1253 lines
32 KiB
Plaintext
1253 lines
32 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# 线性回归的概念"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"1、线性回归的原理 \n",
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" \n",
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"2、线性回归损失函数、代价函数、目标函数 \n",
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" \n",
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"3、优化方法(梯度下降法、牛顿法、拟牛顿法等) \n",
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" \n",
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"4、线性回归的评估指标 \n",
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" \n",
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"5、sklearn参数详解"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## 1、线性回归的原理\n",
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"\n",
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"\n"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"进入一家房产网，可以看到房价、面积、厅室呈现以下数据："
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"\n",
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"<table>\n",
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" <tr>\n",
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" <th>面积($x_1$)</th>\n",
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" <th>厅室数量($x_2)$</th>\n",
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" <th>价格(万元)(y)</th>\n",
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" </tr>\n",
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" <tr>\n",
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" <th>64</th>\n",
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" <th>3</th>\n",
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" <th>225</th>\n",
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" </tr>\n",
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" <tr>\n",
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" <th>59</th>\n",
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" <th>3</th>\n",
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" <th>185</th>\n",
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" </tr>\n",
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" <tr>\n",
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" <th>65</th>\n",
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" <th>3</th>\n",
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" <th>208</th>\n",
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" </tr>\n",
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" <tr>\n",
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" <th>116</th>\n",
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" <th>4</th>\n",
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" <th>508</th>\n",
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" </tr>\n",
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" <tr>\n",
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" <th>……</th>\n",
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" <th>……</th>\n",
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" <th>……</th>\n",
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" </tr>"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"我们可以将价格和面积、厅室数量的关系习得为$f(x)=\\theta_0+\\theta_1x_1+\\theta_2x_2$，使得$f(x)\\approx y$，这就是一个直观的线性回归的样式。"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"<td bgcolor=#87CEEB>\n",
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" <font size =2>\n",
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"小练习：这是国内一个房产网站上任意搜的数据，有兴趣可以找个网站观察一下，还可以获得哪些可能影响到房价的因素？可能会如何影响到实际房价呢？</font>\n",
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"</td>"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### 线性回归的一般形式：\n",
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"有数据集$\\{(x_1,y_1),(x_2,y_2),...,(x_n,y_n)\\}$,其中,$x_i = (x_{i1};x_{i2};x_{i3};...;x_{id}),y_i\\in R$<br> \n",
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"其中n表示变量的数量，d表示每个变量的维度。 \n",
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"可以用以下函数来描述y和x之间的关系："
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"\\begin{align*}\n",
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"f(x) \n",
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"&= \\theta_0 + \\theta_1x_1 + \\theta_2x_2 + ... + \\theta_dx_d \\\\\n",
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"&= \\sum_{i=0}^{d}\\theta_ix_i \\\\\n",
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"\\end{align*}"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"如何来确定$\\theta$的值，使得$f(x)$尽可能接近y的值呢？均方误差是回归中常用的性能度量，即："
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"$$J(\\theta)=\\frac{1}{2}\\sum_{j=1}^{n}(h_{\\theta}(x^{(i)})-y^{(i)})^2$$<br> "
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"我们可以选择$\\theta$，试图让均方误差最小化："
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### 极大似然估计（概率角度的诠释）"
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]
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},
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{
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"cell_type": "markdown",
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||
"metadata": {},
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"source": [
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"下面我们用极大似然估计，来解释为什么要用均方误差作为性能度量"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"我们可以把目标值和变量写成如下等式："
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"$$\n",
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"y^{(i)} = \\theta^T x^{(i)}+\\epsilon^{(i)}\n",
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"$$"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"$\\epsilon$表示我们未观测到的变量的印象，即随机噪音。我们假定$\\epsilon$是独立同分布，服从高斯分布。（根据中心极限定理）"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"$$\n",
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"p(\\epsilon^{(i)}) = \\frac{1}{\\sqrt{2\\pi}\\sigma}exp\\left(-\\frac{(\\epsilon^{(i)})^2}{2\\sigma^2}\\right)$$\n"
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]
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},
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||
{
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||
"cell_type": "markdown",
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||
"metadata": {},
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||
"source": [
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||
"因此，"
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]
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},
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{
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"cell_type": "markdown",
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||
"metadata": {},
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||
"source": [
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||
"$$\n",
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||
"p(y^{(i)}|x^{(i)};\\theta) = \\frac{1}{\\sqrt{2\\pi}\\sigma}exp\\left(-\\frac{(y^{(i)}-\\theta^T x^{(i)})^2}{2\\sigma^2}\\right)\n",
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||
"$$"
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||
]
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},
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||
{
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||
"cell_type": "markdown",
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||
"metadata": {},
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||
"source": [
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||
"我们建立极大似然函数，即描述数据遵从当前样本分布的概率分布函数。由于样本的数据集独立同分布，因此可以写成"
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||
]
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||
},
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||
{
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||
"cell_type": "markdown",
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||
"metadata": {},
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||
"source": [
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||
"$$\n",
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"L(\\theta) = p(\\vec y | X;\\theta) = \\prod^n_{i=1}\\frac{1}{\\sqrt{2\\pi}\\sigma}exp\\left(-\\frac{(y^{(i)}-\\theta^T x^{(i)})^2}{2\\sigma^2}\\right)\n",
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||
"$$"
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||
]
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},
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||
{
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||
"cell_type": "markdown",
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||
"metadata": {},
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||
"source": [
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||
"选择$\\theta$，使得似然函数最大化，这就是极大似然估计的思想。"
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||
]
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||
},
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||
{
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||
"cell_type": "markdown",
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||
"metadata": {},
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||
"source": [
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||
"为了方便计算，我们计算时通常对对数似然函数求最大值："
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||
]
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||
},
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||
{
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"cell_type": "markdown",
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||
"metadata": {},
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||
"source": [
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||
"\\begin{align*}\n",
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"l(\\theta) \n",
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"&= log L(\\theta) = log \\prod^n_{i=1}\\frac{1}{\\sqrt{2\\pi}\\sigma}exp\\left(-\\frac{(y^{(i)}-\\theta^T x^{(i)})^2} {2\\sigma^2}\\right) \\\\\n",
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"& = \\sum^n_{i=1}log\\frac{1}{\\sqrt{2\\pi}\\sigma}exp\\left(-\\frac{(y^{(i)}-\\theta^T x^{(i)})^2}{2\\sigma^2}\\right) \\\\\n",
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||
"& = nlog\\frac{1}{{\\sqrt{2\\pi}\\sigma}} - \\frac{1}{\\sigma^2} \\cdot \\frac{1}{2}\\sum^n_{i=1}((y^{(i)}-\\theta^T x^{(i)})^2\n",
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||
"\\end{align*}"
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||
]
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||
},
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||
{
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||
"cell_type": "markdown",
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||
"metadata": {},
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||
"source": [
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||
"显然，最大化$l(\\theta)$即最小化 $\\frac{1}{2}\\sum^n_{i=1}((y^{(i)}-\\theta^T x^{(i)})^2$。"
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||
]
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||
},
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||
{
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||
"cell_type": "markdown",
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||
"metadata": {},
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||
"source": [
|
||
"这一结果即均方误差，因此用这个值作为代价函数来优化模型在统计学的角度是合理的。"
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||
]
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},
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||
{
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||
"cell_type": "markdown",
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||
"metadata": {},
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||
"source": [
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||
"## 2、线性回归损失函数、代价函数、目标函数\n",
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||
"* 损失函数(Loss Function)：度量单样本预测的错误程度，损失函数值越小，模型就越好。\n",
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||
"* 代价函数(Cost Function)：度量全部样本集的平均误差。\n",
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||
"* 目标函数(Object Function)：代价函数和正则化函数，最终要优化的函数。"
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||
]
|
||
},
|
||
{
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||
"cell_type": "markdown",
|
||
"metadata": {},
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||
"source": [
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||
"常用的损失函数包括：0-1损失函数、平方损失函数、绝对损失函数、对数损失函数等；常用的代价函数包括均方误差、均方根误差、平均绝对误差等。"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"<td bgcolor=#87CEEB>思考题：既然代价函数已经可以度量样本集的平均误差，为什么还要设定目标函数？</td>"
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||
]
|
||
},
|
||
{
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||
"cell_type": "markdown",
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||
"metadata": {},
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||
"source": [
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||
"#### 回答：\n",
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||
"当模型复杂度增加时，有可能对训练集可以模拟的很好，但是预测测试集的效果不好，出现过拟合现象，这就出现了所谓的“结构化风险”。结构风险最小化即为了防止过拟合而提出来的策略，定义模型复杂度为$J(F)$，目标函数可表示为："
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||
]
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||
},
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||
{
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||
"cell_type": "markdown",
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||
"metadata": {},
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||
"source": [
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||
"$$\\underset{f\\in F}{min}\\, \\frac{1}{n}\\sum^{n}_{i=1}L(y_i,f(x_i))+\\lambda J(F)$$"
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||
]
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||
},
|
||
{
|
||
"attachments": {},
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"<img src=\"download.png\">"
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||
]
|
||
},
|
||
{
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||
"cell_type": "markdown",
|
||
"metadata": {},
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||
"source": [
|
||
"例如有以上6个房价和面积关系的数据点，可以看到，当设定$f(x)=\\sum_{j=0}^{5}\\theta_jx_j$时，可以完美拟合训练集数据，但是，真实情况下房价和面积不可能是这样的关系，出现了过拟合现象。当训练集本身存在噪声时，拟合曲线对未知影响因素的拟合往往不是最好的。\n",
|
||
"通常，随着模型复杂度的增加，训练误差会减少；但测试误差会先增加后减小。我们的最终目的时试测试误差达到最小，这就是我们为什么需要选取适合的目标函数的原因。"
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||
]
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||
},
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||
{
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||
"cell_type": "markdown",
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||
"metadata": {},
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||
"source": [
|
||
"## 3、线性回归的优化方法"
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||
]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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||
"### 1、梯度下降法"
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||
]
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||
},
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||
{
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||
"cell_type": "markdown",
|
||
"metadata": {},
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||
"source": [
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||
"设定初始参数$\\theta$,不断迭代，使得$J(\\theta)$最小化：\n",
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||
"$$\\theta_j:=\\theta_j-\\alpha\\frac{\\partial{J(\\theta)}}{\\partial\\theta}$$"
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||
]
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||
},
|
||
{
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||
"cell_type": "markdown",
|
||
"metadata": {},
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||
"source": [
|
||
"\\begin{align*}\n",
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||
"\\frac{\\partial{J(\\theta)}}{\\partial\\theta} \n",
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"&= \\frac{\\partial}{\\partial\\theta_j}\\frac{1}{2}\\sum_{i=1}^{n}(f_\\theta(x)^{(i)}-y^{(i)})^2 \\\\\n",
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||
"&= 2*\\frac{1}{2}\\sum_{i=1}^{n}(f_\\theta(x)^{(i)}-y^{(i)})*\\frac{\\partial}{\\partial\\theta_j}(f_\\theta(x)^{(i)}-y^{(i)}) \\\\\n",
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||
"&= \\sum_{i=1}^{n}(f_\\theta(x)^{(i)}-y^{(i)})*\\frac{\\partial}{\\partial\\theta_j}(\\sum_{j=0}^{d}\\theta_jx_j^{(i)}-y^{(i)}))\\\\\n",
|
||
"&= \\sum_{i=1}^{n}(f_\\theta(x)^{(i)}-y^{(i)})x_j^{(i)} \\\\\n",
|
||
"\\end{align*}"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"即："
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"$$\n",
|
||
"\\theta_j = \\theta_j + \\alpha\\sum_{i=1}^{n}(y^{(i)}-f_\\theta(x)^{(i)})x_j^{(i)}\n",
|
||
"$$"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"注：下标j表示第j个参数，上标i表示第i个数据点。"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"将所有的参数以向量形式表示，可得："
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"$$\n",
|
||
"\\theta = \\theta + \\alpha\\sum_{i=1}^{n}(y^{(i)}-f_\\theta(x)^{(i)})x^{(i)}\n",
|
||
"$$"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"由于这个方法中，参数在每一个数据点上同时进行了移动，因此称为批梯度下降法，对应的，我们可以每一次让参数只针对一个数据点进行移动，即："
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"$$\n",
|
||
"\\theta = \\theta + \\alpha(y^{(i)}-f_\\theta(x)^{(i)})x^{(i)}\n",
|
||
"$$"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"这个算法成为随机梯度下降法，随机梯度下降法的好处是，当数据点很多时，运行效率更高；缺点是，因为每次只针对一个样本更新参数，未必找到最快路径达到最优值，甚至有时候会出现参数在最小值附近徘徊而不是立即收敛。但当数据量很大的时候，随机梯度下降法经常优于批梯度下降法。"
|
||
]
|
||
},
|
||
{
|
||
"attachments": {},
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"<img src=\"download1.png\">"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"当J为凸函数时，梯度下降法相当于让参数$\\theta$不断向J的最小值位置移动"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"梯度下降法的缺陷：如果函数为非凸函数，有可能找到的并非全局最优值，而是局部最优值。"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"### 2、最小二乘法矩阵求解"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"令<br>\n",
|
||
"$$ X = \\left[ \\begin{array} {cccc}\n",
|
||
"(x^{(1)})^T\\\\\n",
|
||
"(x^{(2)})^T\\\\\n",
|
||
"\\ldots \\\\\n",
|
||
"(x^{(n)})^T\n",
|
||
"\\end{array} \\right] $$"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"其中，"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"$$x^{(i)} = \\left[ \\begin{array} {cccc}\n",
|
||
"x_1^{(i)}\\\\\n",
|
||
"x_2^{(i)}\\\\\n",
|
||
"\\ldots \\\\\n",
|
||
"x_d^{(i)}\n",
|
||
"\\end{array} \\right]$$"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"由于"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"$$Y = \\left[ \\begin{array} {cccc}\n",
|
||
"y^{(1)}\\\\\n",
|
||
"y^{(2)}\\\\\n",
|
||
"\\ldots \\\\\n",
|
||
"y^{(n)}\n",
|
||
"\\end{array} \\right]$$"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"$h_\\theta(x)$可以写作"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"$$h_\\theta(x)=X\\theta$$"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"对于向量来说，有"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"$$z^Tz = \\sum_i z_i^2$$"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"因此可以把损失函数写作"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"$$J(\\theta)=\\frac{1}{2}(X\\theta-Y)^T(X\\theta-Y)$$"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"为最小化$J(\\theta)$,对$\\theta$求导可得："
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"\\begin{align*}\n",
|
||
"\\frac{\\partial{J(\\theta)}}{\\partial\\theta} \n",
|
||
"&= \\frac{\\partial}{\\partial\\theta} \\frac{1}{2}(X\\theta-Y)^T(X\\theta-Y) \\\\\n",
|
||
"&= \\frac{1}{2}\\frac{\\partial}{\\partial\\theta} (\\theta^TX^TX\\theta - Y^TX\\theta-\\theta^T X^TY - Y^TY) \\\\\n",
|
||
"\\end{align*}"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"中间两项互为转置，由于求得的值是个标量，矩阵与转置相同，因此可以写成"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"\\begin{align*}\n",
|
||
"\\frac{\\partial{J(\\theta)}}{\\partial\\theta} \n",
|
||
"&= \\frac{1}{2}\\frac{\\partial}{\\partial\\theta} (\\theta^TX^TX\\theta - 2\\theta^T X^TY - Y^TY) \\\\\n",
|
||
"\\end{align*}"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"令偏导数等于零，由于最后一项和$\\theta$无关，偏导数为0。"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"因此，"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"$$\\frac{\\partial{J(\\theta)}}{\\partial\\theta} = \\frac{1}{2}\\frac{\\partial}{\\partial\\theta} \\theta^TX^TX\\theta - \\frac{\\partial}{\\partial\\theta} \\theta^T X^TY\n",
|
||
"$$"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"利用矩阵求导性质，<br> \n",
|
||
"\n",
|
||
"\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"$$\n",
|
||
"\\frac{\\partial \\vec x^T\\alpha}{\\partial \\vec x} =\\alpha \n",
|
||
"$$"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"$$\n",
|
||
"和\n",
|
||
"$$"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"$$\\frac{\\partial A^TB}{\\partial \\vec x} = \\frac{\\partial A^T}{\\partial \\vec x}B + \\frac{\\partial B^T}{\\partial \\vec x}A$$"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
" "
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"\n",
|
||
"\\begin{align*}\n",
|
||
"\\frac{\\partial}{\\partial\\theta} \\theta^TX^TX\\theta \n",
|
||
"&= \\frac{\\partial}{\\partial\\theta}{(X\\theta)^TX\\theta}\\\\\n",
|
||
"&= \\frac{\\partial (X\\theta)^T}{\\partial\\theta}X\\theta + \\frac{\\partial (X\\theta)^T}{\\partial\\theta}X\\theta \\\\\n",
|
||
"&= 2X^TX\\theta\n",
|
||
"\\end{align*}"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"$$\\frac{\\partial{J(\\theta)}}{\\partial\\theta} = X^TX\\theta - X^TY\n",
|
||
"$$"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"令导数等于零，"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"$$X^TX\\theta = X^TY$$"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"$$\\theta = (X^TX)^{(-1)}X^TY\n",
|
||
"$$"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"注：CS229视频中吴恩达的推导利用了矩阵迹的性质，可自行参考学习。"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"### 3、牛顿法"
|
||
]
|
||
},
|
||
{
|
||
"attachments": {},
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"<img src=\"download2.png\">"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"通过图例可知(参考吴恩达CS229),"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"$$f(\\theta)' = \\frac{f(\\theta)}{\\Delta},\\Delta = \\theta_0 - \\theta_1$$"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"$$可求得，\\theta_1 = \\theta_0 - \\frac {f(\\theta_0)}{f(\\theta_0)'}$$"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"重复迭代，可以让逼近取到$f(\\theta)$的最小值"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"当我们对损失函数$l(\\theta)$进行优化的时候，实际上是想要取到$l'(\\theta)$的最小值，因此迭代公式为："
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"$$\n",
|
||
"\\theta :=\\theta-\\frac{l'(\\theta)}{l''(\\theta)}\n",
|
||
"$$"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"$$\n",
|
||
"当\\theta是向量值的时候，\\theta :=\\theta - H^{-1}\\Delta_{\\theta}l(\\theta)\n",
|
||
"$$"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"其中，$\\Delta_{\\theta}l(\\theta)$是$l(\\theta)$对$\\theta_i$的偏导数，$H$是$J(\\theta)$的海森矩阵，<br>\n",
|
||
"$$H_{ij} = \\frac{\\partial ^2l(\\theta)}{\\partial\\theta_i\\partial\\theta_j}$$"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"问题：请用泰勒展开法推导牛顿法公式。"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"Answer：将$f(x)$用泰勒公式展开到第二阶，"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"$f(x) = f(x_0) + f'(x_0)(x - x_0)+\\frac{1}{2}f''(x_0)(x - x_0)^2$"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"对上式求导，并令导数等于0，求得x值"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"$$f'(x) = f'(x_0) + f''(x_0)x -f''(x_0)x_0 = 0$$"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"可以求得，"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"$$x = x_0 - \\frac{f'(x_0)}{f''(x_0)}$$"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"牛顿法的收敛速度非常快，但海森矩阵的计算较为复杂，尤其当参数的维度很多时，会耗费大量计算成本。我们可以用其他矩阵替代海森矩阵，用拟牛顿法进行估计，"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"### 4、拟牛顿法"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"拟牛顿法的思路是用一个矩阵替代计算复杂的海森矩阵H，因此要找到符合H性质的矩阵。"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"要求得海森矩阵符合的条件，同样对泰勒公式求导$f'(x) = f'(x_0) + f''(x_0)x -f''(x_0)x_0$"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"令$x = x_1$，即迭代后的值，代入可得："
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"$$f'(x_1) = f'(x_0) + f''(x_0)x_1 - f''(x_0)x_0$$"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"更一般的，"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"$$f'(x_{k+1}) = f'(x_k) + f''(x_k)x_{k+1} - f''(x_k)x_k$$"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"$$f'(x_{k+1}) - f'(x_k) = f''(x_k)(x_{k+1}- x_k)= H(x_{k+1}- x_k)$$"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"$x_k$为第k个迭代值"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"即找到矩阵G，使得它符合上式。\n",
|
||
"常用的拟牛顿法的算法包括DFP，BFGS等，作为选学内容，有兴趣者可自行查询材料学习。"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"## 4、线性回归的评价指标"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"均方误差(MSE):$\\frac{1}{m}\\sum^{m}_{i=1}(y^{(i)} - \\hat y^{(i)})^2$"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"均方根误差(RMSE)：$\\sqrt{MSE} = \\sqrt{\\frac{1}{m}\\sum^{m}_{i=1}(y^{(i)} - \\hat y^{(i)})^2}$"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"平均绝对误差(MAE)：$\\frac{1}{m}\\sum^{m}_{i=1} | (y^{(i)} - \\hat y^{(i)} | $"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"但以上评价指标都无法消除量纲不一致而导致的误差值差别大的问题，最常用的指标是$R^2$,可以避免量纲不一致问题"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"$$R^2: = 1-\\frac{\\sum^{m}_{i=1}(y^{(i)} - \\hat y^{(i)})^2}{\\sum^{m}_{i=1}(\\bar y - \\hat y^{(i)})^2} =1-\\frac{\\frac{1}{m}\\sum^{m}_{i=1}(y^{(i)} - \\hat y^{(i)})^2}{\\frac{1}{m}\\sum^{m}_{i=1}(\\bar y - \\hat y^{(i)})^2} = 1-\\frac{MSE}{VAR}$$"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"我们可以把$R^2$理解为，回归模型可以成功解释的数据方差部分在数据固有方差中所占的比例，$R^2$越接近1，表示可解释力度越大，模型拟合的效果越好。"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"## 5、sklearn.linear_model参数详解："
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"fit_intercept : 默认为True,是否计算该模型的截距。如果使用中心化的数据，可以考虑设置为False,不考虑截距。注意这里是考虑，一般还是要考虑截距\n",
|
||
"\n",
|
||
"normalize: 默认为false. 当fit_intercept设置为false的时候，这个参数会被自动忽略。如果为True,回归器会标准化输入参数：减去平均值，并且除以相应的二范数。当然啦，在这里还是建议将标准化的工作放在训练模型之前。通过设置sklearn.preprocessing.StandardScaler来实现，而在此处设置为false\n",
|
||
"\n",
|
||
"copy_X : 默认为True, 否则X会被改写\n",
|
||
"\n",
|
||
"n_jobs: int 默认为1. 当-1时默认使用全部CPUs ??(这个参数有待尝试)\n",
|
||
"\n",
|
||
"可用属性：\n",
|
||
"\n",
|
||
"coef_:训练后的输入端模型系数，如果label有两个，即y值有两列。那么是一个2D的array\n",
|
||
"\n",
|
||
"intercept_: 截距\n",
|
||
"\n",
|
||
"可用的methods:\n",
|
||
"\n",
|
||
"fit(X,y,sample_weight=None):\n",
|
||
"X: array, 稀疏矩阵 [n_samples,n_features]\n",
|
||
"y: array [n_samples, n_targets]\n",
|
||
"sample_weight: 权重 array [n_samples]\n",
|
||
"在版本0.17后添加了sample_weight\n",
|
||
"\n",
|
||
"get_params(deep=True)： 返回对regressor 的设置值\n",
|
||
"\n",
|
||
"predict(X): 预测 基于 R^2值\n",
|
||
"\n",
|
||
"score： 评估\n",
|
||
"\n",
|
||
"参考https://blog.csdn.net/weixin_39175124/article/details/79465558"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"<table align =\"left\";background-color=\"#87CEEB\">\n",
|
||
"<tr>\n",
|
||
" <td bgcolor=\"#87CEEB\"><font size=2>练习题：请用以下数据（可自行生成尝试，或用其他已有数据集）</font></td>\n",
|
||
"</tr>\n",
|
||
"<tr>\n",
|
||
"<td bgcolor=\"#87CEEB\"><font size=2>1、首先尝试调用sklearn的线性回归函数进行训练；</font></td>\n",
|
||
"</tr>\n",
|
||
"<tr>\n",
|
||
"<td bgcolor=\"#87CEEB\"><font size=2>2、用最小二乘法的矩阵求解法训练数据；</font></td>\n",
|
||
"</tr>\n",
|
||
"<tr> \n",
|
||
"<td bgcolor=\"#87CEEB\"><font size=2>3、用梯度下降法训练数据；</font></td>\n",
|
||
"</tr>\n",
|
||
"<tr>\n",
|
||
" <td bgcolor=\"#87CEEB\"><font size=2>4、比较各方法得出的结果是否一致。</font></td>\n",
|
||
"</tr>\n",
|
||
"</table>"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"生成数据"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 2,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"#生成数据\n",
|
||
"import numpy as np\n",
|
||
"#生成随机数\n",
|
||
"np.random.seed(1234)\n",
|
||
"x = np.random.rand(500,3)\n",
|
||
"#构建映射关系，模拟真实的数据待预测值,映射关系为y = 4.2 + 5.7*x1 + 10.8*x2，可自行设置值进行尝试\n",
|
||
"y = x.dot(np.array([4.2,5.7,10.8]))"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"#### 1、先尝试调用sklearn的线性回归模型训练数据"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 3,
|
||
"metadata": {},
|
||
"outputs": [
|
||
{
|
||
"name": "stdout",
|
||
"output_type": "stream",
|
||
"text": [
|
||
"估计的参数值为：[ 4.2 5.7 10.8]\n",
|
||
"R2:1.0\n",
|
||
"预测值为: [85.2]\n"
|
||
]
|
||
}
|
||
],
|
||
"source": [
|
||
"import numpy as np\n",
|
||
"from sklearn.linear_model import LinearRegression\n",
|
||
"import matplotlib.pyplot as plt\n",
|
||
"%matplotlib inline\n",
|
||
"\n",
|
||
"# 调用模型\n",
|
||
"lr = LinearRegression(fit_intercept=True)\n",
|
||
"# 训练模型\n",
|
||
"lr.fit(x,y)\n",
|
||
"print(\"估计的参数值为：%s\" %(lr.coef_))\n",
|
||
"# 计算R平方\n",
|
||
"print('R2:%s' %(lr.score(x,y)))\n",
|
||
"# 任意设定变量，预测目标值\n",
|
||
"x_test = np.array([2,4,5]).reshape(1,-1)\n",
|
||
"y_hat = lr.predict(x_test)\n",
|
||
"print(\"预测值为: %s\" %(y_hat))\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"#### 2、最小二乘法的矩阵求解"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 4,
|
||
"metadata": {},
|
||
"outputs": [
|
||
{
|
||
"name": "stdout",
|
||
"output_type": "stream",
|
||
"text": [
|
||
"估计的参数值：[ 4.2 5.7 10.8]\n",
|
||
"预测值为: [85.2]\n"
|
||
]
|
||
}
|
||
],
|
||
"source": [
|
||
"class LR_LS():\n",
|
||
" def __init__(self):\n",
|
||
" self.w = None \n",
|
||
" def fit(self, X, y):\n",
|
||
" # 最小二乘法矩阵求解\n",
|
||
" #============================= show me your code =======================\n",
|
||
" self.w = \n",
|
||
" #============================= show me your code =======================\n",
|
||
" def predict(self, X):\n",
|
||
" # 用已经拟合的参数值预测新自变量\n",
|
||
" #============================= show me your code =======================\n",
|
||
" y_pred = \n",
|
||
" #============================= show me your code =======================\n",
|
||
" return y_pred\n",
|
||
"\n",
|
||
"if __name__ == \"__main__\":\n",
|
||
" lr_ls = LR_LS()\n",
|
||
" lr_ls.fit(x,y)\n",
|
||
" print(\"估计的参数值：%s\" %(lr_ls.w))\n",
|
||
" x_test = np.array([2,4,5]).reshape(1,-1)\n",
|
||
" print(\"预测值为: %s\" %(lr_ls.predict(x_test)))\n",
|
||
"\n",
|
||
" "
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"3、梯度下降法"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 5,
|
||
"metadata": {
|
||
"scrolled": true
|
||
},
|
||
"outputs": [
|
||
{
|
||
"name": "stdout",
|
||
"output_type": "stream",
|
||
"text": [
|
||
"估计的参数值为：[ 4.20000001 5.70000003 10.79999997]\n",
|
||
"预测值为：[85.19999995]\n"
|
||
]
|
||
}
|
||
],
|
||
"source": [
|
||
"class LR_GD():\n",
|
||
" def __init__(self):\n",
|
||
" self.w = None \n",
|
||
" def fit(self,X,y,alpha=0.02,loss = 1e-10): # 设定步长为0.002,判断是否收敛的条件为1e-10\n",
|
||
" y = y.reshape(-1,1) #重塑y值的维度以便矩阵运算\n",
|
||
" [m,d] = np.shape(X) #自变量的维度\n",
|
||
" self.w = np.zeros((d)) #将参数的初始值定为0\n",
|
||
" tol = 1e5\n",
|
||
" #============================= show me your code =======================\n",
|
||
" while tol > loss:\n",
|
||
" # here\n",
|
||
" \n",
|
||
" #============================= show me your code =======================\n",
|
||
" def predict(self, X):\n",
|
||
" # 用已经拟合的参数值预测新自变量\n",
|
||
" y_pred = X.dot(self.w)\n",
|
||
" return y_pred \n",
|
||
"\n",
|
||
"if __name__ == \"__main__\":\n",
|
||
" lr_gd = LR_GD()\n",
|
||
" lr_gd.fit(x,y)\n",
|
||
" print(\"估计的参数值为：%s\" %(lr_gd.w))\n",
|
||
" x_test = np.array([2,4,5]).reshape(1,-1)\n",
|
||
" print(\"预测值为：%s\" %(lr_gd.predict(x_test)))"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"## 参考"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"吴恩达 CS229课程"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"周志华 《机器学习》"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"李航 《统计学习方法》"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"https://hangzhou.anjuke.com/"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"https://www.jianshu.com/p/e0eb4f4ccf3e"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"https://blog.csdn.net/qq_28448117/article/details/79199835"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"https://blog.csdn.net/weixin_39175124/article/details/79465558"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": null,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": []
|
||
}
|
||
],
|
||
"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": "3.6.8"
|
||
},
|
||
"toc": {
|
||
"base_numbering": 1,
|
||
"nav_menu": {},
|
||
"number_sections": true,
|
||
"sideBar": true,
|
||
"skip_h1_title": false,
|
||
"title_cell": "Table of Contents",
|
||
"title_sidebar": "Contents",
|
||
"toc_cell": false,
|
||
"toc_position": {},
|
||
"toc_section_display": true,
|
||
"toc_window_display": false
|
||
}
|
||
},
|
||
"nbformat": 4,
|
||
"nbformat_minor": 2
|
||
}
|