{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Estimador de máxima verossimilhança da distribuição exponencial\n", "Neste notebook do Jupyter vamos demonstrar a dedução matemática do estimador de máxima verossimilhança do parâmetro da distribuição exponencial. Para ver os resultados dos comandos de cada etapa coloque o mouse ao lado esquerdo da célula que tem os comandos e clique em 'Run this cell'\n", "## Preparação\n", "Antes de começar importamos a biblioteca de matemática simbólica do Python. Ela permite criar expressões, como funções e suas derivadas, e manipulá-las simbolicamente." ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "from sympy import *\n", "init_printing()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## A função de densidade probabilística\n", "Para uma dada observação $x_i$ a função de densidade probabilística da distribuição exponecial é" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle \\lambda e^{- \\lambda {x}_{i}}$" ], "text/plain": [ " -\\lambda⋅x[i]\n", "\\lambda⋅ℯ " ] }, "execution_count": 2, "metadata": {}, "output_type": "execute_result" } ], "source": [ "x, L, i, n = symbols('x \\lambda i n')\n", "f = L*exp(-L*Indexed(x,i))\n", "f" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Defina a função de log-verossimilhança negativa\n", "É a soma dos logarítimos dos valores da função acima sobre todos os valores dos dados ($x_i$). Primeiro calculamos o logarítmo da função de massa:" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle - \\lambda {x}_{i} + \\log{\\left(\\lambda \\right)}$" ], "text/plain": [ "-\\lambda⋅x[i] + log(\\lambda)" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "lf = ln(f)\n", "lf=expand_log(lf, force=true)\n", "lf" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "e em seguida definimos a sua soma sobre todos os valores de $x_i$" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle \\log{\\left(\\lambda \\right)} \\sum_{i=1}^{n} 1 + \\sum_{i=1}^{n} - \\lambda {x}_{i}$" ], "text/plain": [ " n n \n", " ___ ___ \n", " ╲ ╲ \n", " ╲ ╲ \n", "log(\\lambda)⋅ ╱ 1 + ╱ -\\lambda⋅x[i]\n", " ╱ ╱ \n", " ‾‾‾ ‾‾‾ \n", " i = 1 i = 1 " ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "LL = Sum(lf, (i,1,n))\n", "LL=simplify(LL)\n", "LL" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "O somatório $\\sum_{i=1}^n 1$ nada mais é que $n$. Substituímos isto na função obtida acima temos:" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle n \\log{\\left(\\lambda \\right)} + \\sum_{i=1}^{n} - \\lambda {x}_{i}$" ], "text/plain": [ " n \n", " ___ \n", " ╲ \n", " ╲ \n", "n⋅log(\\lambda) + ╱ -\\lambda⋅x[i]\n", " ╱ \n", " ‾‾‾ \n", " i = 1 " ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g=Sum(1, (i,1,n))\n", "LL=LL.subs(g,n) \n", "LL" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Estimativa de máxima verossimilhança (mle)\n", "É o valor do parâmetro $\\lambda$ que maximiza a função de log-verossimilhança (ou minimiza a de log-verossimilhança negativa). Sabemos que as funções têm derivadas zero em seus pontos máximos e mínimos.\n", "Então o truque para achar a expressão do mle é encontrar a expressão para o valor de $\\lambda$\n", "que faz a derivada da função de log-verossimilhança ser igual a zero. São duas etapas:\n", "\n", "### Calcule a derivada da função de log-verossimlhança em relação ao parâmetro\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Usamos o comando 'diff' para obter a derivada da log-verossimilhança em função do parâmetro $\\lambda$:" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle - \\sum_{i=1}^{n} {x}_{i} + \\frac{n}{\\lambda}$" ], "text/plain": [ " n \n", " ___ \n", " ╲ \n", " ╲ n \n", "- ╱ x[i] + ───────\n", " ╱ \\lambda\n", " ‾‾‾ \n", " i = 1 " ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "dLL = diff(LL,L)\n", "dLL=simplify(expand(dLL))\n", "dLL" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Resolvemos a derivada para o parâmetro\n", "Agora igualamos a derivada da função de log-verossimilhança a zero e resolvemos esta equação para o parãmetro $\\lambda$:" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle \\left[ \\frac{n}{\\sum_{i=1}^{n} {x}_{i}}\\right]$" ], "text/plain": [ "⎡ n ⎤\n", "⎢──────────⎥\n", "⎢ n ⎥\n", "⎢ ___ ⎥\n", "⎢ ╲ ⎥\n", "⎢ ╲ ⎥\n", "⎢ ╱ x[i]⎥\n", "⎢ ╱ ⎥\n", "⎢ ‾‾‾ ⎥\n", "⎣i = 1 ⎦" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "solve(dLL,L)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Ou seja, o mle do único parâmetro da distribuição exponencial é o inverso da média amostral." ] } ], "metadata": { "kernelspec": { "display_name": "Python 3 (ipykernel)", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.10.8" } }, "nbformat": 4, "nbformat_minor": 4 }