У меня есть различия в значениях коэффициента и погрешностях коэффициента с использованием функций smf.ols и sm.OLS из statsmodels.Несмотря на то, что математически они должны иметь одинаковую формулу регрессии и давать одинаковые результаты.
Я сделал воспроизводимый на 100% пример моего вопроса, отсюда можно скачать фрейм данных df: https://drive.google.com/drive/folders/1i67wztkrAeEZH2tv2hyOlgxG7N80V3pI?usp=sharing
Случай 1: Линейная модель с использованием Patsy из Statsmodels
# First we load the libraries:
import statsmodels.api as sm
import statsmodels.formula.api as smf
import random
import pandas as pd
# We define a specific seed to have the same results:
random.seed(1234)
# Now we read the data that can be downloaded from Google Drive link provided above:
df = pd.read_csv("/Users/user/Documents/example/cars.csv", sep = "|")
# We create the linear regression:
lm1 = smf.ols('price ~ make + fuel_system + engine_type + num_of_doors + bore + compression_ratio + height + peak_rpm + 1', data = df)
# We see the results:
lm1.fit().summary()
Результат lm1:
OLS Regression Results
==============================================================================
Dep. Variable: price R-squared: 0.894
Model: OLS Adj. R-squared: 0.868
Method: Least Squares F-statistic: 35.54
Date: Mon, 18 Feb 2019 Prob (F-statistic): 5.24e-62
Time: 17:19:14 Log-Likelihood: -1899.7
No. Observations: 205 AIC: 3879.
Df Residuals: 165 BIC: 4012.
Df Model: 39
Covariance Type: nonrobust
=========================================================================================
coef std err t P>|t| [0.025 0.975]
-----------------------------------------------------------------------------------------
Intercept 1.592e+04 1.21e+04 1.320 0.189 -7898.396 3.97e+04
make[T.audi] 6519.7045 2371.807 2.749 0.007 1836.700 1.12e+04
make[T.bmw] 1.427e+04 2292.551 6.223 0.000 9740.771 1.88e+04
make[T.chevrolet] -571.8236 2860.026 -0.200 0.842 -6218.788 5075.141
make[T.dodge] -1186.3430 2261.240 -0.525 0.601 -5651.039 3278.353
make[T.honda] 2779.6496 2891.626 0.961 0.338 -2929.709 8489.009
make[T.isuzu] 3098.9677 2592.645 1.195 0.234 -2020.069 8218.004
make[T.jaguar] 1.752e+04 2416.313 7.252 0.000 1.28e+04 2.23e+04
make[T.mazda] 306.6568 2134.567 0.144 0.886 -3907.929 4521.243
make[T.mercedes-benz] 1.698e+04 2320.871 7.318 0.000 1.24e+04 2.16e+04
make[T.mercury] 2958.1002 3605.739 0.820 0.413 -4161.236 1.01e+04
make[T.mitsubishi] -1188.8337 2284.697 -0.520 0.604 -5699.844 3322.176
make[T.nissan] -1211.5463 2073.422 -0.584 0.560 -5305.405 2882.312
make[T.peugot] 3057.0217 4255.809 0.718 0.474 -5345.841 1.15e+04
make[T.plymouth] -894.5921 2332.746 -0.383 0.702 -5500.473 3711.289
make[T.porsche] 9558.8747 3688.038 2.592 0.010 2277.044 1.68e+04
make[T.renault] -2124.9722 2847.536 -0.746 0.457 -7747.277 3497.333
make[T.saab] 3490.5333 2319.189 1.505 0.134 -1088.579 8069.645
make[T.subaru] -1.636e+04 4002.796 -4.087 0.000 -2.43e+04 -8456.659
make[T.toyota] -770.9677 1911.754 -0.403 0.687 -4545.623 3003.688
make[T.volkswagen] 406.9179 2219.714 0.183 0.855 -3975.788 4789.623
make[T.volvo] 5433.7129 2397.030 2.267 0.025 700.907 1.02e+04
fuel_system[T.2bbl] 2142.1594 2232.214 0.960 0.339 -2265.226 6549.545
fuel_system[T.4bbl] 464.1109 3999.976 0.116 0.908 -7433.624 8361.846
fuel_system[T.idi] 1.991e+04 6622.812 3.007 0.003 6837.439 3.3e+04
fuel_system[T.mfi] 3716.5201 3936.805 0.944 0.347 -4056.488 1.15e+04
fuel_system[T.mpfi] 3964.1109 2267.538 1.748 0.082 -513.019 8441.241
fuel_system[T.spdi] 3240.0003 2719.925 1.191 0.235 -2130.344 8610.344
fuel_system[T.spfi] 932.1959 4019.476 0.232 0.817 -7004.041 8868.433
engine_type[T.dohcv] -1.208e+04 4205.826 -2.872 0.005 -2.04e+04 -3773.504
engine_type[T.l] -4833.9860 3763.812 -1.284 0.201 -1.23e+04 2597.456
engine_type[T.ohc] -4038.8848 1213.598 -3.328 0.001 -6435.067 -1642.702
engine_type[T.ohcf] 9618.9281 3504.600 2.745 0.007 2699.286 1.65e+04
engine_type[T.ohcv] 3051.7629 1445.185 2.112 0.036 198.323 5905.203
engine_type[T.rotor] 1403.9928 3217.402 0.436 0.663 -4948.593 7756.579
num_of_doors[T.two] -419.9640 521.754 -0.805 0.422 -1450.139 610.211
bore 3993.4308 1373.487 2.908 0.004 1281.556 6705.306
compression_ratio -1200.5665 460.681 -2.606 0.010 -2110.156 -290.977
height -80.7141 146.219 -0.552 0.582 -369.417 207.988
peak_rpm -0.5903 0.790 -0.747 0.456 -2.150 0.970
==============================================================================
Omnibus: 65.777 Durbin-Watson: 1.217
Prob(Omnibus): 0.000 Jarque-Bera (JB): 399.594
Skew: 1.059 Prob(JB): 1.70e-87
Kurtosis: 9.504 Cond. No. 3.26e+05
==============================================================================
Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
[2] The condition number is large, 3.26e+05. This might indicate that there are
strong multicollinearity or other numerical problems.
"""
Случай 2: Линейная модель, использующая также фиктивные переменные из Statsmodels
# We define a specific seed to have the same results:
random.seed(1234)
# First we check what `object` type variables we have in our dataset:
df.dtypes
# We create a list where we save the `object` type variables names:
object = ['make',
'fuel_system',
'engine_type',
'num_of_doors'
]
# Now we convert those object variables to numeric with get_dummies function to have 1 unique numeric dataframe:
df_num = pd.get_dummies(df, columns = object)
# We ensure the dataframe is numeric casting all values to float64:
df_num = df_num[df_num.columns].apply(pd.to_numeric, errors='coerce', axis = 1)
# We define the predictive variables dataset:
X = df_num.drop('price', axis = 1)
# We define the response variable values:
y = df_num.price.values
# We add a constant as we did in the previous example (adding "+1" to Patsy):
Xc = sm.add_constant(X) # Adds a constant to the model
# We create the linear model and obtain results:
lm2 = sm.OLS(y, Xc)
lm2.fit().summary()
Результат lm2:
OLS Regression Results
==============================================================================
Dep. Variable: y R-squared: 0.894
Model: OLS Adj. R-squared: 0.868
Method: Least Squares F-statistic: 35.54
Date: Mon, 18 Feb 2019 Prob (F-statistic): 5.24e-62
Time: 17:28:16 Log-Likelihood: -1899.7
No. Observations: 205 AIC: 3879.
Df Residuals: 165 BIC: 4012.
Df Model: 39
Covariance Type: nonrobust
======================================================================================
coef std err t P>|t| [0.025 0.975]
--------------------------------------------------------------------------------------
const 1.205e+04 6811.094 1.769 0.079 -1398.490 2.55e+04
bore 3993.4308 1373.487 2.908 0.004 1281.556 6705.306
compression_ratio -1200.5665 460.681 -2.606 0.010 -2110.156 -290.977
height -80.7141 146.219 -0.552 0.582 -369.417 207.988
peak_rpm -0.5903 0.790 -0.747 0.456 -2.150 0.970
make_alfa-romero -2273.9631 1865.185 -1.219 0.225 -5956.669 1408.743
make_audi 4245.7414 1324.140 3.206 0.002 1631.299 6860.184
make_bmw 1.199e+04 1232.635 9.730 0.000 9559.555 1.44e+04
make_chevrolet -2845.7867 1976.730 -1.440 0.152 -6748.733 1057.160
make_dodge -3460.3061 1170.966 -2.955 0.004 -5772.315 -1148.297
make_honda 505.6865 2049.865 0.247 0.805 -3541.661 4553.034
make_isuzu 825.0045 1706.160 0.484 0.629 -2543.716 4193.725
make_jaguar 1.525e+04 1903.813 8.010 0.000 1.15e+04 1.9e+04
make_mazda -1967.3063 982.179 -2.003 0.047 -3906.564 -28.048
make_mercedes-benz 1.471e+04 1423.004 10.338 0.000 1.19e+04 1.75e+04
make_mercury 684.1370 2913.361 0.235 0.815 -5068.136 6436.410
make_mitsubishi -3462.7968 1221.018 -2.836 0.005 -5873.631 -1051.963
make_nissan -3485.5094 946.316 -3.683 0.000 -5353.958 -1617.060
make_peugot 783.0586 3513.296 0.223 0.824 -6153.754 7719.871
make_plymouth -3168.5552 1293.376 -2.450 0.015 -5722.256 -614.854
make_porsche 7284.9115 2853.174 2.553 0.012 1651.475 1.29e+04
make_renault -4398.9354 2037.945 -2.159 0.032 -8422.747 -375.124
make_saab 1216.5702 1487.192 0.818 0.415 -1719.810 4152.950
make_subaru -1.863e+04 3263.524 -5.710 0.000 -2.51e+04 -1.22e+04
make_toyota -3044.9308 776.059 -3.924 0.000 -4577.218 -1512.644
make_volkswagen -1867.0452 1170.975 -1.594 0.113 -4179.072 444.981
make_volvo 3159.7498 1327.405 2.380 0.018 538.862 5780.638
fuel_system_1bbl -2790.4092 2230.161 -1.251 0.213 -7193.740 1612.922
fuel_system_2bbl -648.2498 1094.525 -0.592 0.554 -2809.330 1512.830
fuel_system_4bbl -2326.2983 3094.703 -0.752 0.453 -8436.621 3784.024
fuel_system_idi 1.712e+04 6154.806 2.782 0.006 4971.083 2.93e+04
fuel_system_mfi 926.1109 3063.134 0.302 0.763 -5121.881 6974.102
fuel_system_mpfi 1173.7017 1186.125 0.990 0.324 -1168.238 3515.642
fuel_system_spdi 449.5911 1827.318 0.246 0.806 -3158.349 4057.531
fuel_system_spfi -1858.2133 3111.596 -0.597 0.551 -8001.891 4285.464
engine_type_dohc 2703.6445 1803.080 1.499 0.136 -856.440 6263.729
engine_type_dohcv -9374.0342 3504.717 -2.675 0.008 -1.63e+04 -2454.161
engine_type_l -2130.3416 3357.283 -0.635 0.527 -8759.115 4498.431
engine_type_ohc -1335.2404 1454.047 -0.918 0.360 -4206.177 1535.696
engine_type_ohcf 1.232e+04 2850.883 4.322 0.000 6693.659 1.8e+04
engine_type_ohcv 5755.4074 1669.627 3.447 0.001 2458.820 9051.995
engine_type_rotor 4107.6373 3032.223 1.355 0.177 -1879.323 1.01e+04
num_of_doors_four 6234.8048 3491.722 1.786 0.076 -659.410 1.31e+04
num_of_doors_two 5814.8408 3337.588 1.742 0.083 -775.045 1.24e+04
==============================================================================
Omnibus: 65.777 Durbin-Watson: 1.217
Prob(Omnibus): 0.000 Jarque-Bera (JB): 399.594
Skew: 1.059 Prob(JB): 1.70e-87
Kurtosis: 9.504 Cond. No. 1.01e+16
==============================================================================
Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
[2] The smallest eigenvalue is 5.38e-23. This might indicate that there are
strong multicollinearity problems or that the design matrix is singular.
"""
Как мы видим, некоторые переменные, такие как height, имеют одинаковый коэффициент.Тем не менее некоторые другие этого не делают (уровень isuzu из переменной make, уровень ohc из engine_type или independent term и т. Д.).Разве это не должно быть одинаковым результатом для обоих выходов?Что я здесь упускаю или делаю неправильно?
Заранее благодарен за вашу помощь.
PD Как разъяснил @sukhbinder, даже используя формулу Пэтси без независимого термина (ставя -1)«в формуле, поскольку Пэтси включает его по умолчанию) и исключая независимый термин из фиктивной формулировки, я получаю разные результаты.