Non-Negative MinTrace

Large collections of time series organized into structures at different aggregation levels often require their forecasts to follow their aggregation constraints and to be nonnegative, which poses the challenge of creating novel algorithms capable of coherent forecasts.

The HierarchicalForecast package provides a wide collection of Python implementations of hierarchical forecasting algorithms that follow nonnegative hierarchical reconciliation.

In this notebook, we will show how to use the HierarchicalForecast package to perform nonnegative reconciliation of forecasts on Wiki2 dataset.

You can run these experiments using CPU or GPU with Google Colab.

!pip install hierarchicalforecast statsforecast datasetsforecast

1. Load Data

In this example we will use the Wiki2 dataset. The following cell gets the time series for the different levels in the hierarchy, the summing dataframe S_df which recovers the full dataset from the bottom level hierarchy and the indices of each hierarchy denoted by tags.

import numpy as np
import pandas as pd

from datasetsforecast.hierarchical import HierarchicalData

Y_df, S_df, tags = HierarchicalData.load('./data', 'Wiki2')
Y_df['ds'] = pd.to_datetime(Y_df['ds'])
S_df = S_df.reset_index(names="unique_id")

Y_df.head()

	unique_id	ds	y
0	Total	2016-01-01	156508
1	Total	2016-01-02	129902
2	Total	2016-01-03	138203
3	Total	2016-01-04	115017
4	Total	2016-01-05	126042

S_df.iloc[:5, :5]

	unique_id	de_AAC_AAG_001	de_AAC_AAG_010	de_AAC_AAG_014	de_AAC_AAG_045
0	Total	1	1	1	1
1	de	1	1	1	1
2	en	0	0	0	0
3	fr	0	0	0	0
4	ja	0	0	0	0

tags

{'Views': array(['Total'], dtype=object),
 'Views/Country': array(['de', 'en', 'fr', 'ja', 'ru', 'zh'], dtype=object),
 'Views/Country/Access': array(['de_AAC', 'de_DES', 'de_MOB', 'en_AAC', 'en_DES', 'en_MOB',
        'fr_AAC', 'fr_DES', 'fr_MOB', 'ja_AAC', 'ja_DES', 'ja_MOB',
        'ru_AAC', 'ru_DES', 'ru_MOB', 'zh_AAC', 'zh_DES', 'zh_MOB'],
       dtype=object),
 'Views/Country/Access/Agent': array(['de_AAC_AAG', 'de_AAC_SPD', 'de_DES_AAG', 'de_MOB_AAG',
        'en_AAC_AAG', 'en_AAC_SPD', 'en_DES_AAG', 'en_MOB_AAG',
        'fr_AAC_AAG', 'fr_AAC_SPD', 'fr_DES_AAG', 'fr_MOB_AAG',
        'ja_AAC_AAG', 'ja_AAC_SPD', 'ja_DES_AAG', 'ja_MOB_AAG',
        'ru_AAC_AAG', 'ru_AAC_SPD', 'ru_DES_AAG', 'ru_MOB_AAG',
        'zh_AAC_AAG', 'zh_AAC_SPD', 'zh_DES_AAG', 'zh_MOB_AAG'],
       dtype=object),
 'Views/Country/Access/Agent/Topic': array(['de_AAC_AAG_001', 'de_AAC_AAG_010', 'de_AAC_AAG_014',
        'de_AAC_AAG_045', 'de_AAC_AAG_063', 'de_AAC_AAG_100',
        'de_AAC_AAG_110', 'de_AAC_AAG_123', 'de_AAC_AAG_143',
        'de_AAC_SPD_012', 'de_AAC_SPD_074', 'de_AAC_SPD_080',
        'de_AAC_SPD_105', 'de_AAC_SPD_115', 'de_AAC_SPD_133',
        'de_DES_AAG_064', 'de_DES_AAG_116', 'de_DES_AAG_131',
        'de_MOB_AAG_015', 'de_MOB_AAG_020', 'de_MOB_AAG_032',
        'de_MOB_AAG_059', 'de_MOB_AAG_062', 'de_MOB_AAG_088',
        'de_MOB_AAG_095', 'de_MOB_AAG_109', 'de_MOB_AAG_122',
        'de_MOB_AAG_149', 'en_AAC_AAG_044', 'en_AAC_AAG_049',
        'en_AAC_AAG_075', 'en_AAC_AAG_114', 'en_AAC_AAG_119',
        'en_AAC_AAG_141', 'en_AAC_SPD_004', 'en_AAC_SPD_011',
        'en_AAC_SPD_026', 'en_AAC_SPD_048', 'en_AAC_SPD_067',
        'en_AAC_SPD_126', 'en_AAC_SPD_140', 'en_DES_AAG_016',
        'en_DES_AAG_024', 'en_DES_AAG_042', 'en_DES_AAG_069',
        'en_DES_AAG_082', 'en_DES_AAG_102', 'en_MOB_AAG_018',
        'en_MOB_AAG_022', 'en_MOB_AAG_101', 'en_MOB_AAG_124',
        'fr_AAC_AAG_029', 'fr_AAC_AAG_046', 'fr_AAC_AAG_070',
        'fr_AAC_AAG_087', 'fr_AAC_AAG_098', 'fr_AAC_AAG_104',
        'fr_AAC_AAG_111', 'fr_AAC_AAG_112', 'fr_AAC_AAG_142',
        'fr_AAC_SPD_025', 'fr_AAC_SPD_027', 'fr_AAC_SPD_035',
        'fr_AAC_SPD_077', 'fr_AAC_SPD_084', 'fr_AAC_SPD_097',
        'fr_AAC_SPD_130', 'fr_DES_AAG_023', 'fr_DES_AAG_043',
        'fr_DES_AAG_051', 'fr_DES_AAG_058', 'fr_DES_AAG_061',
        'fr_DES_AAG_091', 'fr_DES_AAG_093', 'fr_DES_AAG_094',
        'fr_DES_AAG_136', 'fr_MOB_AAG_006', 'fr_MOB_AAG_030',
        'fr_MOB_AAG_066', 'fr_MOB_AAG_117', 'fr_MOB_AAG_120',
        'fr_MOB_AAG_121', 'fr_MOB_AAG_135', 'fr_MOB_AAG_147',
        'ja_AAC_AAG_038', 'ja_AAC_AAG_047', 'ja_AAC_AAG_055',
        'ja_AAC_AAG_076', 'ja_AAC_AAG_099', 'ja_AAC_AAG_128',
        'ja_AAC_AAG_132', 'ja_AAC_AAG_134', 'ja_AAC_AAG_137',
        'ja_AAC_SPD_013', 'ja_AAC_SPD_034', 'ja_AAC_SPD_050',
        'ja_AAC_SPD_060', 'ja_AAC_SPD_078', 'ja_AAC_SPD_106',
        'ja_DES_AAG_079', 'ja_DES_AAG_081', 'ja_DES_AAG_113',
        'ja_MOB_AAG_065', 'ja_MOB_AAG_073', 'ja_MOB_AAG_092',
        'ja_MOB_AAG_127', 'ja_MOB_AAG_129', 'ja_MOB_AAG_144',
        'ru_AAC_AAG_008', 'ru_AAC_AAG_145', 'ru_AAC_AAG_146',
        'ru_AAC_SPD_000', 'ru_AAC_SPD_090', 'ru_AAC_SPD_148',
        'ru_DES_AAG_003', 'ru_DES_AAG_007', 'ru_DES_AAG_017',
        'ru_DES_AAG_041', 'ru_DES_AAG_071', 'ru_DES_AAG_072',
        'ru_MOB_AAG_002', 'ru_MOB_AAG_040', 'ru_MOB_AAG_083',
        'ru_MOB_AAG_086', 'ru_MOB_AAG_103', 'ru_MOB_AAG_107',
        'ru_MOB_AAG_118', 'ru_MOB_AAG_125', 'zh_AAC_AAG_021',
        'zh_AAC_AAG_033', 'zh_AAC_AAG_037', 'zh_AAC_AAG_052',
        'zh_AAC_AAG_057', 'zh_AAC_AAG_085', 'zh_AAC_AAG_108',
        'zh_AAC_SPD_039', 'zh_AAC_SPD_096', 'zh_DES_AAG_009',
        'zh_DES_AAG_019', 'zh_DES_AAG_053', 'zh_DES_AAG_054',
        'zh_DES_AAG_056', 'zh_DES_AAG_068', 'zh_DES_AAG_089',
        'zh_DES_AAG_139', 'zh_MOB_AAG_005', 'zh_MOB_AAG_028',
        'zh_MOB_AAG_031', 'zh_MOB_AAG_036', 'zh_MOB_AAG_138'], dtype=object)}

We split the dataframe in train/test splits.

Y_test_df = Y_df.groupby('unique_id', as_index=False).tail(7)
Y_train_df = Y_df.drop(Y_test_df.index)

2. Base Forecasts

The following cell computes the base forecast for each time series using the AutoETS model. Observe that Y_hat_df contains the forecasts but they are not coherent.

from statsforecast.models import AutoETS, Naive
from statsforecast.core import StatsForecast

fcst = StatsForecast(
    models=[AutoETS(season_length=7, model='ZAA'), Naive()], 
    freq='D', 
    n_jobs=-1
)
Y_hat_df = fcst.forecast(df=Y_train_df, h=7)

Observe that the AutoETS model computes negative forecasts for some series.

Y_hat_df.query('AutoETS < 0')

	unique_id	ds	AutoETS	Naive
28	de_AAC_AAG_001	2016-12-25	-523.766907	340.0
29	de_AAC_AAG_001	2016-12-26	-245.337433	340.0
30	de_AAC_AAG_001	2016-12-27	-194.253815	340.0
33	de_AAC_AAG_001	2016-12-30	-315.425659	340.0
34	de_AAC_AAG_001	2016-12-31	-806.920105	340.0
…	…	…	…	…
1217	zh_AAC_AAG_033	2016-12-31	-86.466789	37.0
1345	zh_MOB	2016-12-26	-199.534882	1036.0
1346	zh_MOB	2016-12-27	-69.527260	1036.0
1352	zh_MOB_AAG	2016-12-26	-199.534882	1036.0
1353	zh_MOB_AAG	2016-12-27	-69.527260	1036.0

3. Non-Negative Reconciliation

The following cell makes the previous forecasts coherent and nonnegative using the HierarchicalReconciliation class.

from hierarchicalforecast.methods import MinTrace
from hierarchicalforecast.core import HierarchicalReconciliation

reconcilers = [
    MinTrace(method='ols'),
    MinTrace(method='ols', nonnegative=True)
]
hrec = HierarchicalReconciliation(reconcilers=reconcilers)
Y_rec_df = hrec.reconcile(Y_hat_df=Y_hat_df, Y_df=Y_train_df,
                          S=S_df, tags=tags)

Observe that the nonnegative reconciliation method obtains nonnegative forecasts.

Y_rec_df

	unique_id	ds	AutoETS	Naive	AutoETS/MinTrace_method-ols	Naive/MinTrace_method-ols	AutoETS/MinTrace_method-ols_nonnegative-True	Naive/MinTrace_method-ols_nonnegative-True
0	Total	2016-12-25	94523.164062	95743.0	95852.000421	95743.0	9.664245e+04	95743.0
1	Total	2016-12-26	87734.367188	95743.0	89525.238276	95743.0	9.028857e+04	95743.0
2	Total	2016-12-27	87751.125000	95743.0	89638.119184	95743.0	9.056593e+04	95743.0
3	Total	2016-12-28	133237.968750	95743.0	131051.839057	95743.0	1.314028e+05	95743.0
4	Total	2016-12-29	126501.796875	95743.0	121214.048604	95743.0	1.218000e+05	95743.0
…	…	…	…	…	…	…	…	…
1388	zh_MOB_AAG_138	2016-12-27	62.049744	65.0	-147.399760	65.0	0.000000e+00	65.0
1389	zh_MOB_AAG_138	2016-12-28	54.934032	65.0	7.561682	65.0	4.397229e-15	65.0
1390	zh_MOB_AAG_138	2016-12-29	60.452618	65.0	114.253489	65.0	9.321380e+01	65.0
1391	zh_MOB_AAG_138	2016-12-30	50.356693	65.0	96.446754	65.0	7.565171e+01	65.0
1392	zh_MOB_AAG_138	2016-12-31	66.735626	65.0	208.184648	65.0	1.851130e+02	65.0

Y_rec_df.query('`AutoETS/MinTrace_method-ols_nonnegative-True` < 0')

	unique_id	ds	AutoETS	Naive	AutoETS/MinTrace_method-ols	Naive/MinTrace_method-ols	AutoETS/MinTrace_method-ols_nonnegative-True	Naive/MinTrace_method-ols_nonnegative-True

The free reconciliation method gets negative forecasts.

Y_rec_df.query('`AutoETS/MinTrace_method-ols` < 0')

	unique_id	ds	AutoETS	Naive	AutoETS/MinTrace_method-ols	Naive/MinTrace_method-ols	AutoETS/MinTrace_method-ols_nonnegative-True	Naive/MinTrace_method-ols_nonnegative-True
56	de_DES	2016-12-25	-2553.932861	495.0	-3818.990043	495.0	0.000000e+00	495.0
57	de_DES	2016-12-26	-2155.228271	495.0	-3309.806933	495.0	1.909922e-30	495.0
58	de_DES	2016-12-27	-2720.993896	495.0	-3965.351121	495.0	1.140223e-13	495.0
60	de_DES	2016-12-29	-3429.432617	495.0	-3042.502484	495.0	3.049601e+02	495.0
61	de_DES	2016-12-30	-3963.202637	495.0	-3476.273292	495.0	2.877829e+02	495.0
…	…	…	…	…	…	…	…	…
1380	zh_MOB_AAG_036	2016-12-26	75.298317	115.0	-166.245228	115.0	0.000000e+00	115.0
1381	zh_MOB_AAG_036	2016-12-27	72.895554	115.0	-136.553950	115.0	1.699002e-14	115.0
1386	zh_MOB_AAG_138	2016-12-25	94.796623	65.0	-49.410174	65.0	0.000000e+00	65.0
1387	zh_MOB_AAG_138	2016-12-26	71.293983	65.0	-170.249562	65.0	0.000000e+00	65.0
1388	zh_MOB_AAG_138	2016-12-27	62.049744	65.0	-147.399760	65.0	0.000000e+00	65.0

4. Evaluation

The HierarchicalForecast package includes the evaluate function to evaluate the different hierarchies. We use utilsforecast to compute the mean absolute error.

from hierarchicalforecast.evaluation import evaluate
from utilsforecast.losses import mse

evaluation = evaluate(df = Y_rec_df.merge(Y_test_df, on=['unique_id', 'ds']),
                      tags = tags,
                      train_df = Y_train_df,
                      metrics = [mse],
                      benchmark="Naive")

evaluation.set_index(["level", "metric"]).filter(like='ETS')

		AutoETS	AutoETS/MinTrace_method-ols	AutoETS/MinTrace_method-ols_nonnegative-True
level	metric
Views	mse-scaled	0.735800	0.697371	0.675672
Views/Country	mse-scaled	1.190354	1.053631	0.994758
Views/Country/Access	mse-scaled	1.086102	1.133507	1.172270
Views/Country/Access/Agent	mse-scaled	1.067394	1.100215	1.127960
Views/Country/Access/Agent/Topic	mse-scaled	1.435105	1.381990	1.163428
Overall	mse-scaled	1.010801	0.977667	0.939286

Observe that the nonnegative reconciliation method performs better (lower error) than its unconstrained counterpart.

Getting Started

Tutorials

API Reference

Non-Negative MinTrace

1. Load Data

2. Base Forecasts

3. Non-Negative Reconciliation

4. Evaluation

References

Getting Started

Tutorials

API Reference

​1. Load Data

​2. Base Forecasts

​3. Non-Negative Reconciliation

​4. Evaluation

​References

1. Load Data

2. Base Forecasts

3. Non-Negative Reconciliation

4. Evaluation

References