15. Feature Scaling.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Feature scaling"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Normalization\n",
    "\n",
    "适用于有边界的数据\n",
    "\n",
    "$x_{scale} = \\frac{x-x_{min}}{x_{max}-x_{min}}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = np.random.randint(0, 100, size = 100)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([29, 51, 21, 82, 79, 40, 23, 54, 26, 49, 58, 14, 24, 40,  1, 75, 54,\n",
       "       60, 58, 89, 97, 57, 90, 88, 86, 42, 78, 11, 79, 73, 33,  3, 46, 65,\n",
       "       65, 78, 73, 63, 88, 21, 60, 80, 43, 13, 55, 34, 84, 66, 42, 80, 75,\n",
       "       99, 41, 93, 12, 78, 57, 36, 86, 28, 32, 71, 63, 29, 55, 76, 82, 34,\n",
       "       96, 21, 11, 19, 51, 18, 59, 87,  1, 92, 28, 92, 55, 85, 72, 10, 99,\n",
       "       89,  5, 50, 19, 87, 45, 80, 58, 10, 43, 60, 24, 12, 83,  5])"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = (x - np.min(x))/(np.max(x) - np.min(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.28571429, 0.51020408, 0.20408163, 0.82653061, 0.79591837,\n",
       "       0.39795918, 0.2244898 , 0.54081633, 0.25510204, 0.48979592,\n",
       "       0.58163265, 0.13265306, 0.23469388, 0.39795918, 0.        ,\n",
       "       0.75510204, 0.54081633, 0.60204082, 0.58163265, 0.89795918,\n",
       "       0.97959184, 0.57142857, 0.90816327, 0.8877551 , 0.86734694,\n",
       "       0.41836735, 0.78571429, 0.10204082, 0.79591837, 0.73469388,\n",
       "       0.32653061, 0.02040816, 0.45918367, 0.65306122, 0.65306122,\n",
       "       0.78571429, 0.73469388, 0.63265306, 0.8877551 , 0.20408163,\n",
       "       0.60204082, 0.80612245, 0.42857143, 0.12244898, 0.55102041,\n",
       "       0.33673469, 0.84693878, 0.66326531, 0.41836735, 0.80612245,\n",
       "       0.75510204, 1.        , 0.40816327, 0.93877551, 0.1122449 ,\n",
       "       0.78571429, 0.57142857, 0.35714286, 0.86734694, 0.2755102 ,\n",
       "       0.31632653, 0.71428571, 0.63265306, 0.28571429, 0.55102041,\n",
       "       0.76530612, 0.82653061, 0.33673469, 0.96938776, 0.20408163,\n",
       "       0.10204082, 0.18367347, 0.51020408, 0.17346939, 0.59183673,\n",
       "       0.87755102, 0.        , 0.92857143, 0.2755102 , 0.92857143,\n",
       "       0.55102041, 0.85714286, 0.7244898 , 0.09183673, 1.        ,\n",
       "       0.89795918, 0.04081633, 0.5       , 0.18367347, 0.87755102,\n",
       "       0.44897959, 0.80612245, 0.58163265, 0.09183673, 0.42857143,\n",
       "       0.60204082, 0.23469388, 0.1122449 , 0.83673469, 0.04081633])"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "X = np.random.randint(0, 100, (50, 2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[90, 14],\n",
       "       [41, 88],\n",
       "       [83, 20],\n",
       "       [41, 28],\n",
       "       [23, 46],\n",
       "       [27, 73],\n",
       "       [92, 70],\n",
       "       [27, 97],\n",
       "       [74, 40],\n",
       "       [41, 59]])"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X[:10, :]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "X = np.array(X, dtype = float)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[90., 14.],\n",
       "       [41., 88.],\n",
       "       [83., 20.],\n",
       "       [41., 28.],\n",
       "       [23., 46.],\n",
       "       [27., 73.],\n",
       "       [92., 70.],\n",
       "       [27., 97.],\n",
       "       [74., 40.],\n",
       "       [41., 59.]])"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X[:10, :]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "X[:, 0] = (X[:, 0] - np.min(X[:, 0]))/(np.max(X[:, 0])-np.min(X[:, 0]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "X[:, 1] = (X[:, 1] - np.min(X[:, 1]))/(np.max(X[:, 1])-np.min(X[:, 1]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0.97826087, 0.10638298],\n",
       "       [0.44565217, 0.89361702],\n",
       "       [0.90217391, 0.17021277],\n",
       "       [0.44565217, 0.25531915],\n",
       "       [0.25      , 0.44680851],\n",
       "       [0.29347826, 0.73404255],\n",
       "       [1.        , 0.70212766],\n",
       "       [0.29347826, 0.9893617 ],\n",
       "       [0.80434783, 0.38297872],\n",
       "       [0.44565217, 0.58510638]])"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X[:10, :]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(X[:, 0], X[:, 1])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.523695652173913"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.mean(X[:, 0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.2886534967912737"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.std(X[:, 0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.49851063829787234"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.mean(X[:, 1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.3069292559049654"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.std(X[:, 1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Standardization\n",
    "\n",
    "把所有数据归一到均值为0方差为1的分布中\n",
    "\n",
    "$x_{scale} = \\frac{x-x_{mean}}{s}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "x2 = np.random.randint(0, 100, (50, 2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "x2 = np.array(x2, dtype = float)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "x2[:, 0] = (x2[:, 0] - np.mean(x2[:, 0]))/np.std(x2[:, 0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "x2[:, 1] = (x2[:, 1] - np.mean(x2[:, 1]))/np.std(x2[:, 1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(x2[:, 0], x2[:, 1])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-2.6645352591003756e-17"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.mean(x2[:, 0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.std(x2[:, 0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-8.881784197001253e-18"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.mean(x2[:, 1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.std(x2[:, 1])"
   ]
  }
 ],
 "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": "3.7.2"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}