【キカガク流】人工知能・機械学習 脱ブラックボックス講座 - 中級編 -

20180225_multiple-linear-regression_1.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 行列演算の基礎"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\boldsymbol{w} = (\\boldsymbol{X}^{T}\\boldsymbol{X})^{-1}\\boldsymbol{X}^{T}\\boldsymbol{y}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- ベクトルの定義\n",
    "- 行列の定義\n",
    "- 転置\n",
    "- 逆行列\n",
    "- 行列積"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1]\n",
      " [2]\n",
      " [3]]\n"
     ]
    }
   ],
   "source": [
    "# ベクトルの定義\n",
    "x = np.array([[1], [2], [3]])\n",
    "print(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1 2]\n",
      " [3 4]]\n"
     ]
    }
   ],
   "source": [
    "# 行列の定義\n",
    "X = np.array([[1, 2], [3, 4]])\n",
    "print(X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1 3]\n",
      " [2 4]]\n"
     ]
    }
   ],
   "source": [
    "# 転置\n",
    "Xt = X.T\n",
    "print(Xt)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[-2.   1. ]\n",
      " [ 1.5 -0.5]]\n"
     ]
    }
   ],
   "source": [
    "# 逆行列\n",
    "# linear algebra: 線形代数\n",
    "X_inv = np.linalg.inv(X)\n",
    "print(X_inv)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[  1.00000000e+00   1.11022302e-16]\n",
      " [  0.00000000e+00   1.00000000e+00]]\n"
     ]
    }
   ],
   "source": [
    "# 行列積\n",
    "XX_inv = np.dot(X, X_inv)\n",
    "print(XX_inv)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1],\n",
       "       [2],\n",
       "       [3]])"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1, 2, 3]])"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x.T"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[2 3 4]\n",
      " [1 2 3]]\n"
     ]
    }
   ],
   "source": [
    "X = np.array([\n",
    "    [2, 3, 4],\n",
    "    [1, 2, 3],\n",
    "])\n",
    "print(X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(2, 3)"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "row, col = X.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2 3\n"
     ]
    }
   ],
   "source": [
    "print(row, col)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[2 3 4]\n",
      "[1 2 3]\n"
     ]
    }
   ],
   "source": [
    "for x in X:\n",
    "    print(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 演習問題"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\boldsymbol{w} = (\\boldsymbol{X}^{T}\\boldsymbol{X})^{-1}\\boldsymbol{X}^{T}\\boldsymbol{y}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1 2 3]\n",
      " [1 2 5]\n",
      " [1 3 4]\n",
      " [1 5 9]]\n",
      "[[1]\n",
      " [5]\n",
      " [6]\n",
      " [8]]\n"
     ]
    }
   ],
   "source": [
    "X = np.array([\n",
    "    [1, 2, 3],\n",
    "    [1, 2, 5],\n",
    "    [1, 3, 4],\n",
    "    [1, 5, 9],\n",
    "])\n",
    "print(X)\n",
    "\n",
    "y = np.array([\n",
    "    [1],\n",
    "    [5],\n",
    "    [6],\n",
    "    [8],\n",
    "])\n",
    "print(y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[  4  12  21]\n",
      " [ 12  42  73]\n",
      " [ 21  73 131]]\n"
     ]
    }
   ],
   "source": [
    "Xt = X.T\n",
    "XtX = np.dot(Xt, X)\n",
    "print(XtX)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 1.76530612 -0.39795918 -0.06122449]\n",
      " [-0.39795918  0.84693878 -0.40816327]\n",
      " [-0.06122449 -0.40816327  0.24489796]]\n"
     ]
    }
   ],
   "source": [
    "XtX_inv = np.linalg.inv(XtX)\n",
    "print(XtX_inv)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 20]\n",
      " [ 70]\n",
      " [124]]\n"
     ]
    }
   ],
   "source": [
    "Xty = np.dot(Xt, y)\n",
    "print(Xty)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[-0.14285714]\n",
      " [ 0.71428571]\n",
      " [ 0.57142857]]\n"
     ]
    }
   ],
   "source": [
    "w = np.dot(XtX_inv, Xty)\n",
    "print(w)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Scikit-learn"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from sklearn.linear_model import LinearRegression"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "LinearRegression(copy_X=True, fit_intercept=True, n_jobs=1, normalize=False)"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# モデルの宣言\n",
    "model = LinearRegression()\n",
    "\n",
    "# モデルの学習 ← パラメータの調整\n",
    "model.fit(X, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.        ,  0.71428571,  0.57142857]])"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# 調整後のパラメータ\n",
    "model.coef_"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-0.14285714])"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "model.intercept_"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.69230769230769229"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# 予測精度 ← 決定係数\n",
    "model.score(X, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 3.]]\n"
     ]
    }
   ],
   "source": [
    "# 予測値の計算\n",
    "x = np.array([[1, 2, 3]])\n",
    "y_pred = model.predict(x)\n",
    "print(y_pred)"
   ]
  }
 ],
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20180225_multiple-linear-regression_2.ipynb

20180301_multiple-linear-regression_3.ipynb