{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "## Passeios aleatórios (Random Walks) em 1D \n", "### Teoria e aplicações" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Passeio aleatório em 1D\n", "\n", "Imagine um processo iterativo que segue a fórmula:\n", "\n", "$y_n = y_{n-1} \\pm 1, n>1$\n", "\n", "Assim, dado $y_1$, se somar, ou subtrair 1 fosse regido por um processo aleatório, teríamos a sequência:\n", "\n", "$y_1, y_2, y_3, \\ldots, y_N$ \n", "\n", "que é conhecido como passeio do bêbado, ou passeio aleatório.\n", "\n", "Simular um passeio aleatório é muito simples:" ] }, { "cell_type": "code", "execution_count": 19, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Text(0.5, 1.0, 'Passeio do bêbado')" ] }, "execution_count": 19, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "import random as rng\n", "import matplotlib.pyplot as plt\n", "\n", "def randowWalk(y=0,N=10000):\n", " Y = []\n", " Y.append(y)\n", " for i in range(1,N):\n", " Y.append(Y[i-1]+rng.choice([-1,1]))\n", " return Y\n", "\n", "plt.plot(randowWalk())\n", "plt.ylabel(\"Posição do bêbado\",fontsize=16)\n", "plt.xlabel('tempo',fontsize=16)\n", "plt.title('Passeio do bêbado',fontsize=16)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Modelagem do passeio aleatório em 1D\n", "\n", "Se chamarmos de $d$ a distância do ponto de início do passeio após $N$ passos, isto é, $d=y_1+y_2+\\ldots+y_N$, onde estará o bêbado?\n", "\n", "Para modelarmos esse processo, vamos ter que usar novamente a ideia de conjuntos estatísticos, ou \"ensembles\". Assim, ao invés de modelar $d$ vamos modelar $$. Para facilitar o entendimento e as contas, vamos assumir que $y_1=0$. \n", "\n", "$ = = \\sum\\limits_{i=1}^N $\n", "\n", "Onde $y_i = \\pm 1, i>1$, com a probabilidade de somar 1, ou subtrair 1 igual a $\\frac{1}{2}$. \n", "\n", "Quando $N$ tende a infinito, como as probabilidade de somar 1, ou -1 são as mesmas, $ = y_1 = 0$. Ou seja, no limite, o bêbado não sai do lugar. \n", "\n", "E se calcularmos a média do quadrado da distância, isto é, o quanto, na média o bêbado se distancia do ponto inicial em relação à $N$? \n", "\n", "$ = <(y_1+y_2+\\ldots+y_N)^2> = \\sum\\limits_{i=1}^N + 2\\sum\\limits_{i=1}^N\\sum\\limits_{j=2}^{N-1} $\n", "\n", "Vamos primeiro calcular $ = (-1)(-1)$ ou $ = 1.1$, portanto $ = 1$\n", "\n", "Depois, por outro lado, temos $y_iy_j$, que pode ser igual a 1, se $y_i=1, y_j=1$, ou $y_i=-1, y_j=-1$ e pode ser igual a -1 se $y_i=1, y_j=-1$, ou $y_i=-1, y_j=1$. Ou seja, $=0$.\n", "\n", "Portanto, $ = \\sum\\limits_{i=1}^N 1 = N$. \n", "\n", "Finalmente, respondendo a pergunta, na média, o bêbado se distancia de $\\sqrt{} = \\sqrt N$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Exercício: O que aconteceria ser as probabilidades de ir para frente, ou para trás, fossem diferentes? Modifique o código acima para testar isso." ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "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": "3.7.3" } }, "nbformat": 4, "nbformat_minor": 2 }