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【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 -
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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)"
]
}
],
"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": "3.6.3"
}
},
"nbformat": 4,
"nbformat_minor": 2
}
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20180225_multiple-linear-regression_2.ipynb
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20180301_multiple-linear-regression_3.ipynb
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@rakuishi

rakuishi commented Mar 3, 2018

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