{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Introdução à classificação"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Aproximar ou estimar $P(y\\,|\\,\\mathbf{x})$\n",
    "\n",
    "Como visto na introdução, o erro de classificação pode ser minimizado escolhendo-se a classe $argmax_y \\{P(y\\,|\\,\\mathbf{x})\\}$.\n",
    "\n",
    "Uma primeira observação útil é o fato de que não precisamos, necessariamente, estimar o valor exato de $P(y\\,|\\,\\mathbf{x})$. Precisamos, de fato, apenas de <b>funções discriminantes</b> $g_y(\\mathbf{x})$ tal que, para todo $\\mathbf{x} \\in X$ e para todo $z \\in Y, z\\neq y$, $g_y(\\mathbf{x}) > g_z(\\mathbf{x}) \\Longleftrightarrow P(y\\,|\\,\\mathbf{x}) > P(z\\,|\\,\\mathbf{x})$. Ou seja, basta apenas que as relações de ordem entre os valores das funções discriminantes $g_y(\\mathbf{x})$, $y\\in Y$, sejam consistentes com as relações entre as posterioris $P(y\\,|\\,\\mathbf{x})$, $y\\in Y$, para cada um dos elementos $\\mathbf{x}\\in X$.\n",
    "\n",
    "<b>Como podemos escolher essas funções discriminantes ? Isso é possível ?</b>\n",
    "\n",
    "A seguir apresentamos a ideias relacionadas com um método conhecido na literatura como <b>regressão logística</b>, método este utilizado para classificação binária.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<b>(Parênteses)</b> Considere a função $\\displaystyle s(x) = \\frac{1}{1+e^{-x}}$, $x \\in \\mathbb{R}$.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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5iqwLBTO7zswqzSxhZuWdln3FzKrMbJOZLejm9VPMbGXQ7pfBbb97u8Zfmtna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\n6aBQEBGRDv8fpvxTI7yA/1UAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fbdd3c6eba8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt \n",
    "\n",
    "x = np.arange(-10.0, 10.0, 0.1)\n",
    "plt.plot(x, 1/(1+np.exp(-x)))\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('s ( x)')\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Se considerarmos $\\displaystyle s_a(x) = \\frac{1}{1+e^{-ax}}$, $a \\in \\mathbb{R}$, temos uma família de funções."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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LTAHn227LfT5Fx23biilTNKBcIFQYlAGN050UDBAXiL7/j+1FqWvIdinPARtwDoet3Nqq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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fbdd11a5828>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def s_a(a,x):\n",
    "    return 1 / (1 + np.exp(-a*x))\n",
    "\n",
    "x = np.arange(-10.0, 10.0, 0.1)\n",
    "# Plot da função s_a() para três valores distintos de a\n",
    "plt.plot(x, s_a(1,x), 'bo', x, s_a(0.3,x), 'r*', x, s_a(3,x), 'g+', x, s_a(-1,x), 'x')\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('s_a( x )')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note que $0\\leq s_a(x) \\leq 1$, não importa o valor de $a$ e de $x$. \n",
    "\n",
    "Vamos agora considerar $z = z(\\mathbf{x}) = h_{(w_0,w_1)}(\\mathbf{x}) = w_0 + w_1 \\,x$ ( $\\mathbf{x} = (x_1) = x \\in \\mathbb{R}$ ) e plotar $z(\\mathbf{x}) = s(h_{(w_0,w_1)}(\\mathbf{x}))$ para diferentes valores de $(w_0,w_1)$. Experimente alterar $w_0$ e $w_1$ no código abaixo:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fd8bb097a58>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "w_0 = 1\n",
    "w_1 = 0.5\n",
    "\n",
    "def h(w_0,w_1,x):\n",
    "    return w_0 + w_1*x\n",
    "\n",
    "x = np.arange(-10.0, 10.0, 0.1)\n",
    "plt.plot(x, h(w_0,w_1,x), 'r+', x, s(1,h(w_0,w_1,x)), 'bo')\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('s( h(x) )')\n",
    "plt.ylim([-1,2])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "O que $s(z)$ tem a ver com as probabilidades $P(y\\,|\\,\\mathbf{x})$ ?\n",
    "\n",
    "Se tomarmos\n",
    "$$\n",
    "g_0(z) = \\frac{1}{1+e^{z}}\n",
    "$$\n",
    "e\n",
    "$$\n",
    "g_1(z) = \\frac{e^{z}}{1+e^{z}}\n",
    "$$\n",
    "temos $g_0(z) + g_1(z) = 1$. Assim, para o caso de classificação binária (i.e.,$Y=\\{0,1\\}$), podemos considerar\n",
    "\n",
    "$$P(y=0\\,|\\,\\mathbf{x}) = g_0(z(\\mathbf{x}))$$\n",
    "\n",
    "$$P(y=1\\,|\\,\\mathbf{x}) = g_1(z(\\mathbf{x}))$$\n",
    "\n",
    "com \n",
    "$$z = z(\\mathbf{x}) = h_{(w_0,w_1)}(\\mathbf{x}) = w_0 + w_1x\\,.$$\n",
    "Isto é, fazemos $P(y\\,|\\,\\mathbf{x})$ depender de uma função linear $h_{(w_0,w_1)}(\\mathbf{x}) = w_0 + w_1x$. \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Agora, dado um conjunto de observações $\\{ (\\mathbf{x}_i,y_i): i = 1,\\ldots, N \\}$, gostaríamos de saber qual é a distribuição que mais se aproxima da distribuição que deu origem a essas observações. Como estamos supondo que as distribuções são da forma $P(y=0\\,|\\,\\mathbf{x}) = \\frac{1}{1+e^{w_0 + w_1x}}$ e $P(y=1\\,|\\,\\mathbf{x}) = 1 - P(y=0\\,|\\,\\mathbf{x})$, o que queremos de fato é determinar o valor dos parâmetros $(w_0,w_1)$ que definem a distribuição. Ora, levando-se em conta que as amostras $\\{ (\\mathbf{x}_i,y_i): i = 1,\\ldots, N \\}$ foram de fato observadas, uma boa escolha para os parâmetros $(w_0,w_1)$ seriam os valores que maximizam a probabilidade de $\\{ (\\mathbf{x}_i,y_i): i = 1,\\ldots, N \\}$ ser observado. \n",
    "\n",
    "Supondo que a observação de um dado par $(\\mathbf{x}_i,y_i)$ é independente da observação de qualquer outro par, podemos escrever a probabilidade de se observar essas amostras como:\n",
    "$$\n",
    "L(w_0,w_1) = \\prod_{i=1}^{N} P(y_i\\,|\\,\\mathbf{x}_i,w_0,w_1)\n",
    "$$\n",
    "Note que a probabilidade a posteriori é expressa também em função dos parâmetros $(w_0,w_1)$.\n",
    "Além disso, observe que \n",
    "$$\n",
    "P(y\\,|\\,\\mathbf{x},w_0,w_1) = [g_0(z(\\mathbf{x}))]^{(1-y)}[g_1(z(\\mathbf{x}))]^{y}\n",
    "$$\n",
    "\n",
    "Portanto, podemos escrever:\n",
    "$$\n",
    "L(w_0,w_1) = \\prod_{i=1}^{N} [g_0(z(\\mathbf{x}_i))]^{(1-y_i)}[g_1(z(\\mathbf{x}_i))]^{y_i}\n",
    "$$\n",
    "\n",
    "O que gostaríamos de calcular são os parâmetros $(w_0,w_1)$ que maximizam $L(w_0,w_1)$ (isto é, conferem maior verossimilhança ao fato das amostras $\\{ (\\mathbf{x}_i,y_i): i = 1,\\ldots, N \\}$ terem sido observadas).\n",
    "\n",
    "Em vez de maximizar $L$, podemos maximizar $\\ln\\,L$ já que a aplicação de uma função monótona (no caso a função $\\ln$) não altera a relação de ordem entre os valores em dois pontos da função. Isto é, podemos considerar o problema equivalente de encontrar o ponto de máximo da função \n",
    "$$\n",
    "l(w_0,w_1) = \\ln\\, L(w_0,w_1)\n",
    "$$\n",
    "\n",
    "Note que \n",
    "$$\n",
    "l(w_0,w_1) = \\sum_{i=1}^{N} [ ]\n",
    "$$\n",
    "\n",
    "<font color=\"red\"> Falta preencher um pedaço aqui ...</font>\n",
    "O ponto de máximo dessa função pode ser obtida calculando-se o ponto de mínimo da função negativa. Uma técnica que pode ser usada para tal cálculo é a técnica do gradiente descendente."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Fronteira de decisão"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Como estamos tratando um caso binário, e queremos comparar a relação de ordem entre $g_0(\\mathbf{x})$ e $g_1(\\mathbf{x})$, podemos definir\n",
    "$$\n",
    "g(\\mathbf{x}) = g_1(\\mathbf{x}) - g_0(\\mathbf{x})\n",
    "$$\n",
    "e considerar que $\\mathbf{x}$ é da classe $0$ se $g(\\mathbf{x}) < 0$ e que $\\mathbf{x}$ é da classe $1$ se $g(\\mathbf{x}) > 0$.\n",
    "\n",
    "Note, adicionalmente, que se considerarmos $\\tilde{g}_0(\\mathbf{x}) = \\ln g_0(\\mathbf{x})$ e $\\tilde{g}_1(\\mathbf{x}) = \\ln g_1(\\mathbf{x})$, como a função $\\ln$ é monótona, a ordem relativa não é alterada. Ou seja, temos que  $g_1(\\mathbf{x}) - g_0(\\mathbf{x}) > 0 \\Longleftrightarrow \\tilde{g}_1(\\mathbf{x}) - \\tilde{g}_0(\\mathbf{x}) > 0$. \n",
    "Mas,\n",
    "$$\n",
    "\\tilde{g}_1(\\mathbf{x}) - \\tilde{g}_0(\\mathbf{x}) = \\ln \\frac{g_1(\\mathbf{x})}{g_0(\\mathbf{x})} = \\frac{\\frac{z}{1+z}}{\\frac{1}{1+z}} = \\ln z = w_0 + w_1 x\n",
    "$$\n",
    "\n",
    "Logo, verificar se $g(\\mathbf{x}) > 0$ é equivalente a verificar se $w_0 + w_1 x > 0$. Ou seja, a fronteira que as amostras da classe $0$ das amostras da classe $1$ é uma reta (um função linear)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div style=\"float: left;\">Voltar para o <a href=\"../html/main.html\">menu principal</a></div>\n",
    "<div style=\"float: right;\">Ir para: <a href=\"../html/practice_classification.html\">Prática -- classificação</a></div>"
   ]
  }
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