- Here we take values between $0$ and $1$ and get a value of $X$\n", "\n", "- What function do we use?\n", "\n", "- We use the **Inverse cdf**" ] }, { "cell_type": "markdown", "id": "c5b5b89f", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### Continuous Case: Direct Methods\n", "\n", "#### Probability Integral Transform\n", "\n", "- If $Y$ is a continuous random variable with cdf $F(y)$ , then the\n", " random variable $F_y^{-1}(U)$, where $U \\sim uniform(O, 1)$, has\n", " distribution $F(y)$.\n", "\n", "- **Example:** If $Y \\sim exponential(A)$, then the probability\n", " density function (PDF) is\n", " $$f(x;\\lambda) = \\begin{cases} \\lambda e^{-\\lambda x} & x \\ge 0, \\\\ 0 & x < 0. \\end{cases}$$\n", " and the cumulative distribution function (CDF) is given by\n", " $$F(x;\\lambda) = \\begin{cases} 1-e^{-\\lambda x} & x \\ge 0, \\\\ 0 & x < 0. \\end{cases}$$\n", " Then $$F_Y^{-1}(U) = -\\log(1-U)/\\lambda$$ is an exponential random\n", " variable.\n", "\n", "- Thus, if we generate $U_1,..., U_n$ as iid uniform random variables,\n", " $-\\lambda (1-U_i)$, are iid exponential random variables with\n", " parameter $\\lambda$." ] }, { "cell_type": "code", "execution_count": 3, "id": "cc4dc294", "metadata": { "slideshow": { "slide_type": "slide" } }, "outputs": [], "source": [ "u2x <- function(u, lambda)\n", " {\n", " x <- - log(1-u)/lambda\n", " return(x)\n", " }\n", "\n", "rmyexp <- function(n, lambda)\n", " {\n", " u <- runif(n, 0, 1)\n", " x <- u2x(u, lambda)\n", " return(x)\n", " }" ] }, { "cell_type": 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"text/plain": [ "Plot with title “rexp”" ] }, "metadata": { "image/png": { "height": 450, "width": 600 } }, "output_type": "display_data" } ], "source": [ "n = 1000\n", "lambda = 2\n", "\n", "par(mfrow = c(1, 2))\n", "ylim <- c(0, 2)\n", "xlim <- c(0, 4)\n", "hist(rmyexp(n, lambda), breaks = 20, freq = FALSE,\n", " xlim = xlim, ylim = ylim, xlab=\"\", ylab=\"\", main=\"rmyexp\")\n", "xax <- seq(0, max(x), 0.01)\n", "xDens <- lambda*exp(-lambda*xax)\n", "lines(xax, xDens, type = \"l\", col = \"blue\", lwd = 3)\n", "\n", "hist(rexp(n, rate = lambda), breaks = 20, freq = FALSE,\n", " xlim = xlim, ylim = ylim, xlab=\"\", ylab=\"\", main=\"rexp\")\n", "lines(xax, xDens, type = \"l\", col = \"blue\", lwd = 3)" ] }, { "cell_type": "markdown", "id": "e01ebe0f", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## Indirect Methods\n", "\n", "Indirect simulation\n", "\n", "- What if the cumulative distribution function is difficult to invert,\n", " or not even available?\n", "\n", "- How to invert the CDF of a standard normal distribution?\n", "\n", " $$F(x)=\\int_{-\\infty}^{x}(2\\pi)^{-1/2}e^{-x^2/2}dx$$\n", "\n", "\n", "- It is still possible to simulate from this type of distribution?\n", "\n", "- If the **density** is available, then the answer is YES!" ] }, { "cell_type": "markdown", "id": "5cac5f45", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "The mixture of normal distribution\n", "\n", "- The mixture of normal distribution has density function\n", "\n", " $$f(x, \\omega, \\mu_1, \\mu_2, \\sigma_1, \\sigma_2) = \\omega N(x, \\mu_1, \\sigma_1) + (1- \\omega) N(x, \\mu_2, \\sigma_2)$$\n", "\n", " where $0 < \\omega <1$ is the weight.\n", "\n", "- It is based on the combination of the normal but has many features.\n", "\n", "- It is not so easy to simulate from this distribution using the\n", " direct method." ] }, { "cell_type": "code", "execution_count": 6, "id": "87953753", "metadata": { "slideshow": { "slide_type": "slide" } }, "outputs": [ { "data": { "image/png": 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DayNR0FNBG2wkIsWwYPDxw+DIaBoSGmTUNiIo4cQe/eMDCoxKpEIvj6Yvt2PHuG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\np0MkwubNGD2a74JUtWMHkpIAYMEClD+cuBZ5e+PECdSqBakUQ4f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"text/plain": [ "Plot with title “Normal: Red, Mixture: Blue”" ] }, "metadata": { "image/png": { "height": 450, "width": 600 } }, "output_type": "display_data" } ], "source": [ "w = 0.2\n", "mu1 = -1\n", "mu2 = 1\n", "sigma1 = 0.8\n", "sigma2 = 0.5\n", "\n", "xgrid<-seq(-3, 4, 0.01)\n", "nm <- dnorm(xgrid)\n", "mix <-w*dnorm(xgrid, mu1, sigma1) + (1-w)*dnorm(xgrid, mu2, sigma2)\n", "plot(xgrid, nm, \"l\", lwd=2, col='red',main='Normal: Red, Mixture: Blue',\n", " xlab = 'x', ylab = 'y', ylim=c(0, max(nm,mix)))\n", "lines(xgrid, mix, lwd=2,col='blue')" ] }, { "cell_type": "markdown", "id": "6e581b11", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "The idea\n", "\n", "- Let $f_y(x)$ be the **target distribution** and $f_v(v)$ be the\n", " **proposal distribution**\n", "\n", "- Simulate an $x$-coordinate from the proposal $f_v(x)$\n", "\n", "- Simulate a y-coordinate from $U(0, M*f_v(x))$\n", "\n", "- Reject any points that are not ‘inside’ $f_y(x)$" ] }, { "cell_type": "code", "execution_count": 9, "id": "f2d44d6e", "metadata": { "slideshow": { "slide_type": "slide" } }, "outputs": [ { "data": { "image/png": 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"text/plain": [ "plot without title" ] }, "metadata": { "image/png": { "height": 450, "width": 600 } }, "output_type": "display_data" } ], "source": [ "M0 = (w*dnorm(xgrid, mu1, sigma1) + (1-w)*dnorm(xgrid, mu2, sigma2))/dnorm(xgrid) \n", "plot(xgrid, M0, type=\"l\", xlab=\"x\")" ] }, { "cell_type": "code", "execution_count": 10, "id": "4334f112", "metadata": { "slideshow": { "slide_type": "slide" } }, "outputs": [ { "data": { "image/png": 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FFAE3mKicGaNQDQqhUCAnguRh5atcKYMQBw8iQuXOC7GqJqKKCJPC1YgJyc90ea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\nunXw8uK7IN6cP48TJwBg5EgIoVOzKVPg7AwA3+K77FwNDB2KzEy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"text/plain": [ "Plot with title “Scaled normal: Red, Mixture: Blue”" ] }, "metadata": { "image/png": { "height": 450, "width": 600 } }, "output_type": "display_data" } ], "source": [ "M = max(M0)\n", "Mnm <-M*nm\n", "plot(xgrid, Mnm, lwd=2, type=\"l\",col='red',main='Scaled normal: Red, Mixture: Blue',xlab = 'x',ylab = 'y')\n", "lines(xgrid, mix, lwd=2, col='blue')" ] }, { "cell_type": "markdown", "id": "2d18d9ef", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "#### Reject and accept method\n", "\n", "- **Theorem:** Let $Y \\sim f_Y(y)$ and $V \\sim f_v(v)$, where $f_Y$\n", " and $f_V$ have common support with\n", " $$M = sup_y f_Y(y)/f_V(y) < \\infty.$$ To generate a random variable\n", " $Y\\sim f_Y$, we do the following steps\n", "\n", " 1. Generate $U\\sim uniform(0,1)$ and $V\\sim f_V$ independently.\n", "\n", " 2. If $$U < \\frac{1}{M} \\frac{f_Y(V)}{f_V(V)}$$ set $Y = V$;\n", " otherwise, return to step 1." ] }, { "cell_type": "code", "execution_count": 11, "id": "5e442acc", "metadata": { "slideshow": { "slide_type": "slide" } }, "outputs": [ { "data": { "image/png": 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5mOjvzOHtHRGDcuj61kMHJDYcf5FQFYHHQ+8vQpGRsTQK6uJBZTQgKlpFBCAtWr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\niEihGNBERArFgCYiUigGNBGRQv0/eeEBnT/A1z4AAAAASUVORK5CYII=", "text/plain": [ "Plot with title “Scaled Normal: Red, Mixture: Blue”" ] }, "metadata": { "image/png": { "height": 450, "width": 600 } }, "output_type": "display_data" } ], "source": [ "n<-500\n", "xp<-rnorm(n)\n", "yp<-M*dnorm(xp)*runif(n)\n", "\n", "plot(xgrid, Mnm, lwd=2, type=\"l\",col='red',main='Scaled Normal: Red, Mixture: Blue',xlab = 'x', ylab = 'y')\n", "lines(xgrid, mix, lwd=2, col='blue')\n", "\n", "dmix = w*dnorm(xp, mu1, sigma1) + (1-w)*dnorm(xp, mu2, sigma2)\n", "points(xp[yp>=dmix],yp[yp>=dmix], col='red',pch=20, cex=0.2)\n", "points(xp[yp