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Global Forecasting Models: Modeling Multiple Time Series with Machine Learning
More about forecasting
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- Forecasting time series with missing values
Introduction¶
In Single-Series Modeling (Local Forecasting Model), each time series is analyzed individually, modeled as a combination of its own lags and, optionally, exogenous variables. This approach provides detailed insights specific to each series but can become impractical for scaling when dealing with a large number of time series. In contrast, Multi-Series Modeling (Global Forecasting Model) involves building a unified predictive model that learns multiple time series simultaneously. It attempts to capture common dynamics that influence the series as a whole, thereby reducing noise from individual series. This approach is computationally efficient, easy to maintain, and can yield more robust generalizations across time series, albeit potentially at the cost of sacrificing some individual insights. Two strategies of global forecasting models can be distinguished: Independent multi-series and dependent multi-series.
Advantages of multi-series:
It is easier to maintain and monitor a single model than several.
Since all time series are combined during training, the model has a higher learning capacity even if the series are short.
By combining multiple time series, the model can learn more generalizable patterns.
Limitations of multi-series:
If the series do not follow the same internal dynamics, the model may learn a pattern that does not represent any of them.
The series may mask each other, so the model may not predict all of them with the same performance.
It is more computationally demanding (time and resources) to train and backtest a big model than several small ones.
Dependent multi-series forecasting (Multivariate forecasting)
In dependent multi-series forecasting (multivariate time series), all series are modeled together in a single model, considering that each time series depends not only on its past values but also on the past values of the other series. The forecaster is expected not only to learn the information of each series separately but also to relate them. An example is the measurements made by all the sensors (flow, temperature, pressure...) installed on an industrial machine such as a compressor.
💡 Tip
This is the first in a series of documents on global forecasting models:Multiple time series with equal length¶
In this first example, multiple time series with the same length are used. The goal is to compare the forecasting results of a global model with those of an individual model for each series when forecasting the next 7 days of sales for 50 different items in a store using the 5 years of available history. The data has been obtained from the Store Item Demand Forecasting Challenge. This dataset contains 913,000 sales transactions from 01/01/2013 to 31/12/2017 for 50 products (SKU) in 10 stores.
Libraries¶
# Data manipulation
# ==============================================================================
import numpy as np
import pandas as pd
# Plots
# ==============================================================================
import matplotlib.pyplot as plt
from tqdm.notebook import tqdm
# Modelling and Forecasting
# ==============================================================================
import sklearn
import skforecast
from sklearn.ensemble import HistGradientBoostingRegressor
from sklearn.preprocessing import StandardScaler
from sklearn.feature_selection import RFECV
from skforecast.recursive import ForecasterRecursiveMultiSeries
from skforecast.recursive import ForecasterRecursive
from skforecast.model_selection import TimeSeriesFold, OneStepAheadFold
from skforecast.model_selection import backtesting_forecaster
from skforecast.model_selection import bayesian_search_forecaster
from skforecast.model_selection import backtesting_forecaster_multiseries
from skforecast.model_selection import bayesian_search_forecaster_multiseries
from skforecast.feature_selection import select_features_multiseries
from skforecast.preprocessing import RollingFeatures
from skforecast.exceptions import OneStepAheadValidationWarning
from skforecast.plot import set_dark_theme
from skforecast.preprocessing import series_long_to_dict
from skforecast.preprocessing import exog_long_to_dict
# Warnings configuration
# ==============================================================================
import warnings
color = '\033[1m\033[38;5;208m'
print(f"{color}Version skforecast: {skforecast.__version__}")
print(f"{color}Version scikit-learn: {sklearn.__version__}")
print(f"{color}Version pandas: {pd.__version__}")
print(f"{color}Version numpy: {np.__version__}")
Data¶
# Data loading
# ======================================================================================
data = pd.read_csv('./train_stores_kaggle.csv')
display(data)
print(f"Shape: {data.shape}")
# Data preprocessing
# ======================================================================================
selected_store = 2
selected_items = data.item.unique()
data = data[(data['store'] == selected_store) & (data['item'].isin(selected_items))].copy()
data['date'] = pd.to_datetime(data['date'], format='%Y-%m-%d')
data = pd.pivot_table(
data = data,
values = 'sales',
index = 'date',
columns = 'item'
)
data.columns.name = None
data.columns = [f"item_{col}" for col in data.columns]
data = data.asfreq('1D')
data = data.sort_index()
data.head(4)
The dataset is divided into 3 partitions: one for training, one for validation, and one for testing.
# Split data into train-validation-test
# ======================================================================================
end_train = '2016-05-31 23:59:00'
end_val = '2017-05-31 23:59:00'
data_train = data.loc[:end_train, :].copy()
data_val = data.loc[end_train:end_val, :].copy()
data_test = data.loc[end_val:, :].copy()
print(f"Train dates : {data_train.index.min()} --- {data_train.index.max()} (n={len(data_train)})")
print(f"Validation dates : {data_val.index.min()} --- {data_val.index.max()} (n={len(data_val)})")
print(f"Test dates : {data_test.index.min()} --- {data_test.index.max()} (n={len(data_test)})")
Four of the series are plotted to understand their trends and patterns. The reader is strongly encouraged to plot several more to gain an in-depth understanding of the series.
# Plot time series
# ======================================================================================
set_dark_theme()
fig, axs = plt.subplots(4, 1, figsize=(7, 5), sharex=True)
data.iloc[:, :4].plot(
legend = True,
subplots = True,
title = 'Sales of store 2',
ax = axs,
)
for ax in axs:
ax.axvline(pd.to_datetime(end_train) , color='white', linestyle='--', linewidth=1.5)
ax.axvline(pd.to_datetime(end_val) , color='white', linestyle='--', linewidth=1.5)
fig.tight_layout()
plt.show()
Individual Forecaster for each product¶
A separate Gradient Boosting Machine (GBM) model is trained for each item, using sales over the last 14 days, as well as the average, maximum and minimum sales over the last 7 days as predictive features. The performance of the model over the next 7 days is evaluated using backtesting with the Mean Absolute Error (MAE) as the evaluation metric. Finally, the performance of these individual models is compared with that of a global model trained on all series.
# Train and backtest a model for each item: ForecasterAutoreg
# ======================================================================================
items = []
mae_values = []
predictions = {}
for i, item in enumerate(tqdm(data.columns)):
# Define forecaster
window_features = RollingFeatures(stats=['mean', 'min', 'max'], window_sizes=7)
forecaster = ForecasterRecursive(
regressor = HistGradientBoostingRegressor(random_state=8523),
lags = 14,
window_features = window_features
)
# Backtesting forecaster
cv = TimeSeriesFold(
steps = 7,
initial_train_size = len(data_train) + len(data_val),
refit = False,
)
metric, preds = backtesting_forecaster(
forecaster = forecaster,
y = data[item],
cv = cv,
metric = 'mean_absolute_error',
verbose = False,
show_progress = False
)
items.append(item)
mae_values.append(metric.at[0, 'mean_absolute_error'])
predictions[item] = preds
# Results
uni_series_mae = pd.Series(
data = mae_values,
index = items,
name = 'uni_series_mae'
)
uni_series_mae.head()
Global model¶
A global model is trained on all series simultaneously and evaluated using the same backtesting process as the individual models.
# Train and backtest a model for all items: ForecasterAutoregMultiSeries
# ======================================================================================
items = list(data.columns)
# Define forecaster
window_features = RollingFeatures(stats=['mean', 'min', 'max'], window_sizes=7)
forecaster_ms = ForecasterRecursiveMultiSeries(
regressor = HistGradientBoostingRegressor(random_state=8523),
lags = 14,
encoding = 'ordinal',
transformer_series = StandardScaler(),
window_features = window_features,
)
# Backtesting forecaster for all items
cv = TimeSeriesFold(
steps = 7,
initial_train_size = len(data_train) + len(data_val),
refit = False,
)
multi_series_mae, predictions_ms = backtesting_forecaster_multiseries(
forecaster = forecaster_ms,
series = data,
levels = items,
cv = cv,
metric = 'mean_absolute_error',
add_aggregated_metric = False,
verbose = False,
show_progress = True
)
# Results
display(multi_series_mae.head(3))
print('')
display(predictions_ms.head(3))
Comparison¶
The mean absolute error (MAE) for each item is calculated using the individual and global models. The results are compared to determine which model performs better.
# Difference of backtesting metric for each item
# ======================================================================================
multi_series_mae = multi_series_mae.set_index('levels')
multi_series_mae.columns = ['multi_series_mae']
results = pd.concat((uni_series_mae, multi_series_mae), axis = 1)
results['improvement'] = results.eval('uni_series_mae - multi_series_mae')
results['improvement_(%)'] = 100 * results.eval('(uni_series_mae - multi_series_mae) / uni_series_mae')
results = results.round(2)
results.style.bar(subset=['improvement_(%)'], align='mid', color=['#d65f5f', '#5fba7d'])
# Average improvement for all items
# ======================================================================================
results[['improvement', 'improvement_(%)']].agg(['mean', 'min', 'max'])
# Number of series with positive and negative improvement
# ======================================================================================
pd.Series(np.where(results['improvement_(%)'] < 0, 'negative', 'positive')).value_counts()
The global model achieves an average improvement of 7.4% compared to using an individual model for each series. For all series, the prediction error evaluated by backtesting is lower when the global model is used. This use case demonstrates that a multi-series model can have advantages over multiple individual models when forecasting time series that follow similar dynamics. In addition to the potential improvements in forecasting, it is also important to consider the benefit of having only one model to maintain and the speed of training and prediction.
⚠ Warning
This comparison was made without optimizing the model hyperparameters. See the Case study 3: Hyperparameter tuning and lags selection section to verify that the conclusions hold when the models are tuned with the best combination of hyperparameters and lags.Time Series with different lengths and different exogenous variables¶
When faced with a multi-series forecasting problem, it is common for the series to have varying lengths due to differences in the starting times of data recording. To address this scenario, the ForecasterRecursiveMultiSeries allow the simultaneous modeling of time series of different lengths and using different exogenous variables.
- When the modeled series have different lengths, they must be stored in a Python dictionary. The keys of the dictionary are the names of the series and the values are the series themselves. All series must be of type
pandas.Series, have adatetimeindex and have the same frequency.
| Series values | Allowed |
|---|---|
[NaN, NaN, NaN, NaN, 4, 5, 6, 7, 8, 9] |
✔️ |
[0, 1, 2, 3, 4, 5, 6, 7, 8, NaN] |
✔️ |
[0, 1, 2, 3, 4, NaN, 6, 7, 8, 9] |
✔️ |
[NaN, NaN, 2, 3, 4, NaN, 6, 7, 8, 9] |
✔️ |
- When different exogenous variables are used for each series, or if the exogenous variables are the same but have different values for each series, they must be stored in a dictionary. The keys of the dictionary are the names of the series and the values are the exogenous variables themselves. All exogenous variables must be of type
pandas.DataFrameorpandas.Series.
Data¶
The data for this example is stored in "long format" in a single DataFrame. The series_id column identifies the series to which each observation belongs. The timestamp column contains the date of the observation, and the value column contains the value of the series at that date. Each time series is of a different length. The exogenous variables are stored in a separate DataFrame, also in "long format". The column series_id identifies the series to which each observation belongs. The column timestamp contains the date of the observation, and the remaining columns contain the values of the exogenous variables at that date.
# Load time series of multiple lengths and exogenous variables
# ==============================================================================
series = pd.read_csv(
'https://raw.githubusercontent.com/skforecast/skforecast-datasets/main/data/demo_multi_series.csv'
)
exog = pd.read_csv(
'https://raw.githubusercontent.com/skforecast/skforecast-datasets/main/data/demo_multi_series_exog.csv'
)
series['timestamp'] = pd.to_datetime(series['timestamp'])
exog['timestamp'] = pd.to_datetime(exog['timestamp'])
display(series.head())
print("")
display(exog.head())
When series have different lengths, the data must be transformed into a dictionary. The keys of the dictionary are the names of the series and the values are the series themselves. To do this, the series_long_to_dict function is used, which takes the DataFrame in "long format" and returns a dict of series.
Similarly, when the exogenous variables are different (values or variables) for each series, the data must be transformed into a dictionary. The keys of the dictionary are the names of the series and the values are the exogenous variables themselves. The exog_long_to_dict function is used, which takes the DataFrame in "long format" and returns a dict of exogenous variables.
# Transform series and exog to dictionaries
# ==============================================================================
series_dict = series_long_to_dict(
data = series,
series_id = 'series_id',
index = 'timestamp',
values = 'value',
freq = 'D'
)
exog_dict = exog_long_to_dict(
data = exog,
series_id = 'series_id',
index = 'timestamp',
freq = 'D'
)
Some exogenous variables are omitted for series 1 and 3 to illustrate that different exogenous variables can be used for each series.
# Drop some exogenous variables for series 'id_1000' and 'id_1003'
# ==============================================================================
exog_dict['id_1000'] = exog_dict['id_1000'].drop(columns=['air_temperature', 'wind_speed'])
exog_dict['id_1003'] = exog_dict['id_1003'].drop(columns=['cos_day_of_week'])
# Partition data in train and test
# ==============================================================================
end_train = '2016-07-31 23:59:00'
series_dict_train = {k: v.loc[: end_train,] for k, v in series_dict.items()}
exog_dict_train = {k: v.loc[: end_train,] for k, v in exog_dict.items()}
series_dict_test = {k: v.loc[end_train:,] for k, v in series_dict.items()}
exog_dict_test = {k: v.loc[end_train:,] for k, v in exog_dict.items()}
# Plot series
# ==============================================================================
set_dark_theme()
colors = plt.rcParams['axes.prop_cycle'].by_key()['color']
fig, axs = plt.subplots(5, 1, figsize=(8, 4), sharex=True)
for i, s in enumerate(series_dict.values()):
axs[i].plot(s, label=s.name, color=colors[i])
axs[i].legend(loc='upper right', fontsize=8)
axs[i].tick_params(axis='both', labelsize=8)
axs[i].axvline(pd.to_datetime(end_train), color='white', linestyle='--', linewidth=1) # End train
# Description of each series
# ==============================================================================
for k in series_dict.keys():
print(f"{k}:")
try:
print(
f"\tTrain: len={len(series_dict_train[k])}, {series_dict_train[k].index[0]}"
f" --- {series_dict_train[k].index[-1]} "
f" (missing={series_dict_train[k].isnull().sum()})"
)
except:
print(f"\tTrain: len=0")
try:
print(
f"\tTest : len={len(series_dict_test[k])}, {series_dict_test[k].index[0]}"
f" --- {series_dict_test[k].index[-1]} "
f" (missing={series_dict_test[k].isnull().sum()})"
)
except:
print(f"\tTest : len=0")
# Exogenous variables for each series
# ==============================================================================
for k in series_dict.keys():
print(f"{k}:")
try:
print(f"\t{exog_dict[k].columns.to_list()}")
except:
print(f"\tNo exogenous variables")
Global model¶
# Fit forecaster
# ==============================================================================
regressor = HistGradientBoostingRegressor(random_state=123, max_depth=5)
window_features = RollingFeatures(stats=['mean', 'min', 'max'], window_sizes=7)
forecaster = ForecasterRecursiveMultiSeries(
regressor = regressor,
lags = 14,
window_features = window_features,
encoding = "ordinal",
dropna_from_series = False
)
forecaster.fit(series=series_dict_train, exog=exog_dict_train, suppress_warnings=True)
forecaster
Only series whose last window of data ends at the same datetime index can be predicted together. If levels = None, series that do not reach the maximum index are excluded from prediction. In this example, series 'id_1002' is excluded.
# Predict
# ==============================================================================
predictions = forecaster.predict(steps=5, exog=exog_dict_test, suppress_warnings=True)
predictions
Backtesting¶
When series have different lengths, the backtesting process only returns predictions for the date-times that are present in the series.
# Backtesting
# ==============================================================================
cv = TimeSeriesFold(
steps = 24,
initial_train_size = len(series_dict_train["id_1000"]),
refit = False,
)
metrics_levels, backtest_predictions = backtesting_forecaster_multiseries(
forecaster = forecaster,
series = series_dict,
exog = exog_dict,
cv = cv,
metric = "mean_absolute_error",
add_aggregated_metric = False,
n_jobs = "auto",
verbose = True,
show_progress = True,
suppress_warnings = True
)
display(metrics_levels)
print("")
display(backtest_predictions)
# Plot backtesting predictions
# ==============================================================================
colors = plt.rcParams['axes.prop_cycle'].by_key()['color']
fig, axs = plt.subplots(5, 1, figsize=(8, 4), sharex=True)
for i, s in enumerate(series_dict.keys()):
axs[i].plot(series_dict[s], label=series_dict[s].name, color=colors[i])
axs[i].axvline(pd.to_datetime(end_train), color='white', linestyle='--', linewidth=1)
try:
axs[i].plot(backtest_predictions[s], label='prediction', color="white")
except:
pass
axs[i].legend(loc='upper left', fontsize=8)
axs[i].tick_params(axis='both', labelsize=8)
By allowing the modeling of time series of different lengths and with different exogenous variables, the ForecasterRecursiveMultiSeries class provides a flexible and powerful tool for using all available information to train the forecasting models.
Hyperparameter tuning and lags selection¶
In first example of this document, the comparison between forecasters was done without optimizing the hyperparameters of the regressors. To make a fair comparison, a grid search strategy is used in order to select the best configuration for each forecaster. See more information on hyperparameter tuning and lags selection.
🖉 Note
Hyperparameter tuning for several models can be computationally expensive. In order to speed up the process, the evaluation of each candidate configuration is done using *one-step-ahead* instead of backtesting. For more details of the advantages and limitations of this approach, see the [One-step-ahead validation](https://skforecast.org/latest/user_guides/hyperparameter-tuning-and-lags-selection#one-step-ahead-validation.html).# Hyperparameter search for each single series model
# ======================================================================================
items = []
mae_values = []
def search_space(trial):
search_space = {
'lags' : trial.suggest_categorical('lags', [7, 14]),
'max_iter' : trial.suggest_int('max_iter', 100, 500),
'max_depth' : trial.suggest_int('max_depth', 5, 10),
'learning_rate' : trial.suggest_float('learning_rate', 0.01, 0.1)
}
return search_space
for item in tqdm(data.columns):
window_features = RollingFeatures(stats=['mean', 'min', 'max'], window_sizes=7)
forecaster = ForecasterRecursive(
regressor = HistGradientBoostingRegressor(random_state=123),
lags = 14,
window_features = window_features
)
cv_search = OneStepAheadFold(initial_train_size = len(data_train))
warnings.simplefilter('ignore', category=OneStepAheadValidationWarning)
results_bayesian = bayesian_search_forecaster(
forecaster = forecaster,
y = data.loc[:end_val, item],
cv = cv_search,
search_space = search_space,
n_trials = 20,
metric = 'mean_absolute_error',
return_best = True,
verbose = False,
show_progress = False
)
cv_backtesting = TimeSeriesFold(
steps = 7,
initial_train_size = len(data_train) + len(data_val),
refit = False,
)
metric, preds = backtesting_forecaster(
forecaster = forecaster,
y = data[item],
cv = cv_backtesting,
metric = 'mean_absolute_error',
verbose = False,
show_progress = False
)
items.append(item)
mae_values.append(metric.at[0, 'mean_absolute_error'])
uni_series_mae = pd.Series(
data = mae_values,
index = items,
name = 'uni_series_mae'
)
# Hyperparameter search for the multi-series model and backtesting for each item
# ======================================================================================
def search_space(trial):
search_space = {
'lags' : trial.suggest_categorical('lags', [7, 14]),
'max_iter' : trial.suggest_int('max_iter', 100, 500),
'max_depth' : trial.suggest_int('max_depth', 5, 10),
'learning_rate' : trial.suggest_float('learning_rate', 0.01, 0.1)
}
return search_space
window_features = RollingFeatures(stats=['mean', 'min', 'max'], window_sizes=7)
forecaster_ms = ForecasterRecursiveMultiSeries(
regressor = HistGradientBoostingRegressor(random_state=123),
lags = 14,
window_features = window_features,
transformer_series = StandardScaler(),
encoding = 'ordinal'
)
warnings.simplefilter('ignore', category=OneStepAheadValidationWarning)
results_bayesian_ms = bayesian_search_forecaster_multiseries(
forecaster = forecaster_ms,
series = data.loc[:end_val, :],
levels = None, # Si es None se seleccionan todos los niveles
cv = cv_search,
search_space = search_space,
n_trials = 20,
metric = 'mean_absolute_error',
return_best = True,
verbose = False,
show_progress = False
)
multi_series_mae, predictions_ms = backtesting_forecaster_multiseries(
forecaster = forecaster_ms,
series = data,
levels = None, # Si es None se seleccionan todos los niveles
cv = cv_backtesting,
metric = 'mean_absolute_error',
add_aggregated_metric = False,
verbose = False
)
# Difference in backtesting metric for each item
# ======================================================================================
multi_series_mae = multi_series_mae.set_index('levels')
multi_series_mae.columns = ['multi_series_mae']
results = pd.concat((uni_series_mae, multi_series_mae), axis = 1)
results['improvement'] = results.eval('uni_series_mae - multi_series_mae')
results['improvement_(%)'] = 100 * results.eval('(uni_series_mae - multi_series_mae) / uni_series_mae')
results = results.round(2)
# Average improvement for all items
# ======================================================================================
results[['improvement', 'improvement_(%)']].agg(['mean', 'min', 'max'])
# Number of series with positive and negative improvement
# ======================================================================================
pd.Series(np.where(results['improvement_(%)'] < 0, 'negative', 'positive')).value_counts()
After identifying the combination of lags and hyperparameters that achieve the best predictive performance for each forecaster, more single-series models have achieved higher predictive ability. Even so, the multi-series model provides better results for most of the items.
Feature Selection¶
Feature selection is the process of selecting a subset of relevant features (variables, predictors) for use in model construction. Feature selection techniques are used for several reasons: to simplify models to make them easier to interpret, to reduce training time, to avoid the curse of dimensionality, to improve generalization by reducing overfitting (formally, variance reduction), and others.
Skforecast is compatible with the feature selection methods implemented in the scikit-learn library. There are several methods for feature selection, but the most common are:
Recursive feature elimination (RFE)
Sequential Feature Selection (SFS)
Feature selection based on threshold (SelectFromModel)
💡 Tip
Feature selection is a powerful tool for improving the performance of machine learning models. However, it is computationally expensive and can be time-consuming. Since the goal is to find the best subset of features, not the best model, it is not necessary to use the entire data set or a highly complex model. Instead, it is recommended to use a small subset of the data and a simple model. Once the best subset of features has been identified, the model can then be trained using the entire dataset and a more complex configuration.Weights in multi-series¶
The weights are used to control the influence that each observation has on the training of the model. ForecasterAutoregMultiseries accepts two types of weights:
series_weightscontrols the relative importance of each series. If a series has twice as much weight as the others, the observations of that series influence the training twice as much. The higher the weight of a series relative to the others, the more the model will focus on trying to learn that series.weight_funccontrols the relative importance of each observation according to its index value. For example, a function that assigns a lower weight to certain dates.
If the two types of weights are indicated, they are multiplied to create the final weights as shown in the figure. The resulting sample_weight cannot have negative values.
Learn more about weights in multi-series forecasting and weighted time series forecasting with skforecast.
In this example, item_1 has higher relative importance among series (it weighs 3 times more than the rest of the series), and observations between '2013-12-01' and '2014-01-31' are considered non-representative and a weight of 0 is applied to them.
# Weights in ForecasterAutoregMultiSeries
# ======================================================================================
# Weights for each series
series_weights = {'item_1': 3.0} # Series not presented in the dict will have weight 1
# Weights for each index
def custom_weights(index):
"""
Return 0 if index is between '2013-12-01' and '2014-01-31', 1 otherwise.
"""
weights = np.where(
(index >= '2013-12-01') & (index <= '2014-01-31'),
0,
1
)
return weights
forecaster = ForecasterRecursiveMultiSeries(
regressor = HistGradientBoostingRegressor(random_state=123),
lags = 14,
transformer_series = StandardScaler(),
encoding = 'ordinal',
transformer_exog = None,
weight_func = custom_weights,
series_weights = series_weights
)
forecaster.fit(series=data)
forecaster.predict(steps=7).head(3)
🖉 Note
A dictionary can be passed to `weight_func` to apply different functions for each series. If a series is not presented in the dictionary, it will have weight 1.Conclusions¶
This use case shows that a multi-series model may have advantages over multiple individual models when forecasting time series that follow similar dynamics. Beyond the potential improvements in forecasting, it is also important to take into consideration the benefit of having only one model to maintain.
Session information¶
import session_info
session_info.show(html=False)
Citation¶
How to cite this document
If you use this document or any part of it, please acknowledge the source, thank you!
Global Forecasting Models: Modeling Multiple Time Series with Machine Learning by Joaquín Amat Rodrigo and Javier Escobar Ortiz, available under a CC BY-NC-SA 4.0 at https://www.cienciadedatos.net/documentos/py44-multi-series-forecasting-skforecast.html
How to cite skforecast
If you use skforecast for a scientific publication, we would appreciate it if you cite the published software.
Zenodo:
Amat Rodrigo, Joaquin, & Escobar Ortiz, Javier. (2024). skforecast (v0.14.0). Zenodo. https://doi.org/10.5281/zenodo.8382788
APA:
Amat Rodrigo, J., & Escobar Ortiz, J. (2024). skforecast (Version 0.14.0) [Computer software]. https://doi.org/10.5281/zenodo.8382788
BibTeX:
@software{skforecast, author = {Amat Rodrigo, Joaquin and Escobar Ortiz, Javier}, title = {skforecast}, version = {0.14.0}, month = {11}, year = {2024}, license = {BSD-3-Clause}, url = {https://skforecast.org/}, doi = {10.5281/zenodo.8382788} }
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This work by Joaquín Amat Rodrigo and Javier Escobar Ortiz is licensed under a Attribution-NonCommercial-ShareAlike 4.0 International.
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