{
"cells": [
{
"cell_type": "code",
"execution_count": 1,
"id": "329987f1",
"metadata": {},
"outputs": [],
"source": [
"import numpy as np\n",
"from numpy import sqrt\n",
"import matplotlib.pyplot as plt\n",
"import scipy.io as sio\n",
"import os\n",
"import random\n",
"import time\n",
"import sys\n",
"from matplotlib import rcParams\n",
"from scipy.linalg import eigh\n",
"from sklearn.linear_model import LinearRegression\n",
"from sklearn.preprocessing import PolynomialFeatures\n",
"from scipy import stats\n",
"import statsmodels.api as sm\n",
"from statsmodels.formula.api import ols\n",
"import pandas as pd\n",
"import matplotlib as mpl\n",
"import seaborn as sns\n",
"\n",
"\n",
"\n",
"import statsmodels.formula.api as smf\n",
"\n",
"mpl.rcParams['figure.dpi'] = 400 \n",
"plt.rc('text',usetex=True)"
]
},
{
"cell_type": "markdown",
"id": "b9b501de",
"metadata": {},
"source": [
"## Alguns exemplos do *Introduction to Probability for Data Science*, Stanley H. Chan"
]
},
{
"cell_type": "markdown",
"id": "654cd502",
"metadata": {},
"source": [
"Ajustando o modelo $Y = \\beta_0 + \\beta_1 x + \\beta_2 x^2 + \\varepsilon$ com `numpy`."
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "b152351e",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([[1.00000000e+00, 9.32316068e-01, 8.69213251e-01],\n",
" [1.00000000e+00, 4.00667314e-01, 1.60534296e-01],\n",
" [1.00000000e+00, 3.57947608e-01, 1.28126490e-01],\n",
" [1.00000000e+00, 6.65188911e-01, 4.42476287e-01],\n",
" [1.00000000e+00, 4.57034606e-01, 2.08880631e-01],\n",
" [1.00000000e+00, 4.52698089e-01, 2.04935560e-01],\n",
" [1.00000000e+00, 6.79445639e-01, 4.61646377e-01],\n",
" [1.00000000e+00, 1.91098308e-01, 3.65185632e-02],\n",
" [1.00000000e+00, 4.77749380e-01, 2.28244470e-01],\n",
" [1.00000000e+00, 3.02821866e-01, 9.17010826e-02],\n",
" [1.00000000e+00, 4.24069130e-01, 1.79834627e-01],\n",
" [1.00000000e+00, 3.09252769e-01, 9.56372748e-02],\n",
" [1.00000000e+00, 5.00669784e-01, 2.50670233e-01],\n",
" [1.00000000e+00, 7.45686720e-01, 5.56048684e-01],\n",
" [1.00000000e+00, 9.43347814e-01, 8.89905098e-01],\n",
" [1.00000000e+00, 6.19081748e-01, 3.83262211e-01],\n",
" [1.00000000e+00, 4.17625267e-02, 1.74410864e-03],\n",
" [1.00000000e+00, 9.78771178e-01, 9.57993019e-01],\n",
" [1.00000000e+00, 2.52993410e-01, 6.40056656e-02],\n",
" [1.00000000e+00, 5.28084907e-02, 2.78873669e-03],\n",
" [1.00000000e+00, 2.46939168e-01, 6.09789528e-02],\n",
" [1.00000000e+00, 6.72628813e-01, 4.52429520e-01],\n",
" [1.00000000e+00, 7.29534614e-01, 5.32220753e-01],\n",
" [1.00000000e+00, 9.90588832e-01, 9.81266233e-01],\n",
" [1.00000000e+00, 9.21709384e-01, 8.49548188e-01],\n",
" [1.00000000e+00, 3.01062703e-02, 9.06387509e-04],\n",
" [1.00000000e+00, 4.39344301e-01, 1.93023415e-01],\n",
" [1.00000000e+00, 4.99937605e-01, 2.49937609e-01],\n",
" [1.00000000e+00, 7.57026514e-01, 5.73089142e-01],\n",
" [1.00000000e+00, 2.01148486e-01, 4.04607133e-02],\n",
" [1.00000000e+00, 8.54662926e-01, 7.30448716e-01],\n",
" [1.00000000e+00, 8.17021753e-01, 6.67524544e-01],\n",
" [1.00000000e+00, 1.37711890e-01, 1.89645646e-02],\n",
" [1.00000000e+00, 8.22719210e-01, 6.76866899e-01],\n",
" [1.00000000e+00, 8.00383695e-01, 6.40614059e-01],\n",
" [1.00000000e+00, 3.61067019e-02, 1.30369392e-03],\n",
" [1.00000000e+00, 3.32710339e-01, 1.10696170e-01],\n",
" [1.00000000e+00, 6.96406506e-01, 4.84982022e-01],\n",
" [1.00000000e+00, 2.36852735e-01, 5.60992180e-02],\n",
" [1.00000000e+00, 5.88420489e-01, 3.46238671e-01],\n",
" [1.00000000e+00, 6.12605288e-01, 3.75285239e-01],\n",
" [1.00000000e+00, 3.69473939e-01, 1.36510992e-01],\n",
" [1.00000000e+00, 2.25020908e-01, 5.06344091e-02],\n",
" [1.00000000e+00, 4.94507111e-01, 2.44537283e-01],\n",
" [1.00000000e+00, 4.84044217e-01, 2.34298804e-01],\n",
" [1.00000000e+00, 2.40400462e-01, 5.77923820e-02],\n",
" [1.00000000e+00, 8.63991480e-01, 7.46481278e-01],\n",
" [1.00000000e+00, 7.06783807e-01, 4.99543349e-01],\n",
" [1.00000000e+00, 1.02481086e-01, 1.05023730e-02],\n",
" [1.00000000e+00, 8.27714196e-01, 6.85110791e-01]])"
]
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"\n",
"N = 50\n",
"x = np.random.rand(N)\n",
"a = -2.5\n",
"b = 1.3\n",
"c = 1.2\n",
"\n",
"\n",
"y = a*x**2 + b*x + c + 0.2*np.random.randn(N)\n",
"\n",
"X = np.column_stack((np.ones(N), x, x**2)) # criando a base de funções\n",
"\n",
"X"
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "28230f25",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[ 1.27225644 1.05854855 -2.37886215]\n"
]
}
],
"source": [
"\n",
"beta = np.linalg.lstsq(X, y, rcond=None)[0]\n",
"\n",
"\n",
"t = np.linspace(0,1,200)\n",
"\n",
"yhat = beta[0] + beta[1]*t + beta[2]*t**2\n",
"\n",
"print(beta)"
]
},
{
"cell_type": "code",
"execution_count": 8,
"id": "4c80b1ae",
"metadata": {},
"outputs": [],
"source": [
"np.linalg.lstsq?"
]
},
{
"cell_type": "markdown",
"id": "2efba63a",
"metadata": {},
"source": [
"## Com o `statsmodel`"
]
},
{
"cell_type": "code",
"execution_count": 5,
"id": "00d43776",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
" OLS Regression Results \n",
"==============================================================================\n",
"Dep. Variable: y R-squared: 0.833\n",
"Model: OLS Adj. R-squared: 0.826\n",
"Method: Least Squares F-statistic: 117.3\n",
"Date: Tue, 14 Nov 2023 Prob (F-statistic): 5.32e-19\n",
"Time: 10:06:58 Log-Likelihood: 13.014\n",
"No. Observations: 50 AIC: -20.03\n",
"Df Residuals: 47 BIC: -14.29\n",
"Df Model: 2 \n",
"Covariance Type: nonrobust \n",
"==============================================================================\n",
" coef std err t P>|t| [0.025 0.975]\n",
"------------------------------------------------------------------------------\n",
"const 1.2723 0.088 14.505 0.000 1.096 1.449\n",
"x1 1.0585 0.390 2.715 0.009 0.274 1.843\n",
"x2 -2.3789 0.372 -6.399 0.000 -3.127 -1.631\n",
"==============================================================================\n",
"Omnibus: 10.168 Durbin-Watson: 1.999\n",
"Prob(Omnibus): 0.006 Jarque-Bera (JB): 2.832\n",
"Skew: 0.045 Prob(JB): 0.243\n",
"Kurtosis: 1.838 Cond. No. 23.6\n",
"==============================================================================\n",
"\n",
"Notes:\n",
"[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n"
]
}
],
"source": [
"model = sm.OLS(y,X)\n",
"results = model.fit()\n",
"\n",
"print(results.summary())"
]
},
{
"cell_type": "markdown",
"id": "35fd9562",
"metadata": {},
"source": [
"### Com o `statsmodels.formula.api` "
]
},
{
"cell_type": "code",
"execution_count": 6,
"id": "150921d0",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"
\n",
"OLS Regression Results\n",
"\n",
" Dep. Variable: | y | R-squared: | 0.833 | \n",
"
\n",
"\n",
" Model: | OLS | Adj. R-squared: | 0.826 | \n",
"
\n",
"\n",
" Method: | Least Squares | F-statistic: | 117.3 | \n",
"
\n",
"\n",
" Date: | Tue, 14 Nov 2023 | Prob (F-statistic): | 5.32e-19 | \n",
"
\n",
"\n",
" Time: | 10:07:00 | Log-Likelihood: | 13.014 | \n",
"
\n",
"\n",
" No. Observations: | 50 | AIC: | -20.03 | \n",
"
\n",
"\n",
" Df Residuals: | 47 | BIC: | -14.29 | \n",
"
\n",
"\n",
" Df Model: | 2 | | | \n",
"
\n",
"\n",
" Covariance Type: | nonrobust | | | \n",
"
\n",
"
\n",
"\n",
"\n",
" | coef | std err | t | P>|t| | [0.025 | 0.975] | \n",
"
\n",
"\n",
" Intercept | 1.2723 | 0.088 | 14.505 | 0.000 | 1.096 | 1.449 | \n",
"
\n",
"\n",
" x | 1.0585 | 0.390 | 2.715 | 0.009 | 0.274 | 1.843 | \n",
"
\n",
"\n",
" I(x ** 2) | -2.3789 | 0.372 | -6.399 | 0.000 | -3.127 | -1.631 | \n",
"
\n",
"
\n",
"\n",
"\n",
" Omnibus: | 10.168 | Durbin-Watson: | 1.999 | \n",
"
\n",
"\n",
" Prob(Omnibus): | 0.006 | Jarque-Bera (JB): | 2.832 | \n",
"
\n",
"\n",
" Skew: | 0.045 | Prob(JB): | 0.243 | \n",
"
\n",
"\n",
" Kurtosis: | 1.838 | Cond. No. | 23.6 | \n",
"
\n",
"
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified."
],
"text/plain": [
"\n",
"\"\"\"\n",
" OLS Regression Results \n",
"==============================================================================\n",
"Dep. Variable: y R-squared: 0.833\n",
"Model: OLS Adj. R-squared: 0.826\n",
"Method: Least Squares F-statistic: 117.3\n",
"Date: Tue, 14 Nov 2023 Prob (F-statistic): 5.32e-19\n",
"Time: 10:07:00 Log-Likelihood: 13.014\n",
"No. Observations: 50 AIC: -20.03\n",
"Df Residuals: 47 BIC: -14.29\n",
"Df Model: 2 \n",
"Covariance Type: nonrobust \n",
"==============================================================================\n",
" coef std err t P>|t| [0.025 0.975]\n",
"------------------------------------------------------------------------------\n",
"Intercept 1.2723 0.088 14.505 0.000 1.096 1.449\n",
"x 1.0585 0.390 2.715 0.009 0.274 1.843\n",
"I(x ** 2) -2.3789 0.372 -6.399 0.000 -3.127 -1.631\n",
"==============================================================================\n",
"Omnibus: 10.168 Durbin-Watson: 1.999\n",
"Prob(Omnibus): 0.006 Jarque-Bera (JB): 2.832\n",
"Skew: 0.045 Prob(JB): 0.243\n",
"Kurtosis: 1.838 Cond. No. 23.6\n",
"==============================================================================\n",
"\n",
"Notes:\n",
"[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n",
"\"\"\""
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"#x = np.random.rand(N)\n",
"#y = a*x**2 + b*x + c + 0.2*np.random.randn(N)\n",
"\n",
"df = pd.DataFrame([])\n",
"df['y'] = y\n",
"df['x'] = x\n",
"\n",
"model = smf.ols(formula='y ~ x + I(x**2)', data=df)\n",
"\n",
"result = model.fit()\n",
"result.summary()\n"
]
},
{
"cell_type": "markdown",
"id": "9f440bb2",
"metadata": {},
"source": [
"## Polinômios ortogonais"
]
},
{
"cell_type": "markdown",
"id": "950d8e36",
"metadata": {},
"source": [
"Exemplo com $Y = \\beta_0 + \\beta_1 x+ \\beta_2 x^2+ \\beta_3 x^3+ \\beta_4 x^4 + \\varepsilon$"
]
},
{
"cell_type": "code",
"execution_count": 7,
"id": "6a785767",
"metadata": {},
"outputs": [
{
"data": {
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