More about forecasting in cienciadedatos.net


Introduction

Gradient boosting models have gained popularity in the machine learning community due to their ability to achieve excellent results in a wide range of use cases, including both regression and classification. Although these models have traditionally been less common in forecasting, they can be highly effective in this domain. Some of the key benefits of using gradient boosting models for forecasting include:

  • The ease with which exogenous variables can be included in the model, in addition to autoregressive variables.

  • The ability to capture non-linear relationships between variables.

  • High scalability, allowing models to handle large volumes of data.

  • Some implementations allow the inclusion of categorical variables without the need for additional encoding, such as one-hot encoding.

Despite these benefits, the use of machine learning models for forecasting can present several challenges that can make analysts reluctant to use them, the main ones being:

  • Transforming the data so that it can be used as a regression problem.

  • Depending on how many future predictions are needed (prediction horizon), an iterative process may be required where each new prediction is based on previous ones.

  • Model validation requires specific strategies such as backtesting, walk-forward validation or time series cross-validation. Traditional cross-validation cannot be used.

The skforecast library provides automated solutions to these challenges, making it easier to apply and validate machine learning models to forecasting problems. The library supports several advanced gradient boosting models, including XGBoost, LightGBM, Catboost and scikit-learn HistGradientBoostingRegressor. This document shows how to use them to build accurate forecasting models. To ensure a smooth learning experience, an initial exploration of the data is performed. Then, the modeling process is explained step by step, starting with a recursive model utilizing a LightGBM regressor and moving on to a model incorporating exogenous variables and various coding strategies. The document concludes by demonstrating the use of other gradient boosting model implementations, including XGBoost, CatBoost, and the scikit-learn HistGradientBoostingRegressor.

✎ Note

Machine learning models do not always outperform statistical learning models such as AR, ARIMA or Exponential Smoothing. Which one works best depends largely on the characteristics of the use case to which it is applied. Visit ARIMA and SARIMAX models with skforecast to learn more about statistical models.

✎ Note

Additional examples of how using gradient boosting models for forecasting can be found in the documents Forecasting energy demand with machine learning and Global Forecasting Models.

Use case

Bicycle sharing is a popular shared transport service that provides bicycles to individuals for short-term use. These systems typically provide bike docks where riders can borrow a bike and return it to any dock belonging to the same system. The docks are equipped with special bike racks that secure the bike and only release it via computer control. One of the major challenges faced by operators of these systems is the need to redistribute bikes to ensure that there are bikes available at all docks, as well as free spaces for returns.

In order to improve the planning and execution of bicycle distribution, it is proposed to create a model capable of forecasting the number of users over the next 36 hours. In this way, at 12:00 every day, the company in charge of managing the system will be able to know the expected users for the rest of the day (12 hours) and the next day (24 hours).

For illustrative purposes, the current example only models a single station, but it can be easily extended to cover multiple stations unsing global multi-series forecasting, thereby improving the management of bike-sharing systems on a larger scale.

Libraries

Libraries used in this document.

# Data processing
# ==============================================================================
import numpy as np
import pandas as pd
from astral.sun import sun
from astral import LocationInfo
from skforecast.datasets import fetch_dataset
from feature_engine.datetime import DatetimeFeatures
from feature_engine.creation import CyclicalFeatures
from feature_engine.timeseries.forecasting import WindowFeatures

# Plots
# ==============================================================================
import matplotlib.pyplot as plt
from statsmodels.graphics.tsaplots import plot_acf
from statsmodels.graphics.tsaplots import plot_pacf
from skforecast.plot import plot_residuals, calculate_lag_autocorrelation
import plotly.graph_objects as go
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
# ==============================================================================
import xgboost
import lightgbm
import catboost
import sklearn
import shap
from xgboost import XGBRegressor
from lightgbm import LGBMRegressor
from catboost import CatBoostRegressor
from sklearn.ensemble import HistGradientBoostingRegressor
from sklearn.preprocessing import (
    OneHotEncoder,
    OrdinalEncoder,
    FunctionTransformer,
    PolynomialFeatures,
)
from sklearn.feature_selection import RFECV
from sklearn.pipeline import make_pipeline
from sklearn.compose import make_column_transformer, make_column_selector

import skforecast
from skforecast.recursive import ForecasterEquivalentDate, ForecasterRecursive
from skforecast.model_selection import (
    TimeSeriesFold,
    OneStepAheadFold,
    bayesian_search_forecaster,
    backtesting_forecaster,
)
from skforecast.preprocessing import RollingFeatures
from skforecast.feature_selection import select_features
from skforecast.metrics import calculate_coverage

# Warnings configuration
# ==============================================================================
import warnings
warnings.filterwarnings('once')

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 lightgbm: {lightgbm.__version__}")
print(f"{color}Version xgboost: {xgboost.__version__}")
print(f"{color}Version catboost: {catboost.__version__}")
print(f"{color}Version pandas: {pd.__version__}")
print(f"{color}Version numpy: {np.__version__}")

Version skforecast: 0.15.0
Version scikit-learn: 1.5.2
Version lightgbm: 4.6.0
Version xgboost: 2.1.4
Version catboost: 1.2.7
Version pandas: 2.2.3
Version numpy: 1.26.4

⚠ Warning

At the time of writing this document, catboost is only compatible with numpy version lower than 2.0. If you have a higher version, you can downgrade it by running the following command: pip install numpy==1.26.4

Data

The data in this document represent the hourly usage of the bike share system in the city of Washington, D.C. during the years 2011 and 2012. In addition to the number of users per hour, information about weather conditions and holidays is available. The original data was obtained from the UCI Machine Learning Repository and has been previously cleaned (code) applying the following modifications:

  • Renamed columns with more descriptive names.

  • Renamed categories of the weather variables. The category of heavy_rain has been combined with that of rain.

  • Denormalized temperature, humidity and wind variables.

  • date_time variable created and set as index.

  • Imputed missing values by forward filling.

# Downloading data
# ==============================================================================
data = fetch_dataset('bike_sharing', raw=True)

bike_sharing
------------
Hourly usage of the bike share system in the city of Washington D.C. during the
years 2011 and 2012. In addition to the number of users per hour, information
about weather conditions and holidays is available.
Fanaee-T,Hadi. (2013). Bike Sharing Dataset. UCI Machine Learning Repository.
https://doi.org/10.24432/C5W894.
Shape of the dataset: (17544, 12)

# Preprocessing data (setting index and frequency)
# ==============================================================================
data = data[['date_time', 'users', 'holiday', 'weather', 'temp', 'atemp', 'hum', 'windspeed']]
data['date_time'] = pd.to_datetime(data['date_time'], format='%Y-%m-%d %H:%M:%S')
data = data.set_index('date_time')
if pd.__version__ < '2.2':
    data = data.asfreq('H')
else:
    data = data.asfreq('h')
data = data.sort_index()
data.head()
users holiday weather temp atemp hum windspeed
date_time
2011-01-01 00:00:00 16.0 0.0 clear 9.84 14.395 81.0 0.0
2011-01-01 01:00:00 40.0 0.0 clear 9.02 13.635 80.0 0.0
2011-01-01 02:00:00 32.0 0.0 clear 9.02 13.635 80.0 0.0
2011-01-01 03:00:00 13.0 0.0 clear 9.84 14.395 75.0 0.0
2011-01-01 04:00:00 1.0 0.0 clear 9.84 14.395 75.0 0.0

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 = '2012-04-30 23:59:00'
end_validation = '2012-08-31 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      : 2011-01-01 00:00:00 --- 2012-04-30 23:00:00  (n=11664)
Dates validacion : 2012-05-01 00:00:00 --- 2012-08-31 23:00:00  (n=2952)
Dates test       : 2012-09-01 00:00:00 --- 2012-12-31 23:00:00  (n=2928)

Data exploration

Graphical exploration of time series can be an effective way of identifying trends, patterns, and seasonal variations. This, in turn, helps to guide the selection of the most appropriate forecasting model.

Plot time series

Full time series

# Interactive plot of time series
# ==============================================================================
fig = go.Figure()
fig.add_trace(go.Scatter(x=data_train.index, y=data_train['users'], mode='lines', name='Train'))
fig.add_trace(go.Scatter(x=data_val.index, y=data_val['users'], mode='lines', name='Validation'))
fig.add_trace(go.Scatter(x=data_test.index, y=data_test['users'], mode='lines', name='Test'))
fig.update_layout(
    title  = 'Number of users',
    xaxis_title="Time",
    yaxis_title="Users",
    legend_title="Partition:",
    width=800,
    height=400,
    margin=dict(l=20, r=20, t=35, b=20),
    legend=dict(orientation="h", yanchor="top", y=1, xanchor="left", x=0.001)
)
#fig.update_xaxes(rangeslider_visible=True)
fig.show()
Jan 2011Apr 2011Jul 2011Oct 2011Jan 2012Apr 2012Jul 2012Oct 2012Jan 201302004006008001000
Partition:TrainValidationTestNumber of usersTimeUsers
# Static plot of time series with zoom
# ==============================================================================
zoom = ('2011-08-01 00:00:00', '2011-08-15 00:00:00')
fig = plt.figure(figsize=(8, 4))
grid = plt.GridSpec(nrows=8, ncols=1, hspace=0.1, wspace=0)
main_ax = fig.add_subplot(grid[1:3, :])

data_train['users'].plot(ax=main_ax, label='train', alpha=0.5)
data_val['users'].plot(ax=main_ax, label='validation', alpha=0.5)
data_test['users'].plot(ax=main_ax, label='test', alpha=0.5)
min_y = min(data['users'])
max_y = max(data['users'])
main_ax.fill_between(zoom, min_y, max_y, facecolor='blue', alpha=0.5, zorder=0)
main_ax.set_title(f'Number of users: {data.index.min()}, {data.index.max()}', fontsize=10)
main_ax.set_xlabel('')
main_ax.legend(loc='lower center', ncol=3, bbox_to_anchor=(0.5, -0.8))
zoom_ax = fig.add_subplot(grid[5:, :])
data.loc[zoom[0]: zoom[1]]['users'].plot(ax=zoom_ax, color='blue', linewidth=1)
zoom_ax.set_title(f'Number of users: {zoom}', fontsize=10)
zoom_ax.set_xlabel('')
plt.subplots_adjust(hspace=1)
plt.show();

Seasonality plots

Seasonal plots are a useful tool for identifying seasonal patterns and trends in a time series. They are created by averaging the values of each season over time and then plotting them against time.

# Annual, weekly and daily seasonality
# ==============================================================================
fig, axs = plt.subplots(2, 2, figsize=(8, 5), sharex=False, sharey=True)
axs = axs.ravel()

# Users distribution by month
data['month'] = data.index.month
data.boxplot(column='users', by='month', ax=axs[0], flierprops={'markersize': 3, 'alpha': 0.3})
data.groupby('month')['users'].median().plot(style='o-', linewidth=0.8, ax=axs[0])
axs[0].set_ylabel('Users')
axs[0].set_title('Users distribution by month', fontsize=9)

# Users distribution by week day
data['week_day'] = data.index.day_of_week + 1
data.boxplot(column='users', by='week_day', ax=axs[1], flierprops={'markersize': 3, 'alpha': 0.3})
data.groupby('week_day')['users'].median().plot(style='o-', linewidth=0.8, ax=axs[1])
axs[1].set_ylabel('Users')
axs[1].set_title('Users distribution by week day', fontsize=9)

# Users distribution by the hour of the day
data['hour_day'] = data.index.hour + 1
data.boxplot(column='users', by='hour_day', ax=axs[2], flierprops={'markersize': 3, 'alpha': 0.3})
data.groupby('hour_day')['users'].median().plot(style='o-', linewidth=0.8, ax=axs[2])
axs[2].set_ylabel('Users')
axs[2].set_title('Users distribution by the hour of the day', fontsize=9)

# Users distribution by week day and hour of the day
mean_day_hour = data.groupby(["week_day", "hour_day"])["users"].mean()
mean_day_hour.plot(ax=axs[3])
axs[3].set(
    title       = "Mean users during week",
    xticks      = [i * 24 for i in range(7)],
    xticklabels = ["Mon", "Tue", "Wed", "Thu", "Fri", "Sat", "Sun"],
    xlabel      = "Day and hour",
    ylabel      = "Number of users"
)
axs[3].title.set_size(9)

fig.suptitle("Seasonality plots", fontsize=12)
fig.tight_layout()

There is a clear difference between weekdays and weekends. There is also a clear intra-day pattern, with a different influx of users depending on the time of day.

Autocorrelation plots

Auto-correlation plots are a useful tool for identifying the order of an autoregressive model. The autocorrelation function (ACF) is a measure of the correlation between the time series and a lagged version of itself. The partial autocorrelation function (PACF) is a measure of the correlation between the time series and a lagged version of itself, controlling for the values of the time series at all shorter lags. These plots are useful for identifying the lags to be included in the autoregressive model.

# Autocorrelation plot
# ==============================================================================
fig, ax = plt.subplots(figsize=(5, 2))
plot_acf(data['users'], ax=ax, lags=72)
plt.show()

# Partial autocorrelation plot
# ==============================================================================
fig, ax = plt.subplots(figsize=(5, 2))
plot_pacf(data['users'], ax=ax, lags=72, method='ywm')
plt.show()

# Top 10 lags with highest absolute partial autocorrelation
# ==============================================================================
calculate_lag_autocorrelation(
    data    = data['users'],
    n_lags  = 60,
    sort_by = "partial_autocorrelation_abs"
).head(10)
lag partial_autocorrelation_abs partial_autocorrelation autocorrelation_abs autocorrelation
0 1 0.845220 0.845220 0.845172 0.845172
1 2 0.408349 -0.408349 0.597704 0.597704
2 23 0.355669 0.355669 0.708470 0.708470
3 22 0.343601 0.343601 0.520804 0.520804
4 25 0.332366 -0.332366 0.711256 0.711256
5 10 0.272649 -0.272649 0.046483 -0.046483
6 17 0.241984 0.241984 0.057267 -0.057267
7 19 0.199286 0.199286 0.159897 0.159897
8 21 0.193404 0.193404 0.373666 0.373666
9 3 0.182068 0.182068 0.409680 0.409680

The results of the autocorrelation study show that there is a significant correlation between the number of users in previous hours, as well as the days before, and the number of users in the future. This means that knowing the number of users during certain periods in the past could be valuable in predicting the number of users in the future.

Baseline

When faced with a forecasting problem, it is important to establish a baseline model. This is usually a very simple model that can be used as a reference to assess whether more complex models are worth implementing.

Skforecast allows you to easily create a baseline with its class ForecasterEquivalentDate. This model, also known as Seasonal Naive Forecasting, simply returns the value observed in the same period of the previous season (e.g., the same working day from the previous week, the same hour from the previous day, etc.).

Based on the exploratory analysis performed, the baseline model will be the one that predicts each hour using the value of the same hour on the previous day.

✎ Note

In the following code cells, a baseline forecaster is trained and its predictive ability is evaluated using a backtesting process. If this concept is new to you, do not worry, it will be explained in detail throughout the document. For now, it is sufficient to know that the backtesting process consists of training the model with a certain amount of data and evaluating its predictive ability with the data that the model has not seen. The error metric will be used as a reference to evaluate the predictive ability of the more complex models that will be implemented later. 
# Create baseline: value of the same hour of the previous day
# ==============================================================================
forecaster = ForecasterEquivalentDate(
                 offset    = pd.DateOffset(days=1),
                 n_offsets = 1
             )

# Train forecaster
# ==============================================================================
forecaster.fit(y=data.loc[:end_validation, 'users'])
forecaster

======================== 
ForecasterEquivalentDate 
======================== 
Offset: <DateOffset: days=1> 
Number of offsets: 1 
Aggregation function: mean 
Window size: 24 
Training range: [Timestamp('2011-01-01 00:00:00'), Timestamp('2012-08-31 23:00:00')] 
Training index type: DatetimeIndex 
Training index frequency: h 
Creation date: 2025-03-12 10:36:15 
Last fit date: 2025-03-12 10:36:15 
Skforecast version: 0.15.0 
Python version: 3.12.9 
Forecaster id: None 
# Backtesting
# ==============================================================================
cv = TimeSeriesFold(steps = 36, initial_train_size = len(data.loc[:end_validation]))
metric_baseline, predictions = backtesting_forecaster(
                                   forecaster = forecaster,
                                   y          = data['users'],
                                   cv         = cv,
                                   metric     = 'mean_absolute_error'
                               )
metric_baseline

  0%|          | 0/82 [00:00<?, ?it/s]
mean_absolute_error
0 91.668716

The MAE of the baseline model is used as a reference to assess whether the more complex models are worth implementing.

Multi-step forecasting with LightGBM

LightGBM is a highly efficient implementation of the stochastic gradient boosting algorithm, which has become a benchmark in the field of machine learning. The LightGBM library includes its own API as well as the scikit-learn API, making it compatible with skforecast.

First, an ForecasterRecursive model is trained using past values (lags) and moving average as predictors. Later, exogenous variables are added to the model and the improvement in its performance is assessed. Since Gradient Boosting models have a large number of hyperparameters, a Bayesian Search is performed using the bayesian_search_forecaster() function to find the best combination of hyperparameters and lags. Finally, the predictive ability of the model is evaluated using a backtesting process.

Forecaster

# Create forecaster
# ==============================================================================
window_features = RollingFeatures(stats=["mean"], window_sizes=24 * 3)
forecaster = ForecasterRecursive(
                regressor       = LGBMRegressor(random_state=15926, verbose=-1),
                lags            = 24,
                window_features = window_features
             )

# Train forecaster
# ==============================================================================
forecaster.fit(y=data.loc[:end_validation, 'users'])
forecaster

ForecasterRecursive

General Information
  • Regressor: LGBMRegressor
  • 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]
  • Window features: ['roll_mean_72']
  • Window size: 72
  • Exogenous included: False
  • Weight function included: False
  • Differentiation order: None
  • Creation date: 2025-03-12 10:36:17
  • Last fit date: 2025-03-12 10:36:18
  • Skforecast version: 0.15.0
  • Python version: 3.12.9
  • Forecaster id: None
Exogenous Variables
    None
Data Transformations
  • Transformer for y: None
  • Transformer for exog: None
Training Information
  • Training range: [Timestamp('2011-01-01 00:00:00'), Timestamp('2012-08-31 23:00:00')]
  • Training index type: DatetimeIndex
  • Training index frequency: h
Regressor Parameters
    {'boosting_type': 'gbdt', 'class_weight': None, 'colsample_bytree': 1.0, 'importance_type': 'split', 'learning_rate': 0.1, 'max_depth': -1, 'min_child_samples': 20, 'min_child_weight': 0.001, 'min_split_gain': 0.0, 'n_estimators': 100, 'n_jobs': None, 'num_leaves': 31, 'objective': None, 'random_state': 15926, 'reg_alpha': 0.0, 'reg_lambda': 0.0, 'subsample': 1.0, 'subsample_for_bin': 200000, 'subsample_freq': 0, 'verbose': -1}
Fit Kwargs
    {}

🛈 API Reference    🗎 User Guide 

# Predict
# ==============================================================================
forecaster.predict(steps=10)

2012-09-01 00:00:00    108.331027
2012-09-01 01:00:00     68.562982
2012-09-01 02:00:00     33.499525
2012-09-01 03:00:00     10.027583
2012-09-01 04:00:00      3.037563
2012-09-01 05:00:00     17.162543
2012-09-01 06:00:00     51.059825
2012-09-01 07:00:00    146.940053
2012-09-01 08:00:00    344.320596
2012-09-01 09:00:00    439.738683
Freq: h, Name: pred, dtype: float64

Backtesting

In order to have a robust estimate of the predictive ability of the model, a backtesting process is carried out. The backtesting process consists of generating a forecast for each observation in the test set, following the same procedure as it would be done in production, and then comparing the predicted value with the actual value.

It is highly recommended to review the documentation for the backtesting_forecaster() function to gain a better understanding of its capabilities. This will help to utilize its full potential to analyze the predictive ability of the model.

# Backtest model on test data
# ==============================================================================
cv = TimeSeriesFold(steps = 36, initial_train_size = len(data.loc[:end_validation]))
metric, predictions = backtesting_forecaster(
                          forecaster = forecaster,
                          y          = data['users'],
                          cv         = cv,
                          metric     = 'mean_absolute_error'
                      )
predictions.head()

  0%|          | 0/82 [00:00<?, ?it/s]
pred
2012-09-01 00:00:00 108.331027
2012-09-01 01:00:00 68.562982
2012-09-01 02:00:00 33.499525
2012-09-01 03:00:00 10.027583
2012-09-01 04:00:00 3.037563 
# Backtesting error
# ==============================================================================
metric
mean_absolute_error
0 76.464247

The prediction error of the autoregressive model achieves a lower MAE than that of the baseline model.

Exogenous Variables

So far, only lagged values of the time series have been used as predictors. However, it is possible to include other variables as predictors. These variables are known as exogenous variables (features) and their use can improve the predictive capacity of the model. A very important point to keep in mind is that the values of the exogenous variables must be known at the time of prediction.

Common examples of exogenous variables are those derived from the calendar, such as the day of the week, month, year, or holidays. Weather variables such as temperature, humidity, and wind also fall into this category, as do economic variables such as inflation and interest rates.

⚠ Warning

Exogenous variables must be known at the time of the forecast. For example, if temperature is used as an exogenous variable, the temperature value for the next hour must be known at the time of the forecast. If the temperature value is not known, the forecast will not be possible.

Weather variables should be used with caution. When the model is deployed into production, future weather conditions are not known, but are predictions made by meteorological services. Because they are predictions, they introduce errors into the forecasting model. As a result, the model's predictions are likely to get worse. One way to anticipate this problem, and to know (not avoid) the expected performance of the model, is to use the weather forecasts available at the time the model is trained, rather than the recorded conditions.

Next, exogenous variables are created based on calendar information, sunrise and sunset times, temperature, and holidays. These new variables are then added to the training, validation and test sets, and used as predictors in the autoregressive model.

Calendar and meteorological features

# Calendar features
# ==============================================================================
features_to_extract = [
    'month',
    'week',
    'day_of_week',
    'hour'
]
calendar_transformer = DatetimeFeatures(
    variables           ='index',
    features_to_extract = features_to_extract,
    drop_original       = True,
)
calendar_features = calendar_transformer.fit_transform(data)[features_to_extract]

# Sunlight features
# ==============================================================================
location = LocationInfo(
    name      = 'Washington DC',
    region    = 'USA',
    timezone  = 'US/Eastern',
    latitude  = 40.516666666666666,
    longitude = -77.03333333333333
)
sunrise_hour = [
    sun(location.observer, date=date, tzinfo=location.timezone)['sunrise'].hour
    for date in data.index
]
sunset_hour = [
    sun(location.observer, date=date, tzinfo=location.timezone)['sunset'].hour
    for date in data.index
]
sun_light_features = pd.DataFrame({
                        'sunrise_hour': sunrise_hour,
                        'sunset_hour': sunset_hour}, 
                        index = data.index
                     )
sun_light_features['daylight_hours'] = (
    sun_light_features['sunset_hour'] - sun_light_features['sunrise_hour']
)
sun_light_features["is_daylight"] = np.where(
    (data.index.hour >= sun_light_features["sunrise_hour"])
    & (data.index.hour < sun_light_features["sunset_hour"]),
    1,
    0,
)

# Holiday features
# ==============================================================================
holiday_features = data[['holiday']].astype(int)
holiday_features['holiday_previous_day'] = holiday_features['holiday'].shift(24)
holiday_features['holiday_next_day'] = holiday_features['holiday'].shift(-24)

# Rolling windows of temperature
# ==============================================================================
wf_transformer = WindowFeatures(
    variables   = ["temp"],
    window      = ["1D", "7D"],
    functions   = ["mean", "max", "min"],
    freq        = "h",
)
temp_features = wf_transformer.fit_transform(data[['temp']])


# Merge all exogenous variables
# ==============================================================================
assert all(calendar_features.index == sun_light_features.index)
assert all(calendar_features.index == temp_features.index)
assert all(calendar_features.index == holiday_features.index)
df_exogenous_features = pd.concat([
    calendar_features,
    sun_light_features,
    temp_features,
    holiday_features
], axis=1)

# Due to the creation of moving averages, there are missing values at the beginning
# of the series. And due to holiday_next_day there are missing values at the end.
df_exogenous_features = df_exogenous_features.iloc[7 * 24:, :]
df_exogenous_features = df_exogenous_features.iloc[:-24, :]
df_exogenous_features.head(3)
month week day_of_week hour sunrise_hour sunset_hour daylight_hours is_daylight temp temp_window_1D_mean temp_window_1D_max temp_window_1D_min temp_window_7D_mean temp_window_7D_max temp_window_7D_min holiday holiday_previous_day holiday_next_day
date_time
2011-01-08 00:00:00 1 1 5 0 7 16 9 0 7.38 8.063333 9.02 6.56 10.127976 18.86 4.92 0 0.0 0.0
2011-01-08 01:00:00 1 1 5 1 7 16 9 0 7.38 8.029167 9.02 6.56 10.113333 18.86 4.92 0 0.0 0.0
2011-01-08 02:00:00 1 1 5 2 7 16 9 0 7.38 7.995000 9.02 6.56 10.103571 18.86 4.92 0 0.0 0.0

Features with a cyclical pattern

Certain aspects of the calendar, such as hours or days, are cyclical. For example, the hour-day cycle ranges from 0 to 23 hours. These types of features can be handled in several ways, each with its advantages and disadvantages.

  • One approach is to use the features directly as numeric values without any transformation. This method avoids creating a lot of new features, but may impose an incorrect linear order on the values. For example, hour 23 of one day and hour 00 of the next are very far apart in their linear representation, when in fact there is only one hour difference between them.

  • Another possibility is to treat cyclical features as categorical variables to avoid imposing a linear order. However, this approach may result in the loss of the cyclical information inherent in the variable.

  • There is a third way to deal with cyclical features that is often preferred to the other two methods. This involves transforming the features using the sine and cosine of their period. This approach generates only two new features that capture the cyclicality of the data more accurately than the previous two methods because it preserves the natural order of the feature and avoids imposing a linear order.

✎ Note

For a more detailed explanation of the different ways of dealing with cyclical features, see the Cyclical features in time series forecasting document. 
# Cliclical encoding of calendar and sunlight features
# ==============================================================================
features_to_encode = [
    "month",
    "week",
    "day_of_week",
    "hour",
    "sunrise_hour",
    "sunset_hour",
]
max_values = {
    "month": 12,
    "week": 52,
    "day_of_week": 7,
    "hour": 24,
    "sunrise_hour": 24,
    "sunset_hour": 24,
}
cyclical_encoder = CyclicalFeatures(
    variables     = features_to_encode,
    max_values    = max_values,
    drop_original = False
)

df_exogenous_features = cyclical_encoder.fit_transform(df_exogenous_features)
df_exogenous_features.head(3)
month week day_of_week hour sunrise_hour sunset_hour daylight_hours is_daylight temp temp_window_1D_mean ... week_sin week_cos day_of_week_sin day_of_week_cos hour_sin hour_cos sunrise_hour_sin sunrise_hour_cos sunset_hour_sin sunset_hour_cos
date_time
2011-01-08 00:00:00 1 1 5 0 7 16 9 0 7.38 8.063333 ... 0.120537 0.992709 -0.974928 -0.222521 0.000000 1.000000 0.965926 -0.258819 -0.866025 -0.5
2011-01-08 01:00:00 1 1 5 1 7 16 9 0 7.38 8.029167 ... 0.120537 0.992709 -0.974928 -0.222521 0.258819 0.965926 0.965926 -0.258819 -0.866025 -0.5
2011-01-08 02:00:00 1 1 5 2 7 16 9 0 7.38 7.995000 ... 0.120537 0.992709 -0.974928 -0.222521 0.500000 0.866025 0.965926 -0.258819 -0.866025 -0.5

3 rows × 30 columns

Feature interaction

In many cases, exogenous variables are not isolated. Rather, their effect on the target variable depends on the value of other variables. For example, the effect of temperature on electricity demand depends on the time of day. The interaction between the exogenous variables can be captured by new variables that are obtained by multiplying existing variables together. This interactions are easily obtained with the PolynomialFeatures class from scikit-learn.

# Interaction between exogenous variables
# ==============================================================================
transformer_poly = PolynomialFeatures(
                        degree           = 2,
                        interaction_only = True,
                        include_bias     = False
                    ).set_output(transform="pandas")
poly_cols = [
    'month_sin', 
    'month_cos',
    'week_sin',
    'week_cos',
    'day_of_week_sin',
    'day_of_week_cos',
    'hour_sin',
    'hour_cos',
    'sunrise_hour_sin',
    'sunrise_hour_cos',
    'sunset_hour_sin',
    'sunset_hour_cos',
    'daylight_hours',
    'is_daylight',
    'holiday_previous_day',
    'holiday_next_day',
    'temp_window_1D_mean',
    'temp_window_1D_min',
    'temp_window_1D_max',
    'temp_window_7D_mean',
    'temp_window_7D_min',
    'temp_window_7D_max',
    'temp',
    'holiday'
]
poly_features = transformer_poly.fit_transform(df_exogenous_features[poly_cols])
poly_features = poly_features.drop(columns=poly_cols)
poly_features.columns = [f"poly_{col}" for col in poly_features.columns]
poly_features.columns = poly_features.columns.str.replace(" ", "__")
assert all(poly_features.index == df_exogenous_features.index)
df_exogenous_features = pd.concat([df_exogenous_features, poly_features], axis=1)
df_exogenous_features.head(3)
month week day_of_week hour sunrise_hour sunset_hour daylight_hours is_daylight temp temp_window_1D_mean ... poly_temp_window_7D_mean__temp_window_7D_min poly_temp_window_7D_mean__temp_window_7D_max poly_temp_window_7D_mean__temp poly_temp_window_7D_mean__holiday poly_temp_window_7D_min__temp_window_7D_max poly_temp_window_7D_min__temp poly_temp_window_7D_min__holiday poly_temp_window_7D_max__temp poly_temp_window_7D_max__holiday poly_temp__holiday
date_time
2011-01-08 00:00:00 1 1 5 0 7 16 9 0 7.38 8.063333 ... 49.829643 191.013631 74.744464 0.0 92.7912 36.3096 0.0 139.1868 0.0 0.0
2011-01-08 01:00:00 1 1 5 1 7 16 9 0 7.38 8.029167 ... 49.757600 190.737467 74.636400 0.0 92.7912 36.3096 0.0 139.1868 0.0 0.0
2011-01-08 02:00:00 1 1 5 2 7 16 9 0 7.38 7.995000 ... 49.709571 190.553357 74.564357 0.0 92.7912 36.3096 0.0 139.1868 0.0 0.0

3 rows × 306 columns

Categorical features

There are several approaches to incorporating categorical variables into LightGBM (and other gradient boosting frameworks):

  • One option is to transform the data by converting categorical values to numerical values using methods such as one-hot encoding or ordinal encoding. This approach is applicable to all machine learning models.

  • Alternatively, LightGBM can handle categorical variables internally without the need for preprocessing. This can be done automatically by setting the parameter categorical_features='auto' and encoding the variables as the category datatype within a Pandas DataFrame. As another option, users can specify the names of the features to be treated as categorical by passing a list of column names to the categorical_features parameter.

There is no one method that is always better than the others. General rules are:

  • When the cardinality of categorical variables is high (many different values), it is better to use the native support for categorical variables than to use one-hot encoding.

  • With one-hot encoded data, more split points (i.e., more depth) are needed to recover an equivalent split that could be obtained with a single split point using native handling.

  • When a categorical variable is converted to multiple dummy variables using one-hot, its importance is diluted, making the analysis of the importance of the features more complex to interpret.

# Store categorical variables as category type
# ==============================================================================
data["weather"] = data["weather"].astype("category")

One hot encoding

ColumnTransformers in scikit-learn provide a powerful way to define transformations and apply them to specific features. By encapsulating the transformations in a ColumnTransformer object, it can be passed to a Forecaster using the transformer_exog argument.

✎ Note

It is possible to apply a transformation to the entire dataset independent of the Forecaster. However, it is crucial to ensure that the transformations are learned only from the training data to avoid information leakage. In addition, the same transformation should be applied to the input data during the prediction. Therefore, it is advisable to include the transformation in the Forecaster so that it is handled internally. This approach ensures consistency in the application of transformations and reduces the likelihood of errors. 
# One hot encoding transformer
# ==============================================================================
one_hot_encoder = make_column_transformer(
    (
        OneHotEncoder(sparse_output=False, drop='if_binary'),
        make_column_selector(dtype_include=['category', 'object']),
    ),
    remainder="passthrough",
    verbose_feature_names_out=False,
).set_output(transform="pandas")  

# Create a forecaster with a transformer for exogenous features
# ==============================================================================
forecaster = ForecasterRecursive(
                regressor        = LGBMRegressor(random_state=15926, verbose=-1),
                lags             = 72,
                window_features  = window_features,
                transformer_exog = one_hot_encoder
             )

To examine how the data is transformed, it is possible to use the create_train_X_y method to generate the matrices that the Forecaster uses to train the model. This approach provides insight into the specific data manipulations that occur during the training process.

# View training matrix
# ==============================================================================
exog_features = ['weather']         
X_train, y_train = forecaster.create_train_X_y(
                        y    = data.loc[:end_validation, 'users'],
                        exog = data.loc[:end_validation, exog_features]
                   )
X_train.head(3)
lag_1 lag_2 lag_3 lag_4 lag_5 lag_6 lag_7 lag_8 lag_9 lag_10 ... lag_67 lag_68 lag_69 lag_70 lag_71 lag_72 roll_mean_72 weather_clear weather_mist weather_rain
date_time
2011-01-04 00:00:00 12.0 20.0 52.0 52.0 110.0 157.0 157.0 76.0 72.0 77.0 ... 1.0 1.0 13.0 32.0 40.0 16.0 43.638889 1.0 0.0 0.0
2011-01-04 01:00:00 5.0 12.0 20.0 52.0 52.0 110.0 157.0 157.0 76.0 72.0 ... 2.0 1.0 1.0 13.0 32.0 40.0 43.486111 1.0 0.0 0.0
2011-01-04 02:00:00 2.0 5.0 12.0 20.0 52.0 52.0 110.0 157.0 157.0 76.0 ... 3.0 2.0 1.0 1.0 13.0 32.0 42.958333 1.0 0.0 0.0

3 rows × 76 columns

The One Hot Encoder strategy has been shown for didactic purposes. For the rest of the document, however, the native support for categorical variables is used.

Native implementation for categorical features

The Gradient Boosting libraries (XGBoost, LightGBM, CatBoost, and HistGradientBoostingRegressor) assume that the input categories are integers starting from 0 up to the number of categories [0, 1, ..., n_categories-1]. In most real cases, categorical variables are not encoded with numbers but with strings, so an intermediate transformation step is necessary. Two options are:

  • Set columns with categorical variables to the type category. Internally, this data structure consists of an array of categories and an array of integer values (codes) that point to the actual value of the array of categories. That is, internally it is a numeric array with a mapping that relates each value with a category. Models are able to automatically identify the columns of type category and access their internal codes. This approach is applicable to XGBoost, LightGBM and CatBoost.

  • Preprocess the categorical columns with an OrdinalEncoder to transform their values to integers and then explicitly indicate which features should be treated as categorical.

To use the the automatic detection in skforecast, categorical variables must first be encoded as integers and then stored as the type category again. This is because skforecast internally uses a numeric numpy array to speed up the calculation.

⚠ Warning

The four gradient boosting frameworks – LightGBM, scikit-learn's HistogramGradientBoosting, XGBoost, and CatBoost – are capable of directly handling categorical features within the model. However, it is important to note that each framework has its own configurations, benefits and potential problems. To fully comprehend how to use these frameworks, it is highly recommended to refer to the skforecast user guide for a detailed understanding.

When deploying models in production, it is strongly recommended to avoid using automatic detection based on pandas category type columns. Although pandas provides an internal coding for these columns, it is not consistent across different datasets and may vary depending on the categories present in each dataset. It is therefore crucial to be aware of this issue and to take appropriate measures to ensure consistency in the coding of categorical features when deploying models in production. More details on this issue can be found in Native implementation for categorical features. 
# Transformer: Ordinal encoding + cast to category type
# ==============================================================================
pipeline_categorical = make_pipeline(
    OrdinalEncoder(
        dtype=int,
        handle_unknown="use_encoded_value",
        unknown_value=-1,
        encoded_missing_value=-1
    ),
    FunctionTransformer(
        func=lambda x: x.astype('category'),
        feature_names_out= 'one-to-one'
    )
)

transformer_exog = make_column_transformer(
    (
        pipeline_categorical,
        make_column_selector(dtype_include=['category', 'object']),
    ),
    remainder="passthrough",
    verbose_feature_names_out=False,
).set_output(transform="pandas")

When creating a Forecaster with LGBMRegressor, it is necessary to specify how to handle the categorical columns using the fit_kwargs argument. This is because the categorical_feature argument is only specified in the fit method of LGBMRegressor, and not during its initialization.

# Create a forecaster with automatic categorical detection
# ==============================================================================
forecaster = ForecasterRecursive(
                regressor        = LGBMRegressor(random_state=15926, verbose=-1),
                lags             = 72,
                window_features  = window_features,
                transformer_exog = transformer_exog,
                fit_kwargs       = {"categorical_feature": "auto"}
             )

This is the strategy that will be used throughout the rest of this document.

Evaluate model with exogenous features

The forecaster is trained again, but this time, the exogenous variables are also included as predictors. For categorical features, the native implementation is used.

# Select exogenous variables to be included in the model
# ==============================================================================
exog_features = []
# Columns that ends with _sin or _cos are selected
exog_features.extend(df_exogenous_features.filter(regex='_sin$|_cos$').columns.tolist())
# columns that start with temp_ are selected
exog_features.extend(df_exogenous_features.filter(regex='^temp_.*').columns.tolist())
# Columns that start with holiday_ are selected
exog_features.extend(df_exogenous_features.filter(regex='^holiday_.*').columns.tolist())
exog_features.extend(['temp', 'holiday', 'weather'])

df_exogenous_features = df_exogenous_features.filter(exog_features, axis=1)

# Merge target and exogenous variables in the same dataframe
# ==============================================================================
data = data[['users', 'weather']].merge(
            df_exogenous_features,
            left_index=True,
            right_index=True,
            how='inner' # To use only dates for which we have all the variables
        )
data = data.astype({col: np.float32 for col in data.select_dtypes("number").columns})
data_train = data.loc[: end_train, :].copy()
data_val   = data.loc[end_train:end_validation, :].copy()
data_test  = data.loc[end_validation:, :].copy()

# Backtesting model with exogenous variables on test data
# ==============================================================================
cv = TimeSeriesFold(steps = 36, initial_train_size = len(data.loc[:end_validation]))

metric, predictions = backtesting_forecaster(
                        forecaster    = forecaster,
                        y             = data['users'],
                        exog          = data[exog_features],
                        cv            = cv,
                        metric        = 'mean_absolute_error',
                        n_jobs        = 'auto',
                        verbose       = False,
                        show_progress = True
                     )
metric

  0%|          | 0/81 [00:00<?, ?it/s]
mean_absolute_error
0 48.663656

The inclusion of exogenous variables as predictors improves the predictive capacity of the model.

Hyperparameter tuning

Hyperparameter tuning is a critical step in developing effective machine learning models. The ForecasterRecursive used in the previous section included the first 24 lags and a LGMBRegressor model with the default hyperparameters. However, there is no reason why these values are the most appropriate. To find the best hyperparameters, a Bayesian Search is performed using the bayesian_search_forecaster() function. The search is carried out using the same backtesting process as before, but each time, the model is trained with different combinations of hyperparameters and lags. It is important to note that the hiperparameter search must be done using the validation set, so the test data is never used.

The search is performed by testing each combination of hyperparameters and lags as follows:

  1. Train the model using only the training set.

  2. The model is evaluated using the validation set via backtesting.

  3. Select the combination of hyperparameters and lags that gives the lowest error.

  4. Train the model again using the best combination found, this time using both the training and validation data.

By following these steps, one can obtain a model with optimized hyperparameters and avoid overfitting.

✎ Note

Searching for hyperparameters can take a long time, especially when using a validation strategy based on backtesting (TimeSeriesFold). A faster alternative is to use a validation strategy based on one-step-ahead predictions (OneStepAheadFold). Although this strategy is faster, it may not be as accurate as validation based on backtesting. For a more detailed description of the pros and cons of each strategy, see the section backtesting vs one-step-ahead. 
# Hyperparameters search
# ==============================================================================
forecaster = ForecasterRecursive(
                regressor        = LGBMRegressor(random_state=15926, verbose=-1),
                lags             = 72,
                window_features  = window_features,
                transformer_exog = transformer_exog,
                fit_kwargs       = {"categorical_feature": "auto"}
             )

# Lags grid
lags_grid = [48, 72, [1, 2, 3, 23, 24, 25, 167, 168, 169]]

# Regressor hyperparameters search space
def search_space(trial):
    search_space  = {
        'n_estimators'    : trial.suggest_int('n_estimators', 300, 1000, step=100),
        'max_depth'       : trial.suggest_int('max_depth', 3, 10),
        'min_data_in_leaf': trial.suggest_int('min_data_in_leaf', 25, 500),
        'learning_rate'   : trial.suggest_float('learning_rate', 0.01, 0.5),
        'feature_fraction': trial.suggest_float('feature_fraction', 0.5, 1),
        'max_bin'         : trial.suggest_int('max_bin', 50, 250),
        'reg_alpha'       : trial.suggest_float('reg_alpha', 0, 1),
        'reg_lambda'      : trial.suggest_float('reg_lambda', 0, 1),
        'lags'            : trial.suggest_categorical('lags', lags_grid)
    } 
    return search_space

# Folds training and validation
cv_search = TimeSeriesFold(steps = 36, initial_train_size = len(data_train))

results_search, frozen_trial = bayesian_search_forecaster(
    forecaster    = forecaster,
    y             = data.loc[:end_validation, 'users'], # Test data not used
    exog          = data.loc[:end_validation, exog_features],
    cv            = cv_search,
    search_space  = search_space,
    metric        = 'mean_absolute_error',
    n_trials      = 20, # Increase this value for a more exhaustive search
    return_best   = True
)

best_params = results_search['params'].iat[0]
best_params = best_params | {'random_state': 15926, 'verbose': -1}
best_lags   = results_search['lags'].iat[0]

  0%|          | 0/20 [00:00<?, ?it/s]
`Forecaster` refitted using the best-found lags and parameters, and the whole data set: 
  Lags: [  1   2   3  23  24  25 167 168 169] 
  Parameters: {'n_estimators': 300, 'max_depth': 8, 'min_data_in_leaf': 84, 'learning_rate': 0.03208367008069185, 'feature_fraction': 0.6966856646946507, 'max_bin': 141, 'reg_alpha': 0.2291498038065134, 'reg_lambda': 0.15917712164349562}
  Backtesting metric: 55.01654818651568

# Search results
# ==============================================================================
results_search.head(3)
lags params mean_absolute_error n_estimators max_depth min_data_in_leaf learning_rate feature_fraction max_bin reg_alpha reg_lambda
0 [1, 2, 3, 23, 24, 25, 167, 168, 169] {'n_estimators': 300, 'max_depth': 8, 'min_dat... 55.016548 300.0 8.0 84.0 0.032084 0.696686 141.0 0.229150 0.159177
1 [1, 2, 3, 23, 24, 25, 167, 168, 169] {'n_estimators': 400, 'max_depth': 8, 'min_dat... 57.497662 400.0 8.0 30.0 0.110618 0.814603 197.0 0.219161 0.014016
2 [1, 2, 3, 23, 24, 25, 167, 168, 169] {'n_estimators': 400, 'max_depth': 8, 'min_dat... 57.891986 400.0 8.0 53.0 0.111434 0.805030 191.0 0.044970 0.222861

Since return_best has been set to True, the forecaster object is automatically updated with the best configuration found and trained on the entire dataset. This final model can then be used for future predictions on new data.

Once the best combination of hyperparameters has been identified using the validation data, the predictive capacity of the model is evaluated when applied to the test set.

# Backtesting model
# ==============================================================================
cv = TimeSeriesFold(steps = 36, initial_train_size = len(data.loc[:end_validation]))
metric, predictions = backtesting_forecaster(
                        forecaster = forecaster,
                        y          = data['users'],
                        exog       = data[exog_features],
                        cv         = cv,
                        metric     = 'mean_absolute_error'
                     )
metric

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mean_absolute_error
0 46.937169 
# Plot predictions vs real value
# ======================================================================================
fig = go.Figure()
trace1 = go.Scatter(x=data_test.index, y=data_test['users'], name="test", mode="lines")
trace2 = go.Scatter(x=predictions.index, y=predictions['pred'], name="prediction", mode="lines")
fig.add_trace(trace1)
fig.add_trace(trace2)
fig.update_layout(
    title="Real value vs predicted in test data",
    xaxis_title="Date time",
    yaxis_title="Users",
    width=800,
    height=400,
    margin=dict(l=20, r=20, t=35, b=20),
    legend=dict(orientation="h", yanchor="top", y=1.1, xanchor="left", x=0.001)
)
fig.show()