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Probabilistic forecasting with machine learning
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Introduction¶
When trying to anticipate future values, most forecasting models try to predict what will be the most likely value. This is called point-forecasting. Although knowing in advance the expected value of a time series is useful in almost every business case, this kind of prediction does not provide any information about the confidence of the model nor the prediction uncertainty.
Probabilistic forecasting, as opposed to point-forecasting, is a family of techniques that allow for predicting the expected distribution of the outcome instead of a single future value. This type of forecasting provides much rich information since it allows for creating prediction intervals, the range of likely values where the true value may fall. More formally, a prediction interval defines the interval within which the true value of the response variable is expected to be found with a given probability.
There are multiple ways to estimate prediction intervals, most of which require that the residuals (errors) of the model follow a normal distribution. When this property cannot be assumed, two alternatives commonly used are bootstrapping and quantile regression. In order to illustrate how skforecast allows estimating prediction intervals for multi-step forecasting, the following examples are shown:
Prediction intervals based on bootstrapped residuals and recursive-multi-step forecaster.
Prediction intervals based on quantile regression and direct-multi-step forecaster.
⚠ Warning
As Rob J Hyndman explains in his blog, in real-world problems, almost all prediction intervals are too narrow. For example, nominal 95% intervals may only provide coverage between 71% and 87%. This is a well-known phenomenon and arises because they do not account for all sources of uncertainty. With forecasting models, there are at least four sources of uncertainty:- The random error term
- The parameter estimates
- The choice of model for the historical data
- The continuation of the historical data generating process into the future
✎ Note
If this is the first time using skforecast, please visit Skforecast: time series forecasting with Python and Scikit-learn for a quick introduction.✎ Note
Conformal prediction is a relatively new framework that allows for the creation of confidence measures for predictions made by machine learning models. This method is on the roadmap of skforecast, but not yet available.Prediction intervals using bootstrapped residuals¶
The error of one-step-ahead forecast is defined as $e_t = y_t - \hat{y}_{t|t-1}$. Assuming future errors will be like past errors, it is possible to simulate different predictions by sampling from the collection of errors previously seen in the past (i.e., the residuals) and adding them to the predictions.
Doing this repeatedly, a collection of slightly different predictions is created (possible future paths), that represent the expected variance in the forecasting process.
Finally, prediction intervals can be computed by calculating the $α/2$ and $1 − α/2$ percentiles of the simulated data at each forecasting horizon.
The main advantage of this strategy is that it only requires a single model to estimate any interval. The drawback is that, running hundreds or thousands of bootstrapping iterations, is computationally very expensive and not always workable.
Libraries¶
# Data processing
# ==============================================================================
import numpy as np
import pandas as pd
from skforecast.datasets import fetch_dataset
# Plots
# ==============================================================================
import matplotlib.pyplot as plt
from statsmodels.graphics.tsaplots import plot_acf
from statsmodels.graphics.tsaplots import plot_pacf
import plotly.graph_objects as go
import plotly.io as pio
import plotly.offline as poff
pio.templates.default = "seaborn"
pio.renderers.default = 'notebook'
poff.init_notebook_mode(connected=True)
plt.style.use('seaborn-v0_8-darkgrid')
from skforecast.plot import plot_residuals
from skforecast.plot import plot_prediction_distribution
from pprint import pprint
# Modelling and Forecasting
# ==============================================================================
import skforecast
import sklearn
import lightgbm
from lightgbm import LGBMRegressor
from sklearn.preprocessing import OneHotEncoder
from sklearn.compose import ColumnTransformer
from skforecast.ForecasterAutoreg import ForecasterAutoreg
from skforecast.ForecasterAutoregDirect import ForecasterAutoregDirect
from skforecast.model_selection import bayesian_search_forecaster
from skforecast.model_selection import backtesting_forecaster
from sklearn.metrics import mean_pinball_loss
from scipy.stats import norm
# 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 pandas: {pd.__version__}")
print(f"{color}Version numpy: {np.__version__}")
Data¶
# Data download
# ==============================================================================
data = fetch_dataset(name='bike_sharing_extended_features')
data.head(2)
# One hot encoding of categorical variables
# ==============================================================================
encoder = ColumnTransformer(
[('one_hot_encoder', OneHotEncoder(sparse_output=False), ['weather'])],
remainder='passthrough',
verbose_feature_names_out=False
).set_output(transform="pandas")
data = encoder.fit_transform(data)
# Selección de las variables exógenas
# ==============================================================================
exog_features = [
'weather_clear', 'weather_mist', 'weather_rain', 'month_sin', 'month_cos',
'week_of_year_sin', 'week_of_year_cos', 'week_day_sin', 'week_day_cos',
'hour_day_sin', 'hour_day_cos', 'sunrise_hour_sin', 'sunrise_hour_cos',
'sunset_hour_sin', 'sunset_hour_cos', 'temp'
]
data = data[['users'] + exog_features]
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
# ==============================================================================
data = data.loc['2011-05-30 23:59:00':, :]
end_train = '2012-08-30 23:59:00'
end_validation = '2012-11-15 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)})")
Graphic 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.
# 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=900,
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.show()
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=(6, 2))
plot_acf(data.users, ax=ax, lags=24*3)
plt.show()
# Partial autocorrelation plot
# ==============================================================================
fig, ax = plt.subplots(figsize=(6, 2))
plot_pacf(data.users, ax=ax, lags=24*3)
plt.show()
The autocorrelation plot demonstrates a strong correlation between the number of users in one hour and prior hours, as well as between users in one hour and the corresponding hour in preceding days. This observed correlation suggests that autoregressive models may be effective in this scenario.
In sample residuals¶
A recursive-multi-step forecaster is trained and its hyperparameters optimized. Then, prediction intervals based on bootstrapped residuals are estimated.
# Create forecaster and hyperparameters search
# ==============================================================================
# Forecaster
forecaster = ForecasterAutoreg(
regressor = LGBMRegressor(random_state=15926, verbose=-1),
lags = 7 # Place holder that will be replaced during the search
)
# Lags used as predictors
lags_grid = [24, 48, (1, 2, 3, 23, 24, 25, 47, 48, 49, 71, 72, 73, 364*24, 365*24)]
# Regressor hyperparameters search space
def search_space(trial):
search_space = {
'lags' : trial.suggest_categorical('lags', lags_grid),
'n_estimators' : trial.suggest_int('n_estimators', 200, 800, step=100),
'max_depth' : trial.suggest_int('max_depth', 3, 8, step=1),
'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, 0.8, step=0.1),
'max_bin' : trial.suggest_int('max_bin', 50, 100, step=25),
'reg_alpha' : trial.suggest_float('reg_alpha', 0, 1, step=0.1),
'reg_lambda' : trial.suggest_float('reg_lambda', 0, 1, step=0.1)
}
return search_space
results_search, frozen_trial = bayesian_search_forecaster(
forecaster = forecaster,
y = data.loc[:end_validation, 'users'],
exog = data.loc[:end_validation, exog_features],
steps = 24,
metric = 'mean_absolute_error',
search_space = search_space,
initial_train_size = len(data[:end_train]),
refit = False,
n_trials = 20, # Increase for more exhaustive search
random_state = 123,
return_best = True,
n_jobs = 'auto',
verbose = False,
show_progress = True
)
best_params = results_search['params'].iloc[0]
best_lags = results_search['lags'].iloc[0]
Once the best hyperparameters have been selected, the backtesting_forecaster() function is used to generate the prediction intervals for the entire test set.
The
intervalargument indicates the desired coverage probability of the prediction intervals. In this case,intervalis set to[10, 90], which means that the prediction intervals are calculated for the 10th and 90th percentiles, resulting in a theoretical coverage probability of 80%.The
n_bootargument is used to specify the number of bootstrap samples to be used in estimating the prediction intervals. The larger the number of samples, the more accurate the prediction intervals will be, but the longer the calculation will take.
By default, intervals are calculated using in-sample residuals (residuals from the training set). However, this can result in intervals that are too narrow (overly optimistic).
# Backtesting with prediction intervals in test data using in-sample residuals
# ==============================================================================
metric, predictions = backtesting_forecaster(
forecaster = forecaster,
y = data['users'],
exog = data[exog_features],
steps = 24,
metric = 'mean_absolute_error',
initial_train_size = len(data.loc[:end_validation]),
refit = False,
interval = [10, 90], # 80% prediction interval
n_boot = 250,
in_sample_residuals = True, # Use in-sample residuals
binned_residuals = False,
n_jobs = 'auto',
verbose = False,
show_progress = True
)
display(metric)
predictions.head(5)
# Function to plot predicted intervals
# ======================================================================================
def plot_predicted_intervals(
predictions: pd.DataFrame,
y_true: pd.DataFrame,
target_variable: str,
initial_x_zoom: list=None,
title: str=None,
xaxis_title: str=None,
yaxis_title: str=None,
):
"""
Plot predicted intervals vs real values
Parameters
----------
predictions : pandas DataFrame
Predicted values and intervals.
y_true : pandas DataFrame
Real values of target variable.
target_variable : str
Name of target variable.
initial_x_zoom : list, default `None`
Initial zoom of x-axis, by default None.
title : str, default `None`
Title of the plot, by default None.
xaxis_title : str, default `None`
Title of x-axis, by default None.
yaxis_title : str, default `None`
Title of y-axis, by default None.
"""
fig = go.Figure([
go.Scatter(name='Prediction', x=predictions.index, y=predictions['pred'], mode='lines'),
go.Scatter(name='Real value', x=y_true.index, y=y_true[target_variable], mode='lines'),
go.Scatter(
name='Upper Bound', x=predictions.index, y=predictions['upper_bound'],
mode='lines', marker=dict(color="#444"), line=dict(width=0), showlegend=False
),
go.Scatter(
name='Lower Bound', x=predictions.index, y=predictions['lower_bound'],
marker=dict(color="#444"), line=dict(width=0), mode='lines',
fillcolor='rgba(68, 68, 68, 0.3)', fill='tonexty', showlegend=False
)
])
fig.update_layout(
title=title,
xaxis_title=xaxis_title,
yaxis_title=yaxis_title,
width=900,
height=400,
margin=dict(l=20, r=20, t=35, b=20),
hovermode="x",
xaxis=dict(range=initial_x_zoom),
legend=dict(
orientation="h",
yanchor="top",
y=1.1,
xanchor="left",
x=0.001
)
)
fig.show()
def empirical_coverage(y, lower_bound, upper_bound):
"""
Calculate coverage of a given interval
"""
return np.mean(np.logical_and(y >= lower_bound, y <= upper_bound))
# Plot intervals (with zoom ['2012-12-01', '2012-12-20'])
# ==============================================================================
plot_predicted_intervals(
predictions = predictions,
y_true = data_test,
target_variable = "users",
initial_x_zoom = ['2012-12-01', '2012-12-20'],
title = "Real value vs predicted in test data",
xaxis_title = "Date time",
yaxis_title = "users",
)
# Predicted interval coverage (on test data)
# ==============================================================================
coverage = empirical_coverage(
y = data.loc[end_validation:, 'users'],
lower_bound = predictions["lower_bound"],
upper_bound = predictions["upper_bound"]
)
print(f"Predicted interval coverage: {round(100*coverage, 2)} %")
# Area of the interval
# ==============================================================================
area = (predictions["upper_bound"] - predictions["lower_bound"]).sum()
print(f"Area of the interval: {round(area, 2)}")
The prediction intervals exhibit overconfidence as they tend to be excessively narrow, resulting in a true coverage that falls below the nominal coverage. This phenomenon arises from the tendency of in-sample residuals to often overestimate the predictive capacity of the model.
Out sample residuals (non-conditioned on predicted values)¶
The set_out_sample_residuals() method is used to specify out-sample residuals computed with a validation set through backtesting. Once the new residuals have been added to the forecaster, set in_sample_residuals to False use them.
# Backtesting on validation data to obtain out-sample residuals
# ==============================================================================
_, predictions_val = backtesting_forecaster(
forecaster = forecaster,
y = data.loc[:end_validation, 'users'],
exog = data.loc[:end_validation, exog_features],
steps = 24,
metric = 'mean_absolute_error',
initial_train_size = len(data.loc[:end_train]),
refit = False,
n_jobs = 'auto',
verbose = False,
show_progress = True
)
residuals = data.loc[predictions_val.index, 'users'] - predictions_val['pred']
residuals = residuals.dropna()
# Out-sample residuals distribution
# ==============================================================================
print(pd.Series(np.where(residuals < 0, 'negative', 'positive')).value_counts())
plt.rcParams.update({'font.size': 8})
_ = plot_residuals(residuals=residuals, figsize=(7, 4))
# Store out-sample residuals in the forecaster
# ==============================================================================
forecaster.set_out_sample_residuals(residuals=residuals)
# Backtesting with prediction intervals in test data using out-sample residuals
# ==============================================================================
metric, predictions = backtesting_forecaster(
forecaster = forecaster,
y = data['users'],
exog = data[exog_features],
steps = 24,
metric = 'mean_absolute_error',
initial_train_size = len(data.loc[:end_validation]),
refit = False,
interval = [10, 90], # 80% prediction interval
n_boot = 250,
in_sample_residuals = False, # Use out-sample residuals
binned_residuals = False,
n_jobs = 'auto',
verbose = False,
show_progress = True
)
predictions.head(5)
# Plot intervals (with zoom ['2012-12-01', '2012-12-20'])
# ==============================================================================
plot_predicted_intervals(
predictions = predictions,
y_true = data_test,
target_variable = "users",
initial_x_zoom = ['2012-12-01', '2012-12-20'],
title = "Real value vs predicted in test data",
xaxis_title = "Date time",
yaxis_title = "users",
)
# Predicted interval coverage (on test data)
# ==============================================================================
coverage = empirical_coverage(
y = data.loc[end_validation:, 'users'],
lower_bound = predictions["lower_bound"],
upper_bound = predictions["upper_bound"]
)
print(f"Predicted interval coverage: {round(100*coverage, 2)} %")
# Area of the interval
# ==============================================================================
area = (predictions["upper_bound"] - predictions["lower_bound"]).sum()
print(f"Area of the interval: {round(area, 2)}")
The resulting prediction intervals derived from the out-of-sample residuals are wider than those generated using the in-sample residuals. This results in an empirical coverage that is closer to the nominal coverage, although still lower. Examining the plot, it is easy to see that the intervals are particularly wide when the predicted values are low, indicating that the model is not able to properly locate the uncertainty of its predictions.
Out sample residuals (conditioned on predicted values)¶
The bootstrapping process assumes that the residuals are independently distributed so that they can be used independently of the predicted value. In reality, this is rarely true; in most cases, the magnitude of the residuals is correlated with the magnitude of the predicted value. In this case, for example, one would hardly expect the error to be the same when the predicted number of users is close to zero as when it is in the hundreds.
To account for the dependence between the residuals and the predicted values, skforecast allows to partition the residuals into K bins, where each bin is associated with a range of predicted values. Using this strategy, the bootstrapping process samples the residuals from different bins depending on the predicted value, which can improve the coverage of the interval while adjusting the width if necessary, allowing the model to better distribute the uncertainty of its predictions.
To enable the forecaster to bin the out-sample residuals, the predicted values are passed to the set_out_sample_residuals() method in addition to the residuals. Internally, skforecast uses a sklearn.preprocessing.KBinsDiscretizer with the parameters: n_bins=15, encode=ordinal, strategy=quantile,
subsample=10000, random_state=789654, dtype=np.float64. The binning process can be adjusted using the argument binner_kwargs of the Forecaster object.
# Create and train forecaster
# ==============================================================================
forecaster = ForecasterAutoreg(
regressor = LGBMRegressor(random_state=15926, verbose=-1, **best_params),
lags = best_lags,
binner_kwargs = {'n_bins': 15}
)
forecaster.fit(
y = data.loc[:end_validation, 'users'],
exog = data.loc[:end_validation, exog_features]
)
During the training process, the forecaster uses the in-sample predictions to define the intervals at which the residuals are stored, depending on the predicted value to which they are related. Although not used in this example, the in-sample residuals are divided into bins and stored in the in_sample_residuals_by_bin attribute.
# Intervals of the residual bins
# ==============================================================================
pprint(forecaster.binner_intervals)
Next, the out-of-sample residuals are stored in the predictor. Since the predicted values are also provided, they are binned according to the intervals learned during the fit. To avoid using too much memory, a maximum of 200 residuals are stored per bin.
# Store out-sample residuals in the forecaster
# ==============================================================================
forecaster.set_out_sample_residuals(residuals=residuals, y_pred=predictions_val['pred'])
# Number of residuals by bin
# ==============================================================================
for k, v in forecaster.out_sample_residuals_by_bin.items():
print(f" Bin {k}: n={len(v)}")
# Distribution of the residual by bin
# ==============================================================================
out_sample_residuals_by_bin_df = pd.DataFrame(
dict([(k, pd.Series(v)) for k,v in forecaster.out_sample_residuals_by_bin.items()])
)
fig, ax = plt.subplots(figsize=(6, 3))
out_sample_residuals_by_bin_df.boxplot(ax=ax)
ax.set_title("Distribution of residuals by bin")
ax.set_xlabel("Bin")
ax.set_ylabel("Residuals");
The box plots show how the spread and magnitude of the residuals differ depending on the predicted value. For example, for bin 0, whose predicted value is in the interval (-8.2, 11.1), the residuals never exceed an absolute value of 100, while for bin 9, for predicted values in the interval (486.2, 970.5), they often do.
Finally, the prediction intervals for the test data are estimated using the backtesting process, with out-of-sample residuals conditioned on the predicted values.
# Backtesting with prediction intervals in test data using out-sample residuals
# ==============================================================================
metric, predictions = backtesting_forecaster(
forecaster = forecaster,
y = data['users'],
exog = data[exog_features],
steps = 24,
metric = 'mean_absolute_error',
initial_train_size = len(data.loc[:end_validation]),
refit = False,
interval = [10, 90], # 80% prediction interval
n_boot = 250,
in_sample_residuals = False, # Use out-sample residuals
binned_residuals = True, # Use binned residuals
n_jobs = 'auto',
verbose = False,
show_progress = True
)
predictions.head(3)
# Plot intervals (with zoom ['2012-12-01', '2012-12-20'])
# ==============================================================================
plot_predicted_intervals(
predictions = predictions,
y_true = data_test,
target_variable = "users",
initial_x_zoom = ['2012-12-01', '2012-12-20'],
title = "Real value vs predicted in test data",
xaxis_title = "Date time",
yaxis_title = "users",
)
# Predicted interval coverage (on test data)
# ==============================================================================
coverage = empirical_coverage(
y = data.loc[end_validation:, 'users'],
lower_bound = predictions["lower_bound"],
upper_bound = predictions["upper_bound"]
)
print(f"Predicted interval coverage: {round(100*coverage, 2)} %")
# Area of the interval
# ==============================================================================
area = (predictions["upper_bound"] - predictions["lower_bound"]).sum()
print(f"Area of the interval: {round(area, 2)}")
When using out-of-sample residuals conditioned on the predicted value, the area of the interval is significantly reduced and the uncertainty is mainly allocated to the predictions with high values. The empirical coverage is slightly above the expected coverage, which means that the estimated intervals are conservative.
Predict bootstraping, quantile and distribution¶
The previous sections have demonstrated the use of the backtesting process to estimate the prediction interval over a given period of time. The goal is to mimic the behavior of the model in production by running predictions at regular intervals, incrementally updating the input data.
Alternatively, it is possible to run a single prediction that forecasts N steps ahead without going through the entire backtesting process. In such cases, skforecast provides four different methods: predict_bootstrapping, predict_interval, predict_quantile and predict_distribution.
Predict Bootstraping
The predict_bootstrapping method performs the n_boot bootstrapping iterations that generate the alternative prediction paths. These are the underlying values used to compute the intervals, quantiles, and distributions.
# Fit forecaster
# ==============================================================================
forecaster.fit(
y = data.loc[:end_validation, 'users'],
exog = data.loc[:end_validation, exog_features]
)
# Predict 10 different forecasting sequences of 7 steps each using bootstrapping
# ==============================================================================
boot_predictions = forecaster.predict_bootstrapping(
exog = data_test[exog_features],
steps = 7,
n_boot = 25
)
boot_predictions
A ridge plot is a useful way to visualize the uncertainty of a forecasting model. This plot estimates a kernel density for each step by using the bootstrapped predictions.
# Ridge plot of bootstrapping predictions
# ==============================================================================
_ = plot_prediction_distribution(boot_predictions, figsize=(7, 4))
Predict Interval
In most cases, the user is interested in a specific interval rather than the entire bootstrapping simulation matrix. To address this need, skforecast provides the predict_interval method. This method internally uses predict_bootstrapping to obtain the bootstrapping matrix and estimates the upper and lower quantiles for each step, thus providing the user with the desired prediction intervals.
# Predict intervals for next 7 steps, quantiles 10th and 90th
# ==============================================================================
predictions = forecaster.predict_interval(
exog = data_test[exog_features],
steps = 7,
interval = [10, 90],
n_boot = 150
)
predictions
Predict Quantile
This method operates identically to predict_interval, with the added feature of enabling users to define a specific list of quantiles for estimation at each step. It's important to remember that these quantiles should be specified within the range of 0 to 1.
# Predict quantiles for next 7 steps, quantiles 5th, 25th, 75th and 95th
# ==============================================================================
predictions = forecaster.predict_quantiles(
exog = data_test[exog_features],
steps = 7,
n_boot = 150,
quantiles = [0.05, 0.25, 0.75, 0.95],
)
predictions
Predict Distribution
The intervals estimated so far are distribution-free, which means that no assumptions are made about a particular distribution. The predict_dist method in skforecast allows fitting a parametric distribution to the bootstrapped prediction samples obtained with predict_bootstrapping. This is useful when there is reason to believe that the forecast errors follow a particular distribution, such as the normal distribution or the student's t-distribution. The predict_dist method allows the user to specify any continuous distribution from the scipy.stats module.
# Predict the parameters of a normal distribution for the next 7 steps
# ==============================================================================
predictions = forecaster.predict_dist(
exog = data_test[exog_features],
steps = 7,
n_boot = 150,
distribution = norm
)
predictions
Prediction intervals using quantile regression models¶
As opposed to ordinal linear regression, which is intended to estimate the conditional mean of the response variable given certain values of the predictor variables, quantile regression aims at estimating the conditional quantiles of the response variable. For a continuous distribution function, the $\alpha$-quantile $Q_{\alpha}(x)$ is defined such that the probability of $Y$ being smaller than $Q_{\alpha}(x)$ is, for a given $X=x$, equal to $\alpha$. For example, 36% of the population values are lower than the quantile $Q=0.36$. The most known quantile is the 50%-quantile, more commonly called the median.
By combining the predictions of two quantile regressors, it is possible to build an interval. Each model estimates one of the limits of the interval. For example, the models obtained for $Q = 0.1$ and $Q = 0.9$ produce an 80% prediction interval (90% - 10% = 80%).
Several machine learning algorithms are capable of modeling quantiles. Some of them are:
Just as the squared-error loss function is used to train models that predict the mean value, a specific loss function is needed in order to train models that predict quantiles. The most common metric used for quantile regression is calles quantile loss or pinball loss:
$$\text{pinball}(y, \hat{y}) = \frac{1}{n_{\text{samples}}} \sum_{i=0}^{n_{\text{samples}}-1} \alpha \max(y_i - \hat{y}_i, 0) + (1 - \alpha) \max(\hat{y}_i - y_i, 0)$$where $\alpha$ is the target quantile, $y$ the real value and $\hat{y}$ the quantile prediction.
It can be seen that loss differs depending on the evaluated quantile. The higher the quantile, the more the loss function penalizes underestimates, and the less it penalizes overestimates. As with MSE and MAE, the goal is to minimize its values (the lower loss, the better).
Two disadvantages of quantile regression, compared to the bootstrap approach to prediction intervals, are that each quantile needs its regressor and quantile regression is not available for all types of regression models. However, once the models are trained, the inference is much faster since no iterative process is needed.
This type of prediction intervals can be easily estimated using quantile regressor inside a Forecaster object.
⚠ Warning
Forecasters of typeForecasterAutoregDirect are slower than ForecasterAutoregRecursive because they require training one model per step. Although they can achieve better performance, their scalability is an important limitation when many steps need to be predicted. To limit the time required to run the following examples, the data is aggregated from hourly frequency to daily frequency and only 7 steps ahead (one week) are predicted.
Data¶
# Data download
# ==============================================================================
data = fetch_dataset(name='bike_sharing_extended_features', verbose=False)
# Aggregate data to daily frequency
# ==============================================================================
data = (
data
.resample(rule="D", closed="left", label="right")
.agg({"users": "sum"})
)
# Split train-validation-test
# ==============================================================================
end_train = '2012-05-31 23:59:00'
end_validation = '2012-09-15 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)})")
Quantile regresion models¶
An 80% prediction interval is estimated for 7 steps-ahead predictions using quantile regression. A LightGBM gradient boosting model is trained in this example, however, the reader may use any other model just replacing the definition of the regressor.
# Create forecasters: one for each limit of the interval
# ==============================================================================
# The forecasters obtained for alpha=0.1 and alpha=0.9 produce a 80% confidence
# interval (90% - 10% = 80%).
# Forecaster for quantile 10%
forecaster_q10 = ForecasterAutoregDirect(
regressor = LGBMRegressor(
objective = 'quantile',
metric = 'quantile',
alpha = 0.1,
random_state = 15926,
verbose = -1
),
lags = 7,
steps = 7
)
# Forecaster for quantile 90%
forecaster_q90 = ForecasterAutoregDirect(
regressor = LGBMRegressor(
objective = 'quantile',
metric = 'quantile',
alpha = 0.9,
random_state = 15926,
verbose = -1
),
lags = 7,
steps = 7
)
When validating a quantile regression model, a custom metric must be provided depending on the quantile being estimated.
# Loss function for each quantile (pinball_loss)
# ==============================================================================
def mean_pinball_loss_q10(y_true, y_pred):
"""
Pinball loss for quantile 10.
"""
return mean_pinball_loss(y_true, y_pred, alpha=0.1)
def mean_pinball_loss_q90(y_true, y_pred):
"""
Pinball loss for quantile 90.
"""
return mean_pinball_loss(y_true, y_pred, alpha=0.9)
# Bayesian search of hyper-parameters and lags for each quantile forecaster
# ==============================================================================
def search_space(trial):
search_space = {
'n_estimators' : trial.suggest_int('n_estimators', 100, 500, step=50),
'max_depth' : trial.suggest_int('max_depth', 3, 10, step=1),
'learning_rate' : trial.suggest_float('learning_rate', 0.01, 0.1)
}
return search_space
results_grid_q10 = bayesian_search_forecaster(
forecaster = forecaster_q10,
y = data.loc[:end_validation, 'users'],
steps = 7,
metric = mean_pinball_loss_q10,
search_space = search_space,
initial_train_size = len(data[:end_train]),
refit = False,
n_trials = 10, # Increase for more exhaustive search
random_state = 123,
return_best = True,
n_jobs = 'auto',
verbose = False,
show_progress = True
)
results_grid_q90 = bayesian_search_forecaster(
forecaster = forecaster_q90,
y = data.loc[:end_validation, 'users'],
steps = 7,
metric = mean_pinball_loss_q90,
search_space = search_space,
initial_train_size = len(data[:end_train]),
refit = False,
n_trials = 10, # Increase for more exhaustive search
random_state = 123,
return_best = True,
n_jobs = 'auto',
verbose = False,
show_progress = True
)
Once the best hyper-parameters have been found for each forecaster, a backtesting process is applied again using the test data.
# Backtesting on test data
# ==============================================================================
metric_q10, predictions_q10 = backtesting_forecaster(
forecaster = forecaster_q10,
y = data['users'],
steps = 7,
metric = mean_pinball_loss_q10,
initial_train_size = len(data.loc[:end_validation]),
refit = False,
n_jobs = 'auto',
verbose = False,
show_progress = True
)
metric_q90, predictions_q90 = backtesting_forecaster(
forecaster = forecaster_q90,
y = data['users'],
steps = 7,
metric = mean_pinball_loss_q90,
initial_train_size = len(data.loc[:end_validation]),
refit = False,
n_jobs = 'auto',
verbose = False,
show_progress = True
)
# Plot
# ==============================================================================
fig = go.Figure([
go.Scatter(name='Real value', x=data_test.index, y=data_test['users'], mode='lines'),
go.Scatter(
name='Upper Bound', x=predictions_q90.index, y=predictions_q90['pred'],
mode='lines', marker=dict(color="#444"), line=dict(width=0), showlegend=False
),
go.Scatter(
name='Lower Bound', x=predictions_q10.index, y=predictions_q10['pred'],
marker=dict(color="#444"), line=dict(width=0), mode='lines',
fillcolor='rgba(68, 68, 68, 0.3)', fill='tonexty', showlegend=False
)
])
fig.update_layout(
title="Real value vs predicted in test data",
xaxis_title="Date time",
yaxis_title="users",
width=900,
height=400,
margin=dict(l=20, r=20, t=35, b=20),
hovermode="x",
legend=dict(
orientation="h",
yanchor="top",
y=1.1,
xanchor="left",
x=0.001
)
)
fig.show()
# Predicted interval coverage (on test data)
# ==============================================================================
coverage = empirical_coverage(
y = data.loc[end_validation:, 'users'],
lower_bound = predictions_q10["pred"],
upper_bound = predictions_q90["pred"]
)
print(f"Predicted interval coverage: {round(100*coverage, 2)} %")
# Area of the interval
# ==============================================================================
area = (predictions_q90["pred"] - predictions_q10["pred"]).sum()
print(f"Area of the interval: {round(area, 2)}")
In this use case, the quantile forecasting strategy does not achieve empirical coverage close to the expected coverage (80 percent).
Session information¶
import session_info
session_info.show(html=False)
Bibliography¶
Hyndman, R.J., & Athanasopoulos, G. (2021) Forecasting: principles and practice, 3rd edition, OTexts: Melbourne, Australia.
Forecasting: theory and practice PDF
Citation¶
How to cite this document
If you use this document or any part of it, please acknowledge the source, thank you!
Probabilistic forecasting with machine learning 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://cienciadedatos.net/documentos/py42-probabilistic-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 (v0.13.0). Zenodo. https://doi.org/10.5281/zenodo.8382788
APA:
Amat Rodrigo, J., & Escobar Ortiz, J. (2024). skforecast (Version 0.13.0) [Computer software]. https://doi.org/10.5281/zenodo.8382788
BibTeX:
@software{skforecast, author = {Amat Rodrigo, Joaquin and Escobar Ortiz, Javier}, title = {skforecast}, version = {0.13.0}, month = {8}, 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.