{
"cells": [
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"import numpy as np\n",
"import scipy.stats\n",
"import matplotlib.pyplot as plt\n",
"%matplotlib inline"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Least-Squares Linear Regression"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"- fit slope and intercept so that the linear regression fit (here: \"line\") minimizes the sum of the residuals (vertical offsets or distances)\n",
"\n",
"![](\"images/linear_regression_scheme.png\")
"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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h92tm51L7N/8HhVQqBQoeJebud7j7Gnf/deA14CfU8qT3B5fcR4XSGa2E3au7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\n",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"rng = np.random.RandomState(123)\n",
"mean = [100, 1000]\n",
"cov = [[1, 0.9], [0.9, 1]]\n",
"sample = rng.multivariate_normal(mean, cov, size=100)\n",
"x, y = sample[:, 0], sample[:, 1]\n",
"\n",
"plt.scatter(x, y)\n",
"plt.xlabel('x')\n",
"plt.ylabel('y')\n",
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"- Pearson correlation coefficient (see the [Covariance and Correlation](cov-corr.ipynb) notebook for details)"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([[ 1. , 0.87552229],\n",
" [ 0.87552229, 1. ]])"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"np.corrcoef(np.vstack([x, y]))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### 1. Least-squares linear regression via \"classic statistic\""
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"- using \"classic statistics\":\n",
"\n",
"$w_1 = \\frac{\\sigma_{x,y}}{\\sigma_{x}^{2}}$\n",
"\n",
"$b = \\bar{y} - w_1\\bar{x}$\n",
"\n",
"where \n",
"\n",
"\n",
"$\\text{covariance: } \\sigma_{xy} = \\frac{1}{n} \\sum_{i=1}^{n} (x_i - \\bar{x})(y_i - \\bar{y})$\n",
"\n",
"$\\text{variance: } \\sigma^{2}_{x} = \\frac{1}{n} \\sum_{i=1}^{n} (x_i - \\bar{x})^2$"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"slope: 0.84\n",
"y-intercept: 915.59\n"
]
}
],
"source": [
"cov_xy = np.cov(np.vstack((x, y)), ddof=0)[0, 1]\n",
"var_x = np.var(x, ddof=0)\n",
"w1 = cov_xy / var_x\n",
"b = np.mean(y) - w1*np.mean(x)\n",
"\n",
"print('slope: %.2f' % w1)\n",
"print('y-intercept: %.2f' % b)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### 2. Least-squares linear regression via linear algebra"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"- closed-form (analytical) solution:\n",
"\n",
"$$w = (X^T X)^{-1} X^T y$$"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"slope: 0.84\n",
"y-intercept: 915.59\n"
]
}
],
"source": [
"X = x[:, np.newaxis]\n",
"\n",
"# adding a column vector of \"ones\"\n",
"Xb = np.hstack((np.ones((X.shape[0], 1)), X))\n",
"w = np.zeros(X.shape[1])\n",
"\n",
"z = np.linalg.inv(np.dot(Xb.T, Xb))\n",
"w = np.dot(z, np.dot(Xb.T, y))\n",
"b, w1 = w[0], w[1]\n",
"print('slope: %.2f' % w1)\n",
"print('y-intercept: %.2f' % b)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### 3. Using a pre-implemented function"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"slope: 0.84\n",
"y-intercept: 915.59\n"
]
}
],
"source": [
"w = np.polyfit(x, y, deg=1)\n",
"b, w1 = w[1], w[0]\n",
"print('slope: %.2f' % w1)\n",
"print('y-intercept: %.2f' % b)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"- note that there are many alternative tools (scikit-learn, statsmodels, ...)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Show line fit"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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oZp4HG9dDv7/BgBHQYtuUfl+6+h40P0MaCwUPyR2l82DqFbByPuzUH46+Azr0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\nmfvNLJ/cMoEErxfYCFwPXJm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"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"extremes = np.array([np.min(x), np.max(x)])\n",
"predict = extremes*w1 + b\n",
"\n",
"plt.plot(x, y, marker='o', linestyle='')\n",
"plt.plot(extremes, predict)\n",
"plt.xlabel('x')\n",
"plt.ylabel('y')\n",
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Evaluate"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### Mean squared error (MSE)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$$MSE = \\frac{1}{n} \\sum_{i=1}^{n} \\big(y_i - \\hat{y_i}\\big)^2$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"- note that MSE is scale-dependent"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0.21920128791624113"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"y_predicted = x*w1 + b\n",
"mse = np.mean((y - y_predicted)**2)\n",
"mse"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0.46818937185314358"
]
},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"rmse = np.sqrt(mse)\n",
"rmse"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### Residuals"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
""
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
"image/png": 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]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"plt.scatter(np.arange(x.shape[0]), y - y_predicted)\n",
"plt.ylabel('vertical offset')\n",
"plt.xlabel('index')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### Coefficient of determination (R^2)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"- Total sum of squares (variability of the reponse or target variable, proportional to variance):\n",
"\n",
"$$SS_{total} = \\sum_{i=1}^{n} \\big( y_i - \\bar{y_i} \\big)^2$$\n",
"\n",
"- Residual sum of squares:\n",
"\n",
"$$SS_{residual} = \\sum_{i=1}^{n} \\big( \\hat{y_i} - \\bar{y_i} \\big)^2$$\n",
"\n",
"- Coefficient of determination\n",
"\n",
"$$R^2 = \\frac{SS_{residual}}{SS_{total}}$$"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0.76653928492766576"
]
},
"execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"mean_y = np.mean(y)\n",
"SS_total = np.sum((y - mean_y)**2)\n",
"SS_residual = np.sum((y_predicted - mean_y)**2)\n",
"r_squared = SS_residual / SS_total\n",
"r_squared"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"- usually, the coefficient of determination can be computed by squaring the pearson correlation coefficient:"
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0.766539284927652"
]
},
"execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"slope, intercept, r_value, p_value, std_err = scipy.stats.linregress(x, y)\n",
"r_value**2"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"- note that if the model fits worse than a horizontal line, the coefficient of determination can be negative (so it is not necessarily always recommended to use a squared pearson coefficient to compute $R^2$)"
]
}
],
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