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A time series is a succession of chronologically ordered data spaced at equal or unequal intervals. The forecasting process consists of predicting the future value of a time series, either by modeling the series solely based on its past behavior (autoregressive) or by using other external variables.

This guide explores the use of **scikit-learn** regression models for time series forecasting. Specifically, it introduces **skforecast**, an intuitive library equipped with essential classes and functions to customize any Scikit-learn regression model to effectively address forecasting challenges.

**✎ Note**

In order to apply machine learning models to forecasting problems, the time series has to be transformed into a matrix in which each value is related to the time window (lags) that precedes it.

In a time series context, a lag with respect to a time step $t$ is defined as the values of the series at previous time steps. For example, lag 1 is the value at time step $t − 1$ and lag $m$ is the value at time step $t − m$.

This type of transformation also allows to include additional variables.

Once data have been rearranged into the new shape, any regression model can be trained to predict the next value (step) of the series. During model training, every row is considered a separate data instance, where values at lags 1, 2, ... $p$ are considered predictors for the target quantity of the time series at time step $t + 1$.

When working with time series, it is seldom needed to predict only the next element in the series ($t_{+1}$). Instead, the most common goal is to predict a whole future interval (($t_{+1}$), ..., ($t_{+n}$)) or a far point in time ($t_{+n}$). Several strategies allow generating this type of prediction.

Since the value $t_{n-1}$ is required to predict $t_{n}$, and $t_{n-1}$ is unknown, a recursive process is applied in which, each new prediction, is based on the previous one. This process is known as recursive forecasting or recursive multi-step forecasting and can be easily generated with the `ForecasterAutoreg`

and `ForecasterAutoregCustom`

classes.

Direct multi-step forecasting consists of training a different model for each step of the forecast horizon. For example, to predict the next 5 values of a time series, 5 different models are trained, one for each step. As a result, the predictions are independent of each other.

The main complexity of this approach is to generate the correct training matrices for each model. The `ForecasterAutoregDirect`

class of the **skforecast** library automates this process. It is also important to bear in mind that this strategy has a higher computational cost since it requires the train of multiple models. The following diagram shows the process for a case in which the response variable and two exogenous variables are available.

Some machine learning models, such as long short-term memory (LSTM) neural networks, can predict multiple values of a sequence simultaneously (one-shot). This strategy is not currently implemented in the **skforecast** library, but is expected to be included in future versions.

The libraries used in this document are:

In [1]:

```
# Data manipulation
# ==============================================================================
import numpy as np
import pandas as pd
from skforecast.datasets import fetch_dataset
# Plots
# ==============================================================================
import matplotlib.pyplot as plt
plt.style.use('fivethirtyeight')
plt.rcParams['lines.linewidth'] = 1.5
plt.rcParams['font.size'] = 10
# Modeling and Forecasting
# ==============================================================================
from sklearn.linear_model import Ridge
from sklearn.ensemble import RandomForestRegressor
from sklearn.metrics import mean_squared_error
from sklearn.metrics import mean_absolute_error
from sklearn.preprocessing import StandardScaler
import skforecast
from skforecast.ForecasterAutoreg import ForecasterAutoreg
from skforecast.ForecasterAutoregCustom import ForecasterAutoregCustom
from skforecast.ForecasterAutoregDirect import ForecasterAutoregDirect
from skforecast.model_selection import grid_search_forecaster
from skforecast.model_selection import backtesting_forecaster
from skforecast.utils import save_forecaster
from skforecast.utils import load_forecaster
import shap
# Warnings configuration
# ==============================================================================
import warnings
warnings.filterwarnings('once')
print('Skforecast version: ', skforecast.__version__)
```

A time series is available with the monthly expenditure (millions of dollars) on corticosteroid drugs that the Australian health system had between 1991 and 2008. It is intended to create an autoregressive model capable of predicting future monthly expenditures. The data used in the examples of this document have been obtained from the magnificent book Forecasting: Principles and Practice by Rob J Hyndman and George Athanasopoulos.

In [2]:

```
# Data download
# ==============================================================================
data = fetch_dataset(name='h2o_exog', raw=True)
```

The column `date`

has been stored as a `string`

. To convert it to `datetime`

the `pd.to_datetime()`

function can be use. Once in `datetime`

format, and to make use of **Pandas** functionalities, it is set as an index. Also, since the data is monthly, the frequency is set as Monthly Started 'MS'.

In [3]:

```
# Data preparation
# ==============================================================================
data = data.rename(columns={'fecha': 'date'})
data['date'] = pd.to_datetime(data['date'], format='%Y-%m-%d')
data = data.set_index('date')
data = data.asfreq('MS')
data = data.sort_index()
data.head()
```

Out[3]:

When using the `asfreq()`

method in Pandas, any gaps in the time series will be filled with `NaN`

values to match the specified frequency. Therefore, it is essential to check for any missing values that may occur after this transformation.

In [4]:

```
# Missing values
# ==============================================================================
print(f'Number of rows with missing values: {data.isnull().any(axis=1).mean()}')
```

Although it is unnecessary, since a frequency has been established, it is possible to verify that the time series is complete.

In [5]:

```
# Verify that a temporary index is complete
# ==============================================================================
(data.index == pd.date_range(start=data.index.min(),
end=data.index.max(),
freq=data.index.freq)).all()
```

Out[5]:

In [6]:

```
# Fill gaps in a temporary index
# ==============================================================================
# data.asfreq(freq='30min', fill_value=np.nan)
```

The last 36 months are used as the test set to evaluate the predictive capacity of the model.

In [7]:

```
# Split data into train-test
# ==============================================================================
steps = 36
data_train = data[:-steps]
data_test = data[-steps:]
print(
f"Train dates : {data_train.index.min()} --- "
f"{data_train.index.max()} (n={len(data_train)})"
)
print(
f"Test dates : {data_test.index.min()} --- "
f"{data_test.index.max()} (n={len(data_test)})"
)
fig, ax = plt.subplots(figsize=(6, 2.5))
data_train['y'].plot(ax=ax, label='train')
data_test['y'].plot(ax=ax, label='test')
ax.legend();
```

With the `ForecasterAutoreg`

class, a forecasting model is created and trained using a `RandomForestRegressor`

regressor with a time window of 6 lags. This means that the model uses the previous 6 months as predictors.

In [8]:

```
# Create and train forecaster
# ==============================================================================
forecaster = ForecasterAutoreg(
regressor = RandomForestRegressor(random_state=123),
lags = 6
)
forecaster.fit(y=data_train['y'])
forecaster
```

Out[8]:

Once the model is trained, the test data is predicted (36 months into the future).

In [9]:

```
# Predictions
# ==============================================================================
steps = 36
predictions = forecaster.predict(steps=steps)
predictions.head(5)
```

Out[9]:

In [10]:

```
# Plot predictions versus test data
# ==============================================================================
fig, ax = plt.subplots(figsize=(6, 2.5))
data_train['y'].plot(ax=ax, label='train')
data_test['y'].plot(ax=ax, label='test')
predictions.plot(ax=ax, label='predictions')
ax.legend();
```

The error that the model makes in its predictions is quantified. In this case, the metric used is the mean squared error (mse).

In [11]:

```
# Test error
# ==============================================================================
error_mse = mean_squared_error(
y_true = data_test['y'],
y_pred = predictions
)
print(f"Test error (MSE): {error_mse}")
```

The trained `ForecasterAutoreg`

uses a 6 lag time window and a Random Forest model with the default hyperparameters. However, there is no reason why these values are the most suitable. **Skforecast** provide several search strategies to find the best combination of hyperparameters and lags. In this case, the `grid_search_forecaster`

function is used. It compares the results obtained with each combinations of hyperparameters and lags, and identify the best one.

**💡 Tip**

The computational cost of hyperparameter tuning depends heavily on the backtesting approach chosen to evaluate each hyperparameter combination. In general, the duration of the tuning process increases with the number of re-trains involved in the backtesting.

To effectively speed up the prototyping phase, it is highly recommended to adopt a two-step strategy. First, use `refit=False`

during the initial search to narrow down the range of values. Then, focus on the identified region of interest and apply a tailored backtesting strategy that meets the specific requirements of the use case. For more tips on backtesting strategies, see Hyperparameter tuning and lags selection.

In [12]:

```
# Hyperparameters: grid search
# ==============================================================================
steps = 36
forecaster = ForecasterAutoreg(
regressor = RandomForestRegressor(random_state=123),
lags = 12 # This value will be replaced in the grid search
)
# Candidate values for lags
lags_grid = [10, 20]
# Candidate values for regressor's hyperparameters
param_grid = {
'n_estimators': [100, 250],
'max_depth': [3, 5, 10]
}
results_grid = grid_search_forecaster(
forecaster = forecaster,
y = data_train['y'],
param_grid = param_grid,
lags_grid = lags_grid,
steps = steps,
refit = False,
metric = 'mean_squared_error',
initial_train_size = int(len(data_train)*0.5),
fixed_train_size = False,
return_best = True,
n_jobs = 'auto',
verbose = False
)
```

In [13]:

```
# Search results
# ==============================================================================
results_grid
```

Out[13]:

The best results are obtained using a time window of `20 lags`

and a Random Forest set up of `{'max_depth': 3, 'n_estimators': 250}`

.

Finally, a `ForecasterAutoreg`

is trained with the optimal configuration found. This step is not necessary if `return_best = True`

is specified in the `grid_search_forecaster`

function.

In [14]:

```
# Create and train forecaster with the best hyperparameters and lags found
# ==============================================================================
regressor = RandomForestRegressor(n_estimators=250, max_depth=3, random_state=123)
forecaster = ForecasterAutoreg(
regressor = regressor,
lags = 20
)
forecaster.fit(y=data_train['y'])
```

In [15]:

```
# Predictions
# ==============================================================================
predictions = forecaster.predict(steps=steps)
```

In [16]:

```
# Plot predictions versus test data
# ==============================================================================
fig, ax = plt.subplots(figsize=(6, 2.5))
data_train['y'].plot(ax=ax, label='train')
data_test['y'].plot(ax=ax, label='test')
predictions.plot(ax=ax, label='predictions')
ax.legend();
```

In [17]:

```
# Test error
# ==============================================================================
error_mse = mean_squared_error(
y_true = data_test['y'],
y_pred = predictions
)
print(f"Test error (MSE): {error_mse}")
```

The optimal combination of hyperparameters significantly reduces test error.

To obtain a robust estimate of the model's predictive capacity, a backtesting process is carried out. The process of backtesting consists of evaluating the performance of a predictive model by applying it retrospectively to historical data. Therefore, it is a special type of cross-validation applied to the previous period(s).

**✎ Note**

To ensure an accurate evaluation of your model and gain confidence in its predictive performance on new data, it is critical to employ an appropriate backtesting strategy. Factors such as use case characteristics, available computing resources and time intervals between predictions need to be considered to determine which strategy to use.

In general, the more closely the backtesting process resembles the actual scenario in which the model is used, the more reliable the estimated metric will be. For more information about backtesting, visit Which strategy should I use?.

**Backtesting with refit and increasing training size (fixed origin)**

The model is trained each time before making predictions. With this configuration, the model uses all the data available so far. It is a variation of the standard cross-validation but, instead of making a random distribution of the observations, the training set increases sequentially, maintaining the temporal order of the data.

**Backtesting with refit and fixed training size (rolling origin)**

A technique similar to the previous one but, in this case, the forecast origin rolls forward, therefore, the size of training remains constant. This is also known as time series cross-validation or walk-forward validation.

**Backtesting with intermittent refit**

The model is retrained every $n$ iterations of predictions.

**💡 Tip**

**Backtesting without refit**

After an initial train, the model is used sequentially without updating it and following the temporal order of the data. This strategy has the advantage of being much faster since the model is trained only once. However, the model does not incorporate the latest data available, so it may lose predictive capacity over time.

**Skforecast** library has multiple backtesting strategies implemented. Regardless of which one is used, it is important not to include test data in the search process to avoid overfitting problems.

For this example, a **backtesting with refit** strategy is followed. Internally, the process that the function applies is:

In the first iteration, the model is trained with the observations selected for the initial training (in this case, 87). Then, the next 36 observations are used to validate the predictions of this first model.

In the second iteration, the model is retrained by adding, to the initial training set, the previous 36 validation observations (87 + 36). In the same way, the next 36 observations are established as the new validation set.

This process is repeated until all available observations are used. Following this strategy, the training set increases in each iteration with as many observations as steps are being predicted.

In [18]:

```
# Backtesting
# ==============================================================================
steps = 36
n_backtesting = 36*3 # The last 9 years are separated for the backtest
metric, predictions_backtest = backtesting_forecaster(
forecaster = forecaster,
y = data['y'],
initial_train_size = len(data) - n_backtesting,
fixed_train_size = False,
steps = steps,
metric = 'mean_squared_error',
refit = True,
verbose = True,
show_progress = True
)
print(f"Backtest metric (MSE): {metric}")
```

In [19]:

```
# Plot backtest predictions vs real values
# ==============================================================================
fig, ax = plt.subplots(figsize=(6, 2.5))
data.loc[predictions_backtest.index, 'y'].plot(ax=ax, label='test')
predictions_backtest.plot(ax=ax, label='predictions')
ax.legend();
```

Due to the complex nature of many modern machine learning models, such as ensemble methods, they often function as black boxes, making it difficult to understand why a particular prediction was made. Explanability techniques aim to demystify these models, providing insight into their inner workings and helping to build trust, improve transparency, and meet regulatory requirements in various domains. Enhancing model explainability not only helps to understand model behavior, but also helps to identify biases, improve model performance, and enable stakeholders to make more informed decisions based on machine learning insights.

Skforecast is compatible with some of the most popular model explainability methods: model-specific feature importances, SHAP values, and partial dependence plots.

**Model-specific feature importance**

In [20]:

```
# Extract feature importance
# ==============================================================================
importance = forecaster.get_feature_importances()
importance.sort_values(by='importance', ascending=False).head(10)
```

Out[20]:

** ⚠ Warning**

`get_feature_importances()`

method will only return values if the forecaster's regressor has either the `coef_`

or `feature_importances_`

attribute, which is the default in scikit-learn.
**Shap values**

SHAP (SHapley Additive exPlanations) values are a popular method for explaining machine learning models, as they help to understand how variables and values influence predictions visually and quantitatively.

It is possible to generate SHAP-values explanations from skforecast models with just two essential elements:

The internal regressor of the forecaster.

The training matrices created from the time series and used to fit the forecaster.

By leveraging these two components, users can create insightful and interpretable explanations for their skforecast models. These explanations can be used to verify the reliability of the model, identify the most significant factors that contribute to model predictions, and gain a deeper understanding of the underlying relationship between the input variables and the target variable.

In [21]:

```
# Training matrices used by the forecaster to fit the internal regressor
# ==============================================================================
X_train, y_train = forecaster.create_train_X_y(y=data_train['y'])
# Create SHAP explainer (for three base models)
# ==============================================================================
explainer = shap.TreeExplainer(forecaster.regressor)
# Sample 50% of the data to speed up the calculation
rng = np.random.default_rng(seed=785412)
sample = rng.choice(X_train.index, size=int(len(X_train)*0.5), replace=False)
X_train_sample = X_train.loc[sample, :]
shap_values = explainer.shap_values(X_train_sample)
# Shap summary plot (top 10)
# ==============================================================================
shap.initjs()
shap.summary_plot(shap_values, X_train_sample, max_display=10, show=False)
fig, ax = plt.gcf(), plt.gca()
ax.set_title("SHAP Summary plot")
ax.tick_params(labelsize=8)
fig.set_size_inches(6, 3.5)
```

**✎ Note**

`shap.TreeExplainer`

explainer is used for tree-based models, such as the `RandomForestRegressor`

used in this example. For more information, see the SHAP documentation.
In the previous example, only lags of the predicted variable itself were used as predictors. In certain scenarios, it is possible to have information about other variables, **whose future value is known**, so could serve as additional predictors in the model.

Continuing with the previous example, a new variable is simulated whose behavior is correlated with the modeled time series and which is to be included as a predictor.

In [22]:

```
# Data download
# ==============================================================================
data = fetch_dataset(name='h2o_exog', raw=True, verbose=False)
```

In [23]:

```
# Data preparation
# ==============================================================================
data = data.rename(columns={'fecha': 'date'})
data['date'] = pd.to_datetime(data['date'], format='%Y-%m-%d')
data = data.set_index('date')
data = data.asfreq('MS')
data = data.sort_index()
fig, ax = plt.subplots(figsize=(6, 2.7))
data['y'].plot(ax=ax, label='y')
data['exog_1'].plot(ax=ax, label='exogenous variable')
ax.legend(loc='upper left');
```

In [24]:

```
# Split data into train-test
# ==============================================================================
steps = 36
data_train = data[:-steps]
data_test = data[-steps:]
print(
f"Train dates : {data_train.index.min()} --- "
f"{data_train.index.max()} (n={len(data_train)})"
)
print(
f"Test dates : {data_test.index.min()} --- "
f"{data_test.index.max()} (n={len(data_test)})"
)
```

In [25]:

```
# Create and train forecaster
# ==============================================================================
forecaster = ForecasterAutoreg(
regressor = RandomForestRegressor(random_state=123),
lags = 8
)
forecaster.fit(y=data_train['y'], exog=data_train['exog_1'])
forecaster
```

Out[25]:

Since the `ForecasterAutoreg`

has been trained with an exogenous variable, the value of this variable must be passed to `predict()`

. The future information about the exogenous variable must be available when making predictions.

In [26]:

```
# Predictions
# ==============================================================================
predictions = forecaster.predict(steps=steps, exog=data_test['exog_1'])
```

In [27]:

```
# Plot
# ==============================================================================
fig, ax = plt.subplots(figsize=(6, 2.5))
data_train['y'].plot(ax=ax, label='train')
data_test['y'].plot(ax=ax, label='test')
predictions.plot(ax=ax, label='predictions')
ax.legend();
```