{
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"source": [
"# 回帰分析\n"
]
},
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"\n",
"## 単回帰分析\n",
"\n",
"ここではまず、説明変数が１つの単回帰（Simple Regression）を考えます。\n",
"\n",
"$y_i = \\beta_0+\\beta_1x_i+\\epsilon_i$\n",
"\n",
"\n",
"* $y_i$： 被説明変数（explained variable）, 従属変数（dependent variable）,アウトカム（outcome variable）など\n",
"\n",
"* $x_i$： 説明変数（explanatory variable）, 独立変数（independent variable）など\n",
"\n",
"* $\\epsilon_i$： 誤差項（erros, error terms) 平均はゼロと仮定\n",
"\n",
"* $i$： 観測値のインデックス\n",
"\n",
"* $\\beta_0$：定数項、切片パラメター\n",
"\n",
"* $\\beta_1$：係数パラメター、傾きパラメター"
]
},
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"最小二乗法推定法（Ordinary Least Squares; OLS）やモーメント法、最尤法などで推定できます。\n",
"\n",
"最小二乗法の場合は、\n",
"\n",
"$\\sum_{i=1}^{n}(y_i-\\beta_0-\\beta_1x_i)^2$ を最小化する$\\beta_0$, $\\beta_1$を求めます"
]
},
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"推定した係数$\\beta_1$は被説明変数$y_i$に対する説明変数$x_i$の影響度合いを表す限界効果、あるいは説明変数が１単位変化した場合の被説明変数の変化（$\\frac{\\delta y_i}{\\delta x_i}$）と解釈することができます。"
]
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{
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"\n",
"\n",
"ここでは`statsmodels` を用いて単回帰分析を行います。\n",
"\n",
"`statsmodels`は線形回帰、一般化線形モデル、制限付き従属変数モデル、ARIMA、VARモデルなどをカバーするパッケージです。\n"
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "75f0d4ed-6a37-42fc-92eb-431249ad332d",
"metadata": {},
"outputs": [],
"source": [
"import numpy as np\n",
"import pandas as pd\n",
"import statsmodels.api as sm\n",
"import matplotlib.pyplot as plt"
]
},
{
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"id": "7775cbd7-ac46-4c43-a91c-5145573d2a9f",
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"source": [
"回帰分析で用いるデータを読み込みます。\n",
"\n",
"ここでは、`statsmodels`に付属する`statecrime`というデータセットの1つで、1996年`American National Elections Study`のデータを用いて回帰分析を行います。"
]
},
{
"cell_type": "code",
"execution_count": 18,
"id": "6eda48b5-9d42-4f0f-9273-719ff44d925c",
"metadata": {},
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"source": [
"data = sm.datasets.statecrime.load_pandas().data"
]
},
{
"cell_type": "code",
"execution_count": 19,
"id": "73638e8b-8435-4103-b1b7-107530831332",
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"outputs": [
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"\n",
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"
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" \n",
" \n",
" | \n",
" violent | \n",
" murder | \n",
" hs_grad | \n",
" poverty | \n",
" single | \n",
" white | \n",
" urban | \n",
"
\n",
" \n",
" | state | \n",
" | \n",
" | \n",
" | \n",
" | \n",
" | \n",
" | \n",
" | \n",
"
\n",
" \n",
" \n",
" \n",
" | Alabama | \n",
" 459.9 | \n",
" 7.1 | \n",
" 82.1 | \n",
" 17.5 | \n",
" 29.0 | \n",
" 70.0 | \n",
" 48.65 | \n",
"
\n",
" \n",
" | Alaska | \n",
" 632.6 | \n",
" 3.2 | \n",
" 91.4 | \n",
" 9.0 | \n",
" 25.5 | \n",
" 68.3 | \n",
" 44.46 | \n",
"
\n",
" \n",
"
\n",
"