[time-series]
3일차 복습 풀이
데이터 및 전처리
import numpy as np
import pandas as pd
from statsmodels.formula.api import ols
df = pd.read_csv('tute1.csv', parse_dates=True, index_col='Date')
n = len(df)
df['t'] = np.arange(n)
years = int(np.ceil(n/4))
df['season'] = np.tile([1,2,3,4], years)[:n]
df.head()
:::note[output]
Sales AdBudget GDP t season
Date
1981-03-01 1020.2 659.2 251.8 0 1
1981-06-01 889.2 589.0 290.9 1 2
1981-09-01 795.0 512.5 290.8 2 3
1981-12-01 1003.9 614.1 292.4 3 4
1982-03-01 1057.7 647.2 279.1 4 1
:::
선형 추세
ols('Sales ~ t', df).fit().summary()
:::note[output]
<class 'statsmodels.iolib.summary.Summary'>
"""
OLS Regression Results
==============================================================================
Dep. Variable: Sales R-squared: 0.007
Model: OLS Adj. R-squared: -0.003
Method: Least Squares F-statistic: 0.6817
Date: Thu, 08 Aug 2024 Prob (F-statistic): 0.411
Time: 20:55:38 Log-Likelihood: -599.80
No. Observations: 100 AIC: 1204.
Df Residuals: 98 BIC: 1209.
Df Model: 1
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
Intercept 934.8032 19.535 47.852 0.000 896.036 973.571
t 0.2815 0.341 0.826 0.411 -0.395 0.958
==============================================================================
Omnibus: 12.605 Durbin-Watson: 2.002
Prob(Omnibus): 0.002 Jarque-Bera (JB): 5.652
Skew: -0.350 Prob(JB): 0.0592
Kurtosis: 2.068 Cond. No. 114.
==============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
"""
:::
계절성
ols('Sales ~ C(season)', df).fit().summary()
:::note[output]
<class 'statsmodels.iolib.summary.Summary'>
"""
OLS Regression Results
==============================================================================
Dep. Variable: Sales R-squared: 0.787
Model: OLS Adj. R-squared: 0.780
Method: Least Squares F-statistic: 118.1
Date: Thu, 08 Aug 2024 Prob (F-statistic): 4.19e-32
Time: 20:55:58 Log-Likelihood: -522.86
No. Observations: 100 AIC: 1054.
Df Residuals: 96 BIC: 1064.
Df Model: 3
Covariance Type: nonrobust
==================================================================================
coef std err t P>|t| [0.025 0.975]
----------------------------------------------------------------------------------
Intercept 1050.0000 9.213 113.972 0.000 1031.713 1068.287
C(season)[T.2] -84.1040 13.029 -6.455 0.000 -109.966 -58.242
C(season)[T.3] -239.5920 13.029 -18.389 0.000 -265.454 -213.730
C(season)[T.4] -81.3560 13.029 -6.244 0.000 -107.218 -55.494
==============================================================================
Omnibus: 2.308 Durbin-Watson: 1.595
Prob(Omnibus): 0.315 Jarque-Bera (JB): 1.712
Skew: 0.279 Prob(JB): 0.425
Kurtosis: 3.316 Cond. No. 4.79
==============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
"""
:::
독립변수 조합 찾기
ols('Sales ~ AdBudget', df).fit().summary()
:::note[output]
<class 'statsmodels.iolib.summary.Summary'>
"""
OLS Regression Results
==============================================================================
Dep. Variable: Sales R-squared: 0.824
Model: OLS Adj. R-squared: 0.822
Method: Least Squares F-statistic: 458.9
Date: Thu, 08 Aug 2024 Prob (F-statistic): 9.37e-39
Time: 20:57:12 Log-Likelihood: -513.27
No. Observations: 100 AIC: 1031.
Df Residuals: 98 BIC: 1036.
Df Model: 1
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
Intercept -22.7772 45.538 -0.500 0.618 -113.146 67.592
AdBudget 1.6413 0.077 21.423 0.000 1.489 1.793
==============================================================================
Omnibus: 6.582 Durbin-Watson: 1.534
Prob(Omnibus): 0.037 Jarque-Bera (JB): 7.810
Skew: 0.342 Prob(JB): 0.0201
Kurtosis: 4.186 Cond. No. 6.53e+03
==============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
[2] The condition number is large, 6.53e+03. This might indicate that there are
strong multicollinearity or other numerical problems.
"""
:::
ols('Sales ~ GDP', df).fit().summary()
:::note[output]
<class 'statsmodels.iolib.summary.Summary'>
"""
OLS Regression Results
==============================================================================
Dep. Variable: Sales R-squared: 0.404
Model: OLS Adj. R-squared: 0.398
Method: Least Squares F-statistic: 66.55
Date: Thu, 08 Aug 2024 Prob (F-statistic): 1.17e-12
Time: 20:57:17 Log-Likelihood: -574.23
No. Observations: 100 AIC: 1152.
Df Residuals: 98 BIC: 1158.
Df Model: 1
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
Intercept 2171.3527 150.061 14.470 0.000 1873.561 2469.144
GDP -4.3481 0.533 -8.158 0.000 -5.406 -3.290
==============================================================================
Omnibus: 3.708 Durbin-Watson: 1.734
Prob(Omnibus): 0.157 Jarque-Bera (JB): 2.165
Skew: -0.098 Prob(JB): 0.339
Kurtosis: 2.306 Cond. No. 5.54e+03
==============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
[2] The condition number is large, 5.54e+03. This might indicate that there are
strong multicollinearity or other numerical problems.
"""
:::
ols('Sales ~ AdBudget + GDP', df).fit().summary()
:::note[output]
<class 'statsmodels.iolib.summary.Summary'>
"""
OLS Regression Results
==============================================================================
Dep. Variable: Sales R-squared: 0.833
Model: OLS Adj. R-squared: 0.830
Method: Least Squares F-statistic: 242.7
Date: Thu, 08 Aug 2024 Prob (F-statistic): 1.77e-38
Time: 20:57:37 Log-Likelihood: -510.53
No. Observations: 100 AIC: 1027.
Df Residuals: 97 BIC: 1035.
Df Model: 2
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
Intercept -438.9770 183.405 -2.393 0.019 -802.985 -74.969
AdBudget 1.8520 0.117 15.806 0.000 1.619 2.085
GDP 1.0365 0.443 2.339 0.021 0.157 1.916
==============================================================================
Omnibus: 6.293 Durbin-Watson: 1.641
Prob(Omnibus): 0.043 Jarque-Bera (JB): 6.895
Skew: 0.364 Prob(JB): 0.0318
Kurtosis: 4.060 Cond. No. 2.97e+04
==============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
[2] The condition number is large, 2.97e+04. This might indicate that there are
strong multicollinearity or other numerical problems.
"""
:::
ols('Sales ~ AdBudget + C(season)', df).fit().summary()
:::note[output]
<class 'statsmodels.iolib.summary.Summary'>
"""
OLS Regression Results
==============================================================================
Dep. Variable: Sales R-squared: 0.829
Model: OLS Adj. R-squared: 0.822
Method: Least Squares F-statistic: 115.1
Date: Thu, 08 Aug 2024 Prob (F-statistic): 1.49e-35
Time: 20:57:46 Log-Likelihood: -511.84
No. Observations: 100 AIC: 1034.
Df Residuals: 95 BIC: 1047.
Df Model: 4
Covariance Type: nonrobust
==================================================================================
coef std err t P>|t| [0.025 0.975]
----------------------------------------------------------------------------------
Intercept -201.4519 258.738 -0.779 0.438 -715.112 312.208
C(season)[T.2] 7.3204 22.238 0.329 0.743 -36.828 51.469
C(season)[T.3] 38.3021 58.611 0.653 0.515 -78.056 154.660
C(season)[T.4] -5.1703 19.633 -0.263 0.793 -44.147 33.807
AdBudget 1.9260 0.398 4.839 0.000 1.136 2.716
==============================================================================
Omnibus: 6.063 Durbin-Watson: 1.475
Prob(Omnibus): 0.048 Jarque-Bera (JB): 6.789
Skew: 0.336 Prob(JB): 0.0336
Kurtosis: 4.085 Cond. No. 3.82e+04
==============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
[2] The condition number is large, 3.82e+04. This might indicate that there are
strong multicollinearity or other numerical problems.
"""
:::
ols('Sales ~ AdBudget + GDP + C(season)', df).fit().summary()
:::note[output]
<class 'statsmodels.iolib.summary.Summary'>
"""
OLS Regression Results
==============================================================================
Dep. Variable: Sales R-squared: 0.840
Model: OLS Adj. R-squared: 0.832
Method: Least Squares F-statistic: 98.78
Date: Thu, 08 Aug 2024 Prob (F-statistic): 7.42e-36
Time: 20:58:55 Log-Likelihood: -508.48
No. Observations: 100 AIC: 1029.
Df Residuals: 94 BIC: 1045.
Df Model: 5
Covariance Type: nonrobust
==================================================================================
coef std err t P>|t| [0.025 0.975]
----------------------------------------------------------------------------------
Intercept -737.5609 327.561 -2.252 0.027 -1387.941 -87.181
C(season)[T.2] 8.6108 21.624 0.398 0.691 -34.324 51.546
C(season)[T.3] 54.9637 57.349 0.958 0.340 -58.904 168.831
C(season)[T.4] -1.8296 19.131 -0.096 0.924 -39.814 36.155
AdBudget 2.2766 0.411 5.546 0.000 1.462 3.092
GDP 1.1497 0.450 2.555 0.012 0.256 2.043
==============================================================================
Omnibus: 5.514 Durbin-Watson: 1.581
Prob(Omnibus): 0.063 Jarque-Bera (JB): 5.710
Skew: 0.339 Prob(JB): 0.0575
Kurtosis: 3.955 Cond. No. 5.40e+04
==============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
[2] The condition number is large, 5.4e+04. This might indicate that there are
strong multicollinearity or other numerical problems.
"""
:::
STL 분해
from statsmodels.tsa.seasonal import STL
m = STL(df.Sales, period=4).fit()
m.plot();
:::note[output]
<Figure size 640x480 with 4 Axes>
:::
추세의 강도
1 - m.resid.var() / (m.trend + m.resid).var()
:::note[output]
0.4999592147323638
:::
계절성의 강도
1 - m.resid.var() / (m.seasonal + m.resid).var()
:::note[output]
0.9141388728480302
:::