第7章 非定常時系列の予測

In [27]:

from typing import Union
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
from itertools import product
from statsmodels.tsa.stattools import adfuller
from statsmodels.stats.diagnostic import acorr_ljungbox
from statsmodels.tsa.statespace.sarimax import SARIMAX
from tqdm.auto import tqdm

from typing import Union import pandas as pd import numpy as np import matplotlib.pyplot as plt from itertools import product from statsmodels.tsa.stattools import adfuller from statsmodels.stats.diagnostic import acorr_ljungbox from statsmodels.tsa.statespace.sarimax import SARIMAX from tqdm.auto import tqdm

自己回帰和分移動平均(ARIMA)モデルを定義する¶

非定常時系列を考慮の対処にするために一般的なモデル化手続きを変更する¶

非定常時系列を予測する¶

In [40]:

url = 'https://raw.githubusercontent.com/marcopeix/TimeSeriesForecastingInPython/master/data/jj.csv'
df = pd.read_csv(url, parse_dates=['date'], index_col='date')

train = df.iloc[:-4]
test = df.iloc[-4:]

# plot
fig, ax = plt.subplots()
ax.set_ylabel('EPS')
df['data'].plot(ax=ax)
ax.axvspan(test.index[0], test.index[-1], color='tab:orange', alpha=0.5)
plt.show()

url = 'https://raw.githubusercontent.com/marcopeix/TimeSeriesForecastingInPython/master/data/jj.csv' df = pd.read_csv(url, parse_dates=['date'], index_col='date') train = df.iloc[:-4] test = df.iloc[-4:] # plot fig, ax = plt.subplots() ax.set_ylabel('EPS') df['data'].plot(ax=ax) ax.axvspan(test.index[0], test.index[-1], color='tab:orange', alpha=0.5) plt.show()

In [41]:

for d in [0, 1, 2]:
    ADF_result = adfuller(np.diff(df['data'], n=d))
    print(f'EPS({d}次差分)ADF検定')
    print(f'ADF Statistic: {ADF_result[0]:.3f}')
    print(f'p-value: {ADF_result[1]:.3f}')

for d in [0, 1, 2]: ADF_result = adfuller(np.diff(df['data'], n=d)) print(f'EPS({d}次差分)ADF検定') print(f'ADF Statistic: {ADF_result[0]:.3f}') print(f'p-value: {ADF_result[1]:.3f}')

EPS(0次差分)ADF検定
ADF Statistic: 2.742
p-value: 1.000
EPS(1次差分)ADF検定
ADF Statistic: -0.407
p-value: 0.909
EPS(2次差分)ADF検定
ADF Statistic: -3.585
p-value: 0.006

In [42]:

# 一意なARIMA(p,q)モデルをすべて適合させる関数
def optimize_ARIMA(endog: Union[pd.Series, list], orders: list, d: int) -> pd.DataFrame:
    """
    endog: 時系列データ
    orders: ARIMA(p, d, q)のpとqの値の組み合わせ
    d: 和分次数
    """
    res = []
    for order in tqdm(orders):
        try:
            model = SARIMAX(
                endog, 
                order=(order[0], d, order[1]),
                simple_differencing=False
            ).fit(disp=False)
            res.append([order, model.aic])
        except:
            continue
    df_res = (
        pd.DataFrame(res, columns=['(p,q)', 'AIC'])
        .sort_values(by='AIC', ascending=True)
        .reset_index(drop=True)
    )
    return df_res

# 一意なARIMA(p,q)モデルをすべて適合させる関数 def optimize_ARIMA(endog: Union[pd.Series, list], orders: list, d: int) -> pd.DataFrame: """ endog: 時系列データ orders: ARIMA(p, d, q)のpとqの値の組み合わせ d: 和分次数 """ res = [] for order in tqdm(orders): try: model = SARIMAX( endog, order=(order[0], d, order[1]), simple_differencing=False ).fit(disp=False) res.append([order, model.aic]) except: continue df_res = ( pd.DataFrame(res, columns=['(p,q)', 'AIC']) .sort_values(by='AIC', ascending=True) .reset_index(drop=True) ) return df_res

In [43]:

d = 2 # 上のADF検定から和分次数は2に決定
orders = list(product(range(4), range(4))) # p,qをそれぞれ0~3まで動かす
optimize_ARIMA(df['data'], orders, 2)

d = 2 # 上のADF検定から和分次数は2に決定 orders = list(product(range(4), range(4))) # p,qをそれぞれ0~3まで動かす optimize_ARIMA(df['data'], orders, 2)

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/home/yoneda/.local/lib/python3.10/site-packages/statsmodels/tsa/base/tsa_model.py:473: ValueWarning: A date index has been provided, but it has no associated frequency information and so will be ignored when e.g. forecasting.
  self._init_dates(dates, freq)
/home/yoneda/.local/lib/python3.10/site-packages/statsmodels/tsa/base/tsa_model.py:473: ValueWarning: A date index has been provided, but it has no associated frequency information and so will be ignored when e.g. forecasting.
  self._init_dates(dates, freq)
/home/yoneda/.local/lib/python3.10/site-packages/statsmodels/tsa/base/tsa_model.py:473: ValueWarning: A date index has been provided, but it has no associated frequency information and so will be ignored when e.g. forecasting.
  self._init_dates(dates, freq)
/home/yoneda/.local/lib/python3.10/site-packages/statsmodels/tsa/base/tsa_model.py:473: ValueWarning: A date index has been provided, but it has no associated frequency information and so will be ignored when e.g. forecasting.
  self._init_dates(dates, freq)
/home/yoneda/.local/lib/python3.10/site-packages/statsmodels/tsa/base/tsa_model.py:473: ValueWarning: A date index has been provided, but it has no associated frequency information and so will be ignored when e.g. forecasting.
  self._init_dates(dates, freq)
/home/yoneda/.local/lib/python3.10/site-packages/statsmodels/tsa/base/tsa_model.py:473: ValueWarning: A date index has been provided, but it has no associated frequency information and so will be ignored when e.g. forecasting.
  self._init_dates(dates, freq)
/home/yoneda/.local/lib/python3.10/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.'
/home/yoneda/.local/lib/python3.10/site-packages/statsmodels/tsa/base/tsa_model.py:473: ValueWarning: A date index has been provided, but it has no associated frequency information and so will be ignored when e.g. forecasting.
  self._init_dates(dates, freq)
/home/yoneda/.local/lib/python3.10/site-packages/statsmodels/tsa/base/tsa_model.py:473: ValueWarning: A date index has been provided, but it has no associated frequency information and so will be ignored when e.g. forecasting.
  self._init_dates(dates, freq)
/home/yoneda/.local/lib/python3.10/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.'
/home/yoneda/.local/lib/python3.10/site-packages/statsmodels/tsa/base/tsa_model.py:473: ValueWarning: A date index has been provided, but it has no associated frequency information and so will be ignored when e.g. forecasting.
  self._init_dates(dates, freq)
/home/yoneda/.local/lib/python3.10/site-packages/statsmodels/tsa/base/tsa_model.py:473: ValueWarning: A date index has been provided, but it has no associated frequency information and so will be ignored when e.g. forecasting.
  self._init_dates(dates, freq)
/home/yoneda/.local/lib/python3.10/site-packages/statsmodels/tsa/base/tsa_model.py:473: ValueWarning: A date index has been provided, but it has no associated frequency information and so will be ignored when e.g. forecasting.
  self._init_dates(dates, freq)
/home/yoneda/.local/lib/python3.10/site-packages/statsmodels/tsa/base/tsa_model.py:473: ValueWarning: A date index has been provided, but it has no associated frequency information and so will be ignored when e.g. forecasting.
  self._init_dates(dates, freq)
/home/yoneda/.local/lib/python3.10/site-packages/statsmodels/tsa/base/tsa_model.py:473: ValueWarning: A date index has been provided, but it has no associated frequency information and so will be ignored when e.g. forecasting.
  self._init_dates(dates, freq)
/home/yoneda/.local/lib/python3.10/site-packages/statsmodels/tsa/base/tsa_model.py:473: ValueWarning: A date index has been provided, but it has no associated frequency information and so will be ignored when e.g. forecasting.
  self._init_dates(dates, freq)
/home/yoneda/.local/lib/python3.10/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.'
/home/yoneda/.local/lib/python3.10/site-packages/statsmodels/tsa/base/tsa_model.py:473: ValueWarning: A date index has been provided, but it has no associated frequency information and so will be ignored when e.g. forecasting.
  self._init_dates(dates, freq)
/home/yoneda/.local/lib/python3.10/site-packages/statsmodels/tsa/base/tsa_model.py:473: ValueWarning: A date index has been provided, but it has no associated frequency information and so will be ignored when e.g. forecasting.
  self._init_dates(dates, freq)
/home/yoneda/.local/lib/python3.10/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.'
/home/yoneda/.local/lib/python3.10/site-packages/statsmodels/tsa/base/tsa_model.py:473: ValueWarning: A date index has been provided, but it has no associated frequency information and so will be ignored when e.g. forecasting.
  self._init_dates(dates, freq)
/home/yoneda/.local/lib/python3.10/site-packages/statsmodels/tsa/base/tsa_model.py:473: ValueWarning: A date index has been provided, but it has no associated frequency information and so will be ignored when e.g. forecasting.
  self._init_dates(dates, freq)
/home/yoneda/.local/lib/python3.10/site-packages/statsmodels/tsa/base/tsa_model.py:473: ValueWarning: A date index has been provided, but it has no associated frequency information and so will be ignored when e.g. forecasting.
  self._init_dates(dates, freq)
/home/yoneda/.local/lib/python3.10/site-packages/statsmodels/tsa/base/tsa_model.py:473: ValueWarning: A date index has been provided, but it has no associated frequency information and so will be ignored when e.g. forecasting.
  self._init_dates(dates, freq)
/home/yoneda/.local/lib/python3.10/site-packages/statsmodels/tsa/statespace/sarimax.py:966: UserWarning: Non-stationary starting autoregressive parameters found. Using zeros as starting parameters.
  warn('Non-stationary starting autoregressive parameters'
/home/yoneda/.local/lib/python3.10/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.'
/home/yoneda/.local/lib/python3.10/site-packages/statsmodels/tsa/base/tsa_model.py:473: ValueWarning: A date index has been provided, but it has no associated frequency information and so will be ignored when e.g. forecasting.
  self._init_dates(dates, freq)
/home/yoneda/.local/lib/python3.10/site-packages/statsmodels/tsa/base/tsa_model.py:473: ValueWarning: A date index has been provided, but it has no associated frequency information and so will be ignored when e.g. forecasting.
  self._init_dates(dates, freq)
/home/yoneda/.local/lib/python3.10/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.'
/home/yoneda/.local/lib/python3.10/site-packages/statsmodels/tsa/base/tsa_model.py:473: ValueWarning: A date index has been provided, but it has no associated frequency information and so will be ignored when e.g. forecasting.
  self._init_dates(dates, freq)
/home/yoneda/.local/lib/python3.10/site-packages/statsmodels/tsa/base/tsa_model.py:473: ValueWarning: A date index has been provided, but it has no associated frequency information and so will be ignored when e.g. forecasting.
  self._init_dates(dates, freq)
/home/yoneda/.local/lib/python3.10/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.'
/home/yoneda/.local/lib/python3.10/site-packages/statsmodels/tsa/base/tsa_model.py:473: ValueWarning: A date index has been provided, but it has no associated frequency information and so will be ignored when e.g. forecasting.
  self._init_dates(dates, freq)
/home/yoneda/.local/lib/python3.10/site-packages/statsmodels/tsa/base/tsa_model.py:473: ValueWarning: A date index has been provided, but it has no associated frequency information and so will be ignored when e.g. forecasting.
  self._init_dates(dates, freq)
/home/yoneda/.local/lib/python3.10/site-packages/statsmodels/tsa/statespace/sarimax.py:966: UserWarning: Non-stationary starting autoregressive parameters found. Using zeros as starting parameters.
  warn('Non-stationary starting autoregressive parameters'
/home/yoneda/.local/lib/python3.10/site-packages/statsmodels/tsa/base/tsa_model.py:473: ValueWarning: A date index has been provided, but it has no associated frequency information and so will be ignored when e.g. forecasting.
  self._init_dates(dates, freq)
/home/yoneda/.local/lib/python3.10/site-packages/statsmodels/tsa/base/tsa_model.py:473: ValueWarning: A date index has been provided, but it has no associated frequency information and so will be ignored when e.g. forecasting.
  self._init_dates(dates, freq)
/home/yoneda/.local/lib/python3.10/site-packages/statsmodels/tsa/statespace/sarimax.py:966: UserWarning: Non-stationary starting autoregressive parameters found. Using zeros as starting parameters.
  warn('Non-stationary starting autoregressive parameters'
/home/yoneda/.local/lib/python3.10/site-packages/statsmodels/tsa/base/tsa_model.py:473: ValueWarning: A date index has been provided, but it has no associated frequency information and so will be ignored when e.g. forecasting.
  self._init_dates(dates, freq)
/home/yoneda/.local/lib/python3.10/site-packages/statsmodels/tsa/base/tsa_model.py:473: ValueWarning: A date index has been provided, but it has no associated frequency information and so will be ignored when e.g. forecasting.
  self._init_dates(dates, freq)
/home/yoneda/.local/lib/python3.10/site-packages/statsmodels/tsa/statespace/sarimax.py:966: UserWarning: Non-stationary starting autoregressive parameters found. Using zeros as starting parameters.
  warn('Non-stationary starting autoregressive parameters'
/home/yoneda/.local/lib/python3.10/site-packages/statsmodels/tsa/base/tsa_model.py:473: ValueWarning: A date index has been provided, but it has no associated frequency information and so will be ignored when e.g. forecasting.
  self._init_dates(dates, freq)
/home/yoneda/.local/lib/python3.10/site-packages/statsmodels/tsa/base/tsa_model.py:473: ValueWarning: A date index has been provided, but it has no associated frequency information and so will be ignored when e.g. forecasting.
  self._init_dates(dates, freq)
/home/yoneda/.local/lib/python3.10/site-packages/statsmodels/tsa/statespace/sarimax.py:966: UserWarning: Non-stationary starting autoregressive parameters found. Using zeros as starting parameters.
  warn('Non-stationary starting autoregressive parameters'
/home/yoneda/.local/lib/python3.10/site-packages/statsmodels/base/model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "

Out[43]:

	(p,q)	AIC
0	(3, 3)	117.898033
1	(3, 1)	119.338719
2	(3, 2)	119.504218
3	(3, 0)	160.819665
4	(0, 3)	218.838417
5	(0, 2)	241.466413
6	(2, 3)	246.507250
7	(1, 3)	254.710421
8	(1, 2)	258.390204
9	(2, 2)	258.974897
10	(2, 1)	268.110244
11	(1, 1)	268.199857
12	(0, 1)	299.164245
13	(1, 0)	316.439752
14	(2, 0)	316.822281
15	(0, 0)	382.017235

In [44]:

# (3, 3)モデルの残差分析を行う
# 残差分析
model = SARIMAX(train['data'], order=(3, d, 3), simple_differencing=False).fit(disp=False)
model.plot_diagnostics(figsize=(10, 8))
plt.show()

# (3, 3)モデルの残差分析を行う # 残差分析 model = SARIMAX(train['data'], order=(3, d, 3), simple_differencing=False).fit(disp=False) model.plot_diagnostics(figsize=(10, 8)) plt.show()

/home/yoneda/.local/lib/python3.10/site-packages/statsmodels/tsa/base/tsa_model.py:473: ValueWarning: A date index has been provided, but it has no associated frequency information and so will be ignored when e.g. forecasting.
  self._init_dates(dates, freq)
/home/yoneda/.local/lib/python3.10/site-packages/statsmodels/tsa/base/tsa_model.py:473: ValueWarning: A date index has been provided, but it has no associated frequency information and so will be ignored when e.g. forecasting.
  self._init_dates(dates, freq)
/home/yoneda/.local/lib/python3.10/site-packages/statsmodels/tsa/statespace/sarimax.py:966: UserWarning: Non-stationary starting autoregressive parameters found. Using zeros as starting parameters.
  warn('Non-stationary starting autoregressive parameters'
/home/yoneda/.local/lib/python3.10/site-packages/statsmodels/base/model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "

In [45]:

# 残差が無相関かの評価
acorr_ljungbox(model.resid, np.arange(1, 11))

# 残差が無相関かの評価 acorr_ljungbox(model.resid, np.arange(1, 11))

Out[45]:

	lb_stat	lb_pvalue
1	1.652020	0.198684
2	1.654544	0.437241
3	7.276173	0.063597
4	9.231226	0.055573
5	9.853425	0.079497
6	10.097125	0.120621
7	10.346926	0.169751
8	10.379005	0.239426
9	10.721222	0.295303
10	11.159917	0.345196

In [46]:

test['naive_seasonal'] = train.iloc[-4:].values
test['pred_ARIMA'] = model.get_prediction(train.shape[0], train.shape[0] + 3).predicted_mean.values

test['naive_seasonal'] = train.iloc[-4:].values test['pred_ARIMA'] = model.get_prediction(train.shape[0], train.shape[0] + 3).predicted_mean.values

/tmp/ipykernel_1794/416178706.py:1: SettingWithCopyWarning: 
A value is trying to be set on a copy of a slice from a DataFrame.
Try using .loc[row_indexer,col_indexer] = value instead

See the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy
  test['naive_seasonal'] = train.iloc[-4:].values
/home/yoneda/.local/lib/python3.10/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(
/home/yoneda/.local/lib/python3.10/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(
/tmp/ipykernel_1794/416178706.py:2: SettingWithCopyWarning: 
A value is trying to be set on a copy of a slice from a DataFrame.
Try using .loc[row_indexer,col_indexer] = value instead

See the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy
  test['pred_ARIMA'] = model.get_prediction(train.shape[0], train.shape[0] + 3).predicted_mean.values

In [51]:

fig, ax = plt.subplots()
df['data'].iloc[-20:].plot(ax=ax, label='actual')
ax.axvspan(test.index[0], test.index[-1], color='tab:orange', alpha=0.5)
methods = ['naive_seasonal', 'pred_ARIMA']
for method in methods:
    test[method].plot(ax=ax, label=method, ls='dashed')
ax.legend()

fig, ax = plt.subplots() df['data'].iloc[-20:].plot(ax=ax, label='actual') ax.axvspan(test.index[0], test.index[-1], color='tab:orange', alpha=0.5) methods = ['naive_seasonal', 'pred_ARIMA'] for method in methods: test[method].plot(ax=ax, label=method, ls='dashed') ax.legend()

Out[51]:

<matplotlib.legend.Legend at 0x7f0544ea64d0>

In [54]:

def mape(y_true, y_pred):
    return np.mean(np.abs((y_pred - y_true) / y_true))

plt.xlabel('Methods')
plt.ylabel('MAPE')
plt.bar(
    methods,
    [mape(test['data'], test[method]) for method in methods]
)
plt.gca().yaxis.set_major_formatter(lambda y, _: '{:.0%}'.format(y))
plt.show()

def mape(y_true, y_pred): return np.mean(np.abs((y_pred - y_true) / y_true)) plt.xlabel('Methods') plt.ylabel('MAPE') plt.bar( methods, [mape(test['data'], test[method]) for method in methods] ) plt.gca().yaxis.set_major_formatter(lambda y, _: '{:.0%}'.format(y)) plt.show()

In [ ]: