Prophetのモデルの中身
Prophetがどのようにしてモデルを作成しているのか、もう少し詳細に見てみたいと思います。
Taylor, Sean J., and Benjamin Letham. “Forecasting at scale.” The American Statistician 72.1 (2018): 37-45.
import numpy as np
import pandas as pd
import random
import seaborn as sns
import matplotlib.pyplot as plt
import japanize_matplotlib
from prophet import Prophet
実験に使用するデータ
１年間のデータを用意します。このデータには以下の特徴があります。
- 2020/1/1 ~ 2020/12/31 の期間のデータ
- 週ごとに周期的な数値を取る
- 土日は大きな値をとる
- 不定期にイベントが発生し、その時は非常に大きな値を取る(
horidays_indexに該当する日付を指しています)
date = pd.date_range("2020-01-01", periods=365, freq="D")
horidays_index = random.sample([i for i in range(365)], 10)
y = [
np.log(10 + i + 10 * np.cos(i)) # トレンド
+ np.cos(di.weekday() * np.pi / 28) * 3 # 週ごとに周期性あり
+ (di.weekday() in {5, 6}) * 0.5 # 〃
+ (i in horidays_index) * 2 # 祝日だけ数値を増やす
+ np.random.rand() / 10 # ノイズ
for i, di in enumerate(date)
]
df = pd.DataFrame({"ds": date, "y": y})
df.index = date
plt.title("サンプルデータ")
sns.lineplot(data=df)
plt.show()
トレンドの指定
growth=“linear”
# モデルを作成
m = Prophet(
yearly_seasonality=False,
weekly_seasonality=True,
daily_seasonality=False,
growth="linear",
)
m.fit(df)
# 将来を予測
future = m.make_future_dataframe(periods=90)
forecast = m.predict(future)
fig = m.plot(forecast)
plt.axvspan(18627, 18627 + 90, color="coral", alpha=0.4, label="予測期間")
plt.legend()
plt.show()
Initial log joint probability = -2.99674
Iter log prob ||dx|| ||grad|| alpha alpha0 # evals Notes
99 1040.17 0.000845008 95.4398 0.3546 0.3546 128
Iter log prob ||dx|| ||grad|| alpha alpha0 # evals Notes
166 1045.08 8.28451e-05 66.0092 1.087e-06 0.001 252 LS failed, Hessian reset
184 1045.09 1.58284e-05 61.1258 3.535e-07 0.001 307 LS failed, Hessian reset
199 1045.09 1.20349e-07 45.161 0.0243 1 330
Iter log prob ||dx|| ||grad|| alpha alpha0 # evals Notes
264 1045.71 0.000316932 140.641 3.665e-06 0.001 456 LS failed, Hessian reset
292 1045.82 5.70302e-05 48.9256 1.155e-06 0.001 533 LS failed, Hessian reset
299 1045.82 7.71897e-06 40.8043 0.3543 1 544
Iter log prob ||dx|| ||grad|| alpha alpha0 # evals Notes
305 1045.82 5.91182e-07 46.3158 2.336e-08 0.001 594 LS failed, Hessian reset
317 1045.82 2.63213e-06 51.4268 5.486e-08 0.001 649 LS failed, Hessian reset
343 1045.82 1.59952e-05 25.621 2.535e-07 0.001 728 LS failed, Hessian reset
353 1045.82 6.70942e-07 31.5639 1.825e-08 0.001 779 LS failed, Hessian reset
355 1045.82 3.11455e-08 40.1124 0.08424 1 783
Optimization terminated normally:
Convergence detected: relative gradient magnitude is below tolerance
growth=“flat”
# モデルを作成
m = Prophet(
yearly_seasonality=False,
weekly_seasonality=True,
daily_seasonality=False,
growth="flat",
)
m.fit(df)
# 将来を予測
future = m.make_future_dataframe(periods=90)
forecast = m.predict(future)
fig = m.plot(forecast)
plt.axvspan(18627, 18627 + 90, color="coral", alpha=0.4, label="予測期間")
plt.legend()
plt.show()
Initial log joint probability = -3.30635
Iter log prob ||dx|| ||grad|| alpha alpha0 # evals Notes
8 725.322 0.00030891 0.0789655 0.955 0.955 14
Optimization terminated normally:
Convergence detected: relative gradient magnitude is below tolerance
季節変化の指定
seasonalityが全てFalse
# モデルを作成
m = Prophet(
yearly_seasonality=False,
weekly_seasonality=False,
daily_seasonality=False,
growth="linear",
)
m.fit(df)
# 将来を予測
future = m.make_future_dataframe(periods=90)
forecast = m.predict(future)
fig = m.plot(forecast)
plt.axvspan(18627, 18627 + 90, color="coral", alpha=0.4, label="予測期間")
plt.legend()
plt.show()
Initial log joint probability = -2.99674
Iter log prob ||dx|| ||grad|| alpha alpha0 # evals Notes
99 1014.78 0.000250992 97.5427 1.468 0.01468 134
Iter log prob ||dx|| ||grad|| alpha alpha0 # evals Notes
199 1020 0.00045602 60.8104 1 1 268
Iter log prob ||dx|| ||grad|| alpha alpha0 # evals Notes
208 1020.1 0.000167489 76.8606 1.772e-06 0.001 314 LS failed, Hessian reset
299 1020.33 0.000353008 41.1849 2.43 0.243 446
Iter log prob ||dx|| ||grad|| alpha alpha0 # evals Notes
352 1020.86 8.23232e-05 43.9782 1.438e-06 0.001 586 LS failed, Hessian reset
399 1020.99 0.000244078 56.987 1 1 657
Iter log prob ||dx|| ||grad|| alpha alpha0 # evals Notes
488 1021.04 3.90795e-05 57.3821 6.974e-07 0.001 856 LS failed, Hessian reset
499 1021.04 1.23592e-05 41.4144 1 1 876
Iter log prob ||dx|| ||grad|| alpha alpha0 # evals Notes
510 1021.04 4.89524e-06 59.8544 1.187e-07 0.001 929 LS failed, Hessian reset
571 1021.06 1.29427e-06 39.655 2.478e-08 0.001 1071 LS failed, Hessian reset
598 1021.07 1.94887e-05 41.0613 3.113e-07 0.001 1157 LS failed, Hessian reset
599 1021.07 4.80514e-06 27.1307 0.643 0.643 1158
Iter log prob ||dx|| ||grad|| alpha alpha0 # evals Notes
645 1021.07 8.81062e-06 74.3981 1.365e-07 0.001 1264 LS failed, Hessian reset
660 1021.07 2.61181e-07 41.0723 5.687e-09 0.001 1321 LS failed, Hessian reset
661 1021.07 9.2387e-08 26.7092 0.8838 0.8838 1322
Optimization terminated normally:
Convergence detected: relative gradient magnitude is below tolerance
weekly_seasonality=True
# モデルを作成
m = Prophet(
yearly_seasonality=False,
weekly_seasonality=True,
daily_seasonality=False,
growth="linear",
)
m.fit(df)
# 将来を予測
future = m.make_future_dataframe(periods=90)
forecast = m.predict(future)
fig = m.plot(forecast)
plt.axvspan(18627, 18627 + 90, color="coral", alpha=0.4, label="予測期間")
plt.legend()
plt.show()
Initial log joint probability = -2.99674
Iter log prob ||dx|| ||grad|| alpha alpha0 # evals Notes
99 1040.17 0.000845008 95.4398 0.3546 0.3546 128
Iter log prob ||dx|| ||grad|| alpha alpha0 # evals Notes
166 1045.08 8.28451e-05 66.0092 1.087e-06 0.001 252 LS failed, Hessian reset
184 1045.09 1.58284e-05 61.1258 3.535e-07 0.001 307 LS failed, Hessian reset
199 1045.09 1.20349e-07 45.161 0.0243 1 330
Iter log prob ||dx|| ||grad|| alpha alpha0 # evals Notes
264 1045.71 0.000316932 140.641 3.665e-06 0.001 456 LS failed, Hessian reset
292 1045.82 5.70302e-05 48.9256 1.155e-06 0.001 533 LS failed, Hessian reset
299 1045.82 7.71897e-06 40.8043 0.3543 1 544
Iter log prob ||dx|| ||grad|| alpha alpha0 # evals Notes
305 1045.82 5.91182e-07 46.3158 2.336e-08 0.001 594 LS failed, Hessian reset
317 1045.82 2.63213e-06 51.4268 5.486e-08 0.001 649 LS failed, Hessian reset
343 1045.82 1.59952e-05 25.621 2.535e-07 0.001 728 LS failed, Hessian reset
353 1045.82 6.70942e-07 31.5639 1.825e-08 0.001 779 LS failed, Hessian reset
355 1045.82 3.11455e-08 40.1124 0.08424 1 783
Optimization terminated normally:
Convergence detected: relative gradient magnitude is below tolerance
yearly_seasonality=True
# モデルを作成
m = Prophet(
yearly_seasonality=True,
weekly_seasonality=True,
daily_seasonality=True,
growth="linear",
)
m.fit(df)
# 将来を予測
future = m.make_future_dataframe(periods=90)
forecast = m.predict(future)
fig = m.plot(forecast)
plt.axvspan(18627, 18627 + 90, color="coral", alpha=0.4, label="予測期間")
plt.legend()
plt.show()
Initial log joint probability = -2.99674
Iter log prob ||dx|| ||grad|| alpha alpha0 # evals Notes
99 1076.96 1.35095e-06 94.027 0.3775 0.3775 136
Iter log prob ||dx|| ||grad|| alpha alpha0 # evals Notes
125 1076.96 2.65686e-06 89.152 2.895e-08 0.001 203 LS failed, Hessian reset
164 1076.96 2.28191e-08 91.6592 0.4114 0.4114 258
Optimization terminated normally:
Convergence detected: relative gradient magnitude is below tolerance
休日・イベント効果
休日・イベントの指定なし(holidays=None)
# モデルを作成
m = Prophet(
yearly_seasonality=False,
weekly_seasonality=True,
daily_seasonality=False,
growth="linear",
holidays=None, # 休日・イベントの指定なし
)
m.fit(df)
# 将来を予測
future = m.make_future_dataframe(periods=90)
forecast = m.predict(future)
fig = m.plot(forecast)
plt.axvspan(18627, 18627 + 90, color="coral", alpha=0.4, label="予測期間")
plt.legend()
plt.show()
Initial log joint probability = -2.99674
Iter log prob ||dx|| ||grad|| alpha alpha0 # evals Notes
99 1040.17 0.000845008 95.4398 0.3546 0.3546 128
Iter log prob ||dx|| ||grad|| alpha alpha0 # evals Notes
166 1045.08 8.28451e-05 66.0092 1.087e-06 0.001 252 LS failed, Hessian reset
184 1045.09 1.58284e-05 61.1258 3.535e-07 0.001 307 LS failed, Hessian reset
199 1045.09 1.20349e-07 45.161 0.0243 1 330
Iter log prob ||dx|| ||grad|| alpha alpha0 # evals Notes
264 1045.71 0.000316932 140.641 3.665e-06 0.001 456 LS failed, Hessian reset
292 1045.82 5.70302e-05 48.9256 1.155e-06 0.001 533 LS failed, Hessian reset
299 1045.82 7.71897e-06 40.8043 0.3543 1 544
Iter log prob ||dx|| ||grad|| alpha alpha0 # evals Notes
305 1045.82 5.91182e-07 46.3158 2.336e-08 0.001 594 LS failed, Hessian reset
317 1045.82 2.63213e-06 51.4268 5.486e-08 0.001 649 LS failed, Hessian reset
343 1045.82 1.59952e-05 25.621 2.535e-07 0.001 728 LS failed, Hessian reset
353 1045.82 6.70942e-07 31.5639 1.825e-08 0.001 779 LS failed, Hessian reset
355 1045.82 3.11455e-08 40.1124 0.08424 1 783
Optimization terminated normally:
Convergence detected: relative gradient magnitude is below tolerance
休日・イベントの指定あり
# モデルを作成
df_holidays = pd.DataFrame(
{"holiday": "event", "ds": [di for i, di in enumerate(date) if i in horidays_index]}
)
m = Prophet(
yearly_seasonality=False,
weekly_seasonality=True,
daily_seasonality=False,
growth="linear",
holidays=df_holidays, # 休日・イベントの指定あり
)
m.fit(df)
# 将来を予測
future = m.make_future_dataframe(periods=90)
forecast = m.predict(future)
fig = m.plot(forecast)
plt.axvspan(18627, 18627 + 90, color="coral", alpha=0.4, label="予測期間")
plt.legend()
plt.show()
Initial log joint probability = -2.99674
Iter log prob ||dx|| ||grad|| alpha alpha0 # evals Notes
99 1343.59 0.00181622 336.302 0.9914 0.9914 120
Iter log prob ||dx|| ||grad|| alpha alpha0 # evals Notes
199 1366.61 0.00129331 915.041 0.2388 0.9805 232
Iter log prob ||dx|| ||grad|| alpha alpha0 # evals Notes
265 1372.99 0.000111116 289.963 4.926e-07 0.001 351 LS failed, Hessian reset
299 1374.36 0.00236219 89.8611 1 1 393
Iter log prob ||dx|| ||grad|| alpha alpha0 # evals Notes
399 1375.61 0.00707432 82.6526 1 1 509
Iter log prob ||dx|| ||grad|| alpha alpha0 # evals Notes
431 1375.93 8.82293e-05 240.658 4.05e-07 0.001 581 LS failed, Hessian reset
453 1376.02 6.12661e-05 95.4391 1.44e-06 0.001 644 LS failed, Hessian reset
471 1376.06 1.2062e-05 14.2904 2.239e-07 0.001 709 LS failed, Hessian reset
485 1376.1 1.9227e-05 69.5783 2.446e-07 0.001 772 LS failed, Hessian reset
493 1376.12 3.43778e-05 51.3583 1.873e-06 0.001 837 LS failed, Hessian reset
499 1376.13 2.95479e-05 34.8029 0.5524 0.5524 847
Iter log prob ||dx|| ||grad|| alpha alpha0 # evals Notes
509 1376.13 8.88412e-06 44.6026 5.452e-07 0.001 897 LS failed, Hessian reset
521 1376.14 1.6737e-05 27.0944 1.441e-07 0.001 947 LS failed, Hessian reset
531 1376.14 7.82708e-06 47.5548 1.997e-07 0.001 1000 LS failed, Hessian reset
567 1376.28 3.47594e-05 113.402 3.767e-07 0.001 1079 LS failed, Hessian reset
599 1376.35 0.000359311 95.2959 1 1 1118
Iter log prob ||dx|| ||grad|| alpha alpha0 # evals Notes
604 1376.35 4.84989e-06 13.344 1.606e-07 0.001 1168 LS failed, Hessian reset
612 1376.35 2.91053e-08 12.4075 0.04535 1 1191
Optimization terminated normally:
Convergence detected: relative gradient magnitude is below tolerance
