Introduction

In machine learning, stacking is an ensemble technique that combines multiple models to reduce their biases and improve predictive performance. More specifically, the predictions of each model (base models) are stacked and used as input to a final model (meta model) to compute the prediction.

Stacking is effective because it leverages the strengths of different algorithms and attempts to mitigate their individual weaknesses. By combining several models, it can capture complex patterns in the data and improve prediction accuracy.

However, stacking can be computationally expensive and requires careful tuning to avoid overfitting. To this end, it is highly recommended to train the final estimator through cross-validation. In addition, obtaining diverse and well-performing base models is critical to the success of the stacking technique.

The following example shows how to use scikit-learn and skforecast to create a forecasting model that combines several individual regressors to achieve better results.

Libraries

Libraries used in this document.

# Data processing
# ==============================================================================
import numpy as np
import pandas as pd

# Plots
# ==============================================================================
import matplotlib.pyplot as plt
import plotly.express as px
import plotly.io as pio
import plotly.offline as poff
pio.templates.default = "seaborn"
poff.init_notebook_mode(connected=True)
plt.style.use('seaborn-v0_8-darkgrid')

# Modelling and Forecasting
# ==============================================================================
from lightgbm import LGBMRegressor
from sklearn.linear_model import Ridge
from sklearn.ensemble  import StackingRegressor
from sklearn.model_selection  import KFold
from sklearn.preprocessing  import StandardScaler
import skforecast
from skforecast.recursive import ForecasterRecursive
from skforecast.model_selection import TimeSeriesFold
from skforecast.model_selection import grid_search_forecaster
from skforecast.model_selection import backtesting_forecaster
from skforecast.datasets import fetch_dataset

# Configuration warnings
# ==============================================================================
import warnings
warnings.filterwarnings('once')

color = '\033[1m\033[38;5;208m' 
print(f"{color}Version skforecast: {skforecast.__version__}")

Version skforecast: 0.15.1

Data

The data in this document represent monthly fuel consumption in Spain from 1969-01-01 to 2022-08-01. The goal is to create a model capable of forecasting the consumption over the next 12 month.

# Downloading data
# ==============================================================================
data = fetch_dataset(name = 'fuel_consumption')
data = data.loc[:"2019-01-01", ['Gasolinas']]
data = data.rename(columns = {'Gasolinas':'consumption'})
data.index.name = 'date'
data['consumption'] = data['consumption']/100000
data.head(3)

fuel_consumption
----------------
Monthly fuel consumption in Spain from 1969-01-01 to 2022-08-01.
Obtained from Corporación de Reservas Estratégicas de Productos Petrolíferos and
Corporación de Derecho Público tutelada por el Ministerio para la Transición
Ecológica y el Reto Demográfico. https://www.cores.es/es/estadisticas
Shape of the dataset: (644, 5)

	consumption
date
1969-01-01	1.668752
1969-02-01	1.554668
1969-03-01	1.849837

In addition to the past values of the series (lags), an additional variable indicating the month of the year is added. This variable is included in the model to capture the seasonality of the series.

# Calendar features
# ==============================================================================
data['month_of_year'] = data.index.month
data.head(3)

	consumption	month_of_year
date
1969-01-01	1.668752	1
1969-02-01	1.554668	2
1969-03-01	1.849837	3

To facilitate the training of the models, the search for optimal hyperparameters, and the evaluation of their predictive accuracy, the data are divided into three separate sets: training, validation, and test.

# Split train-validation-test
# ==============================================================================
end_train = '2007-12-01 23:59:00'
end_validation = '2012-12-01 23:59:00'
data_train = data.loc[: end_train, :]
data_val   = data.loc[end_train:end_validation, :]
data_test  = data.loc[end_validation:, :]

print(f"Dates train      : {data_train.index.min()} --- {data_train.index.max()}  (n={len(data_train)})")
print(f"Dates validacion : {data_val.index.min()} --- {data_val.index.max()}  (n={len(data_val)})")
print(f"Dates test       : {data_test.index.min()} --- {data_test.index.max()}  (n={len(data_test)})")

Dates train      : 1969-01-01 00:00:00 --- 2007-12-01 00:00:00  (n=468)
Dates validacion : 2008-01-01 00:00:00 --- 2012-12-01 00:00:00  (n=60)
Dates test       : 2013-01-01 00:00:00 --- 2019-01-01 00:00:00  (n=73)

# Interactive plot of time series
# ==============================================================================
data.loc[:end_train, 'partition'] = 'train'
data.loc[end_train:end_validation, 'partition'] = 'validation'
data.loc[end_validation:, 'partition'] = 'test'

fig = px.line(
    data_frame = data.reset_index(),
    x      = 'date',
    y      = 'consumption',
    color  = 'partition',
    title  = 'Fuel consumption',
    width  = 700,
    height = 350,
)
fig.update_layout(
    width  = 700,
    height = 350,
    margin=dict(l=20, r=20, t=35, b=20),
    legend=dict(
        orientation="h",
        yanchor="top",
        y=1,
        xanchor="left",
        x=0.001
    )
)
fig.show()
data = data.drop(columns='partition')

Individual models

First, two individual models are trained separately - a linear regression model and a gradient boosting model - and their performance is evaluated on the test set.

LightGBM

# Create forecaster
# ==============================================================================
params_lgbm = {'learning_rate': 0.1, 'max_depth': 5, 'n_estimators': 500, 'verbose': -1}
forecaster = ForecasterRecursive(
                 regressor = LGBMRegressor(random_state=123, **params_lgbm),
                 lags = 12
             )

# Backtesting on test data
# ==============================================================================
cv = TimeSeriesFold(
      steps              = 12,
      initial_train_size = len(data.loc[:end_validation]),
      fixed_train_size   = False,
    )
metric, predictions = backtesting_forecaster(
                            forecaster = forecaster,
                            y          = data['consumption'],
                            exog       = data['month_of_year'],
                            cv         = cv,
                            metric     = 'mean_squared_error'
                      )        
metric

  0%|          | 0/7 [00:00<?, ?it/s]

	mean_squared_error
0	0.066196

Linear model

# Create forecaster
# ==============================================================================
params_ridge = {'alpha': 0.001}
forecaster = ForecasterRecursive(
                 regressor = Ridge(random_state=123, **params_ridge),
                 lags = 12,
                 transformer_y = StandardScaler()
             )

# Backtesting on test data
# ==============================================================================
metric, predictions = backtesting_forecaster(
                            forecaster = forecaster,
                            y          = data['consumption'],
                            exog       = data['month_of_year'],
                            cv         = cv,
                            metric     = 'mean_squared_error'
                      )        
metric

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	mean_squared_error
0	0.049092

StackingRegressor

With scikit-learn it is very easy to combine multiple regressors thanks to its StackingRegressor class.

The estimators parameter corresponds to the list of the estimators (base learners) which are stacked together in parallel on the input data. It should be given as a list of names and estimators. The final_estimator (meta model) will use the predictions of the estimators as input.

# Create stacking regressor
# ==============================================================================
estimators = [
    ('ridge', Ridge(random_state=123, **params_ridge)),
    ('lgbm', LGBMRegressor(random_state=123, **params_lgbm)),
]
stacking_regressor = StackingRegressor(
                        estimators = estimators,
                        final_estimator = Ridge(),
                        cv = KFold(n_splits=10, shuffle=False)
                    )
stacking_regressor

StackingRegressor(cv=KFold(n_splits=10, random_state=None, shuffle=False),
                  estimators=[('ridge', Ridge(alpha=0.001, random_state=123)),
                              ('lgbm',
                               LGBMRegressor(max_depth=5, n_estimators=500,
                                             random_state=123, verbose=-1))],
                  final_estimator=Ridge())

In a Jupyter environment, please rerun this cell to show the HTML representation or trust the notebook.
On GitHub, the HTML representation is unable to render, please try loading this page with nbviewer.org.

# Create forecaster
# ==============================================================================
forecaster = ForecasterRecursive(
                 regressor = stacking_regressor,
                 lags = 12
             )

# Backtesting on test data
# ==============================================================================
metric, predictions = backtesting_forecaster(
                            forecaster = forecaster,
                            y          = data['consumption'],
                            exog       = data['month_of_year'],
                            cv         = cv,
                            metric     = 'mean_squared_error'
                      )        
metric

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	mean_squared_error
0	0.044688

The results obtained by stacking the two models - the linear model and the gradient boosting model - are better than the results obtained by each model separately.

Hiperparameters search

When using StackingRegressor, the hyperparameters of individual regressors must be prefixed with the name of the regressor followed by two underlines. For example, the hyperparameter alpha of the Ridge regressor must be specified as ridge__alpha. The hyperparameter of the final estimator must be specified with the final_estimator__ prefix.

# Grid search of hyperparameters and lags
# ==============================================================================
param_grid = {
    'ridge__alpha': [0.001, 0.01, 0.1, 1, 10],
    'lgbm__n_estimators': [100, 500],
    'lgbm__max_depth': [3, 5, 10],
    'lgbm__learning_rate': [0.01, 0.1],
}

lags_grid = [24]

cv_Search = TimeSeriesFold(
      steps              = 12,
      initial_train_size = len(data.loc[:end_train]),
      fixed_train_size   = False,
    )

results_grid = grid_search_forecaster(
                   forecaster  = forecaster,
                   y           = data.loc[:end_validation, 'consumption'],
                   exog        = data.loc[:end_validation, 'month_of_year'],
                   param_grid  = param_grid,
                   lags_grid   = lags_grid,
                   cv          = cv_Search,
                   metric      = 'mean_squared_error',
               )

results_grid.head()

lags grid:   0%|          | 0/1 [00:00<?, ?it/s]

params grid:   0%|          | 0/60 [00:00<?, ?it/s]

`Forecaster` refitted using the best-found lags and parameters, and the whole data set: 
  Lags: [ 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24] 
  Parameters: {'lgbm__learning_rate': 0.1, 'lgbm__max_depth': 5, 'lgbm__n_estimators': 100, 'ridge__alpha': 0.001}
  Backtesting metric: 0.07934431616061044

	lags	lags_label	params	mean_squared_error	lgbm__learning_rate	lgbm__max_depth	lgbm__n_estimators	ridge__alpha
0	[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14...	[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14...	{'lgbm__learning_rate': 0.1, 'lgbm__max_depth'...	0.079344	0.10	5.0	100.0	0.001
1	[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14...	[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14...	{'lgbm__learning_rate': 0.1, 'lgbm__max_depth'...	0.079360	0.10	5.0	100.0	0.010
2	[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14...	[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14...	{'lgbm__learning_rate': 0.01, 'lgbm__max_depth...	0.079488	0.01	3.0	500.0	0.001
3	[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14...	[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14...	{'lgbm__learning_rate': 0.01, 'lgbm__max_depth...	0.079504	0.01	3.0	500.0	0.010
4	[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14...	[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14...	{'lgbm__learning_rate': 0.1, 'lgbm__max_depth'...	0.079512	0.10	5.0	100.0	0.100

Once the best hyperparameters have been determined for each regressor in the ensemble, the test error is computed through back-testing.

# Backtesting the best model on test data
# ==============================================================================
metric, predictions = backtesting_forecaster(
                            forecaster = forecaster,
                            y          = data['consumption'],
                            exog       = data['month_of_year'],
                            cv         = cv,
                            metric     = 'mean_squared_error'
                      )        
metric

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	mean_squared_error
0	0.012739

Feature importance

When a regressor of type StackingRegressor is used as a regressor in a predictor, its get_feature_importances method will not work. This is because objects of type StackingRegressor do not have either the feature_importances or coef_ attribute. Instead, it is necessary to inspect each of the regressors that are part of the stacking.

# Feature importances for each regressor in the stacking
# ==============================================================================
if forecaster.regressor.__class__.__name__ == 'StackingRegressor':
    importancia_pred = []
    for regressor in forecaster.regressor.estimators_:
        try:
            importancia = pd.DataFrame(
                data = {
                    'feature': forecaster.regressor.feature_names_in_,
                    f'importance_{type(regressor).__name__}': regressor.coef_,
                    f'importance_abs_{type(regressor).__name__}': np.abs(regressor.coef_)
                }
            ).set_index('feature')
        except:
            importancia = pd.DataFrame(
                data = {
                    'feature': forecaster.regressor.feature_names_in_,
                    f'importance_{type(regressor).__name__}': regressor.feature_importances_,
                    f'importance_abs_{type(regressor).__name__}': np.abs(regressor.feature_importances_)
                }
            ).set_index('feature')
        importancia_pred.append(importancia)
    
    importancia_pred = pd.concat(importancia_pred, axis=1)
    
else:
    importancia_pred = forecaster.get_feature_importances()
    importancia_pred['importance_abs'] = importancia_pred['importance'].abs()
    importancia_pred = importancia_pred.sort_values(by='importance_abs', ascending=False)

importancia_pred.head(5)

	importance_Ridge	importance_abs_Ridge	importance_LGBMRegressor	importance_abs_LGBMRegressor
feature
lag_1	0.016128	0.016128	58	58
lag_2	0.226245	0.226245	63	63
lag_3	0.185437	0.185437	44	44
lag_4	0.206009	0.206009	54	54
lag_5	0.105056	0.105056	38	38

Session information

import session_info
session_info.show(html=False)

-----
lightgbm            4.6.0
matplotlib          3.10.1
numpy               2.0.2
pandas              2.2.3
plotly              5.23.0
session_info        1.0.0
skforecast          0.15.1
sklearn             1.5.2
-----
IPython             9.0.2
jupyter_client      8.6.3
jupyter_core        5.7.2
notebook            6.4.12
-----
Python 3.12.9 | packaged by Anaconda, Inc. | (main, Feb  6 2025, 18:56:27) [GCC 11.2.0]
Linux-5.15.0-1077-aws-x86_64-with-glibc2.31
-----
Session information updated at 2025-03-27 11:40

Citation

How to cite this document

If you use this document or any part of it, please acknowledge the source, thank you!

Stacking ensemble of machine learning models to improve forecasting by Joaquín Amat Rodrigo and Javier Escobar Ortiz, available under Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0 DEED) at https://www.cienciadedatos.net/documentos/py52-stacking-ensemble-models-forecasting.html

How to cite skforecast

If you use skforecast for a publication, we would appreciate it if you cite the published software.

Zenodo:

Amat Rodrigo, Joaquin, & Escobar Ortiz, Javier. (2024). skforecast (v0.15.1). Zenodo. https://doi.org/10.5281/zenodo.8382788

APA:

Amat Rodrigo, J., & Escobar Ortiz, J. (2024). skforecast (Version 0.15.1) [Computer software]. https://doi.org/10.5281/zenodo.8382788

BibTeX:

@software{skforecast, author = {Amat Rodrigo, Joaquin and Escobar Ortiz, Javier}, title = {skforecast}, version = {0.15.1}, month = {03}, year = {2025}, license = {BSD-3-Clause}, url = {https://skforecast.org/}, doi = {10.5281/zenodo.8382788} }

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Stacking ensemble of machine learning models to improve forecasting

Joaquín Amat Rodrigo, Javier Escobar Ortiz

November, 2023 (last update March 2025)