{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Statsmodels"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Statsmodels is a Python module that allows users to explore data, estimate statistical models, and perform statistical tests. An extensive list of descriptive statistics, statistical tests, plotting functions, and result statistics are available for different types of data and each estimator.\n",
"\n",
"Library documentation: http://statsmodels.sourceforge.net/"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Linear Regression Models"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"# needed to display the graphs\n",
"%matplotlib inline\n",
"from pylab import *"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"import numpy as np\n",
"import pandas as pd\n",
"import statsmodels.api as sm\n",
"from statsmodels.sandbox.regression.predstd import wls_prediction_std\n",
"np.random.seed(9876789)"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"# create some artificial data\n",
"nsample = 100\n",
"x = np.linspace(0, 10, 100)\n",
"X = np.column_stack((x, x**2))\n",
"beta = np.array([1, 0.1, 10])\n",
"e = np.random.normal(size=nsample)"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"# add column of 1s for intercept\n",
"X = sm.add_constant(X)\n",
"y = np.dot(X, beta) + e"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
" OLS Regression Results \n",
"==============================================================================\n",
"Dep. Variable: y R-squared: 1.000\n",
"Model: OLS Adj. R-squared: 1.000\n",
"Method: Least Squares F-statistic: 4.020e+06\n",
"Date: Sun, 16 Nov 2014 Prob (F-statistic): 2.83e-239\n",
"Time: 20:59:31 Log-Likelihood: -146.51\n",
"No. Observations: 100 AIC: 299.0\n",
"Df Residuals: 97 BIC: 306.8\n",
"Df Model: 2 \n",
"==============================================================================\n",
" coef std err t P>|t| [95.0% Conf. Int.]\n",
"------------------------------------------------------------------------------\n",
"const 1.3423 0.313 4.292 0.000 0.722 1.963\n",
"x1 -0.0402 0.145 -0.278 0.781 -0.327 0.247\n",
"x2 10.0103 0.014 715.745 0.000 9.982 10.038\n",
"==============================================================================\n",
"Omnibus: 2.042 Durbin-Watson: 2.274\n",
"Prob(Omnibus): 0.360 Jarque-Bera (JB): 1.875\n",
"Skew: 0.234 Prob(JB): 0.392\n",
"Kurtosis: 2.519 Cond. No. 144.\n",
"==============================================================================\n"
]
}
],
"source": [
"# fit model and print the summary\n",
"model = sm.OLS(y, X)\n",
"results = model.fit()\n",
"print(results.summary())"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"('Parameters: ', array([ 1.34233516, -0.04024948, 10.01025357]))\n",
"('R2: ', 0.9999879365025871)\n"
]
}
],
"source": [
"# individual results parameters can be accessed\n",
"print('Parameters: ', results.params)\n",
"print('R2: ', results.rsquared)"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
" OLS Regression Results \n",
"==============================================================================\n",
"Dep. Variable: y R-squared: 0.933\n",
"Model: OLS Adj. R-squared: 0.928\n",
"Method: Least Squares F-statistic: 211.8\n",
"Date: Sun, 16 Nov 2014 Prob (F-statistic): 6.30e-27\n",
"Time: 20:59:31 Log-Likelihood: -34.438\n",
"No. Observations: 50 AIC: 76.88\n",
"Df Residuals: 46 BIC: 84.52\n",
"Df Model: 3 \n",
"==============================================================================\n",
" coef std err t P>|t| [95.0% Conf. Int.]\n",
"------------------------------------------------------------------------------\n",
"x1 0.4687 0.026 17.751 0.000 0.416 0.522\n",
"x2 0.4836 0.104 4.659 0.000 0.275 0.693\n",
"x3 -0.0174 0.002 -7.507 0.000 -0.022 -0.013\n",
"const 5.2058 0.171 30.405 0.000 4.861 5.550\n",
"==============================================================================\n",
"Omnibus: 0.655 Durbin-Watson: 2.896\n",
"Prob(Omnibus): 0.721 Jarque-Bera (JB): 0.360\n",
"Skew: 0.207 Prob(JB): 0.835\n",
"Kurtosis: 3.026 Cond. No. 221.\n",
"==============================================================================\n"
]
}
],
"source": [
"# example with non-linear relationship\n",
"nsample = 50\n",
"sig = 0.5\n",
"x = np.linspace(0, 20, nsample)\n",
"X = np.column_stack((x, np.sin(x), (x-5)**2, np.ones(nsample)))\n",
"beta = [0.5, 0.5, -0.02, 5.]\n",
"\n",
"y_true = np.dot(X, beta)\n",
"y = y_true + sig * np.random.normal(size=nsample)\n",
"\n",
"res = sm.OLS(y, X).fit()\n",
"print(res.summary())"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"('Parameters: ', array([ 0.46872448, 0.48360119, -0.01740479, 5.20584496]))\n",
"('Standard errors: ', array([ 0.02640602, 0.10380518, 0.00231847, 0.17121765]))\n",
"('Predicted values: ', array([ 4.77072516, 5.22213464, 5.63620761, 5.98658823,\n",
" 6.25643234, 6.44117491, 6.54928009, 6.60085051,\n",
" 6.62432454, 6.6518039 , 6.71377946, 6.83412169,\n",
" 7.02615877, 7.29048685, 7.61487206, 7.97626054,\n",
" 8.34456611, 8.68761335, 8.97642389, 9.18997755,\n",
" 9.31866582, 9.36587056, 9.34740836, 9.28893189,\n",
" 9.22171529, 9.17751587, 9.1833565 , 9.25708583,\n",
" 9.40444579, 9.61812821, 9.87897556, 10.15912843,\n",
" 10.42660281, 10.65054491, 10.8063004 , 10.87946503,\n",
" 10.86825119, 10.78378163, 10.64826203, 10.49133265,\n",
" 10.34519853, 10.23933827, 10.19566084, 10.22490593,\n",
" 10.32487947, 10.48081414, 10.66779556, 10.85485568,\n",
" 11.01006072, 11.10575781]))\n"
]
}
],
"source": [
"# look at some quantities of interest\n",
"print('Parameters: ', res.params)\n",
"print('Standard errors: ', res.bse)\n",
"print('Predicted values: ', res.predict())"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
""
]
},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
},
{
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],
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"# plot the true relationship vs. the prediction\n",
"prstd, iv_l, iv_u = wls_prediction_std(res)\n",
"\n",
"fig, ax = plt.subplots(figsize=(8,6))\n",
"\n",
"ax.plot(x, y, 'o', label=\"data\")\n",
"ax.plot(x, y_true, 'b-', label=\"True\")\n",
"ax.plot(x, res.fittedvalues, 'r--.', label=\"OLS\")\n",
"ax.plot(x, iv_u, 'r--')\n",
"ax.plot(x, iv_l, 'r--')\n",
"ax.legend(loc='best')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Time-Series Analysis"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"from statsmodels.tsa.arima_process import arma_generate_sample"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"# generate some data\n",
"np.random.seed(12345)\n",
"arparams = np.array([.75, -.25])\n",
"maparams = np.array([.65, .35])"
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"# set parameters\n",
"arparams = np.r_[1, -arparams]\n",
"maparam = np.r_[1, maparams]\n",
"nobs = 250\n",
"y = arma_generate_sample(arparams, maparams, nobs)"
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"# add some dates information\n",
"dates = sm.tsa.datetools.dates_from_range('1980m1', length=nobs)\n",
"y = pd.TimeSeries(y, index=dates)\n",
"arma_mod = sm.tsa.ARMA(y, order=(2,2))\n",
"arma_res = arma_mod.fit(trend='nc', disp=-1)"
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
" ARMA Model Results \n",
"==============================================================================\n",
"Dep. Variable: y No. Observations: 250\n",
"Model: ARMA(2, 2) Log Likelihood -245.887\n",
"Method: css-mle S.D. of innovations 0.645\n",
"Date: Sun, 16 Nov 2014 AIC 501.773\n",
"Time: 20:59:32 BIC 519.381\n",
"Sample: 01-31-1980 HQIC 508.860\n",
" - 10-31-2000 \n",
"==============================================================================\n",
" coef std err z P>|z| [95.0% Conf. Int.]\n",
"------------------------------------------------------------------------------\n",
"ar.L1.y 0.8411 0.403 2.089 0.038 0.052 1.630\n",
"ar.L2.y -0.2693 0.247 -1.092 0.276 -0.753 0.214\n",
"ma.L1.y 0.5352 0.412 1.299 0.195 -0.273 1.343\n",
"ma.L2.y 0.0157 0.306 0.051 0.959 -0.585 0.616\n",
" Roots \n",
"=============================================================================\n",
" Real Imaginary Modulus Frequency\n",
"-----------------------------------------------------------------------------\n",
"AR.1 1.5618 -1.1289j 1.9271 -0.0996\n",
"AR.2 1.5618 +1.1289j 1.9271 0.0996\n",
"MA.1 -1.9835 +0.0000j 1.9835 0.5000\n",
"MA.2 -32.1812 +0.0000j 32.1812 0.5000\n",
"-----------------------------------------------------------------------------\n"
]
}
],
"source": [
"print(arma_res.summary())"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 2",
"language": "python",
"name": "python2"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 2
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython2",
"version": "2.7.9"
}
},
"nbformat": 4,
"nbformat_minor": 0
}