ARIMA 모형

ARMA(p,q) 모형은 AR(p) 모형과 MA(q) 모형이 합쳐진 모형입니다.

import matplotlib.pyplot as plt
import pandas as pd

df = pd.read_excel('beer.xlsx')
df.head()

y = df.production
y.plot()

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ARMA(1, 1)

from  statsmodels.tsa.api import SARIMAX

arma = SARIMAX(y, order=(1, 0, 1)).fit()
arma.summary() 

y.plot()
arma.predict(0, 250).plot()

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from statsmodels.tsa.stattools import kpss
kpss(y)

C:\Users\eupho\AppData\Local\Temp\ipykernel_28656\4027067367.py:2: InterpolationWarning: The test statistic is outside of the range of p-values available in the
look-up table. The actual p-value is smaller than the p-value returned.

  kpss(y)

(1.1416279223770989,
 0.01,
 9,
 {'10%': 0.347, '5%': 0.463, '2.5%': 0.574, '1%': 0.739})

yd = y.diff(1).dropna()
yd.plot()

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kpss(yd)

C:\Users\eupho\AppData\Local\Temp\ipykernel_28656\757044700.py:1: InterpolationWarning: The test statistic is outside of the range of p-values available in the
look-up table. The actual p-value is greater than the p-value returned.

  kpss(yd)

(0.12919635959269035,
 0.1,
 17,
 {'10%': 0.347, '5%': 0.463, '2.5%': 0.574, '1%': 0.739})

d_arma = SARIMAX(yd, order=(1, 0, 1)).fit()
d_arma.summary() 

c:\Users\eupho\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:473: ValueWarning: An unsupported index was provided and will be ignored when e.g. forecasting.
  self._init_dates(dates, freq)
c:\Users\eupho\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:473: ValueWarning: An unsupported index was provided and will be ignored when e.g. forecasting.
  self._init_dates(dates, freq)
c:\Users\eupho\anaconda3\lib\site-packages\statsmodels\tsa\statespace\sarimax.py:978: UserWarning: Non-invertible starting MA parameters found. Using zeros as starting parameters.
  warn('Non-invertible starting MA parameters found.'

yd.plot()
d_arma.predict(0, 250).plot()

c:\Users\eupho\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:836: ValueWarning: No supported index is available. Prediction results will be given with an integer index beginning at `start`.
  return get_prediction_index(
c:\Users\eupho\anaconda3\lib\site-packages\statsmodels\tsa\base\tsa_model.py:836: FutureWarning: No supported index is available. In the next version, calling this method in a model without a supported index will result in an exception.
  return get_prediction_index(

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ARIMA(p, d, q) 모형은 d번 차분한 값을 ARMA(p, q)으로 분석하는 것과 같습니다.

arima = SARIMAX(y, order=(1, 1, 1)).fit()
arima.summary() 

c:\Users\eupho\anaconda3\lib\site-packages\statsmodels\tsa\statespace\sarimax.py:978: UserWarning: Non-invertible starting MA parameters found. Using zeros as starting parameters.
  warn('Non-invertible starting MA parameters found.'

y.plot()
arima.predict(0, 250).plot()

<Axes: >

계절성 ARIMA 모형은 계절성에 대해서 별도의 ARIMA 모형으로 분석합니다. seasonal_order의 마지막 값은 계절성의 주기를 나타냅니다.

sarima = SARIMAX(y, order=(1, 1, 0), seasonal_order=(1, 0, 0, 4)).fit()
sarima.summary()

y.plot()
sarima.predict(0, 250).plot()

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계절성 ARIMA 모형은 계절성의 주기를 제외하고 p, d, q, P, D, Q 모두 6개의 값을 설정해야 합니다. 아래 코드는 이 6개의 값에 0, 1, 2 세 가지를 대입하여 총 729가지 모형을 시도합니다. 그중 AIC가 가장 낮은 모형을 best_sarima에 설정합니다.

from itertools import combinations_with_replacement
import numpy as np
import warnings

warnings.filterwarnings("ignore")

min_aic = np.inf
for ar, ma, seasonal_ar, seasonal_ma in combinations_with_replacement([0, 1, 2], 4):
    try:
        sarima = SARIMAX(y, order=(ar, 1, ma), seasonal_order=(seasonal_ar, 0, seasonal_ma, 4)).fit()
        print(ar, ma, seasonal_ar, seasonal_ma, sarima.aic)
    except:
        pass
    else:
        if sarima.aic < min_aic:
            min_aic = sarima.aic
            best_sarima = sarima
warnings.resetwarnings()

0 0 0 0 2446.993870307465
0 0 0 0 2446.993870307465
0 0 0 1 2269.0870570757484
0 0 0 1 2269.0870570757484
0 0 0 2 2205.6411732862007
0 0 0 2 2205.6411732862007
0 0 1 1 1968.7883160730275
0 0 1 1 1968.7883160730275
0 0 1 2 1970.3592365281647
0 0 2 2 1965.3952928674607
0 0 2 2 1965.3952928674607
0 1 1 1 1840.469834804714
0 1 1 1 1840.469834804714
0 1 1 2 1840.7075395781462
0 1 2 2 1836.4792775263727
0 1 2 2 1836.4792775263727
0 2 2 2 1822.26570408957
0 2 2 2 1822.26570408957
1 1 1 1 1827.969616228781
1 1 1 2 1828.8672615487212
1 1 2 2 1824.7291286441794
1 2 2 2 1824.2634784092993

최적의 모델은 다음과 같습니다.

best_sarima.summary()