Baseline forecaster¶

In forecasting modeling, a baseline serves as a basic, often simplistic model that acts as a fundamental reference for evaluating the performance of more complex models. It provides a baseline forecast using simple techniques that do not rely on sophisticated algorithms or extensive data analysis. Common examples of baseline strategies include:

• Last Observed Value: Uses the last observed value as a forecast for all future periods.

• Moving Average: In this technique, the average of the last few observations is calculated and used as a forecast for the next period. For example, a 3-period moving average would use the average of the last three observations.

• Last Equivalent Date (Seasonal Naive Forecasting): Extends the concept of the last observed value by considering the corresponding period in the previous season (e.g., the same working day from the previous week) as the forecast for the current period.

The primary goal of establishing a baseline is to provide a benchmark against which the performance of more advanced predictive models can be evaluated. If the model does not outperform the baseline, it may indicate that there is a fundamental problem with the approach or that the added complexity is not justified by the available data. This underscores the importance of carefully evaluating the appropriateness of complex models relative to the simplicity and effectiveness of baseline models.

ForecasterEquivalentDate¶

The ForecasterEquivalentDate class from skforecast.recursive allows the creation of a baseline forecast based on the concept of equivalent dates. In this context, an equivalent date is a historical date that has similar characteristics to the target date. The forecast for a given date is based on the value observed on the last n equivalent dates.

The behavior of the forecast is primarily controlled by two arguments:

• offset: This parameter determines how many steps back in time to go to find the most recent equivalent date for the target period. When given as an integer, offset represents the number of steps to go back in time. For example, if the frequency of the time series is daily, offset = 7 means that the most recent data similar to the target period is the value observed 7 days ago. Additionally, it is possible to use Pandas DateOffsets to move back a given number of valid dates. For example, Bday(2) can be used to move back two business days.

• n_offsets: This parameter determines the number of equivalent dates to use in the prediction. If n_offsets is greater than 1, the values at the equivalent dates are aggregated using the specified aggregation function, agg_func. For example, if the frequency of the time series is daily, offset = 7, n_offsets = 2, and agg_func = np.mean, the predicted value will be the mean of the values observed 7 and 14 days ago.

The common baseline strategies map to the following parameter settings:

Baseline strategy	offset	n_offsets	agg_func
Last observed value	1	1	—
Moving average (last k values)	1	k	np.mean
Seasonal naive (same period last season)	season length	1	—
Mean of the last k seasons	season length	k	np.mean

⚠ Warning

Within a forecast horizon the equivalent-date baseline does not roll: it computes its predictors once from the observed history, so a configuration such as '3-month moving average' (offset=1, n_offsets=3) returns the mean of the last three observed values for every step of the fold, not a rolling average that updates step by step. The same applies when offset is a pandas DateOffset.

✏️ Note

ForecasterEquivalentDate is designed to integrate seamlessly with other functionality offered by skforecast, such as backtesting. This makes it easy to obtain the baseline for a given period.

Libraries and data¶

In [1]:

# Libraries
# ==============================================================================
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from sklearn.metrics import mean_absolute_error
from skforecast.datasets import fetch_dataset
from skforecast.recursive import ForecasterEquivalentDate
from skforecast.model_selection import (
    TimeSeriesFold, 
    backtesting_forecaster, 
    grid_search_equivalent_date
)
from skforecast.plot import set_dark_theme, plot_prediction_intervals

# Libraries # ============================================================================== import numpy as np import pandas as pd import matplotlib.pyplot as plt from sklearn.metrics import mean_absolute_error from skforecast.datasets import fetch_dataset from skforecast.recursive import ForecasterEquivalentDate from skforecast.model_selection import ( TimeSeriesFold, backtesting_forecaster, grid_search_equivalent_date ) from skforecast.plot import set_dark_theme, plot_prediction_intervals

In [2]:

# Download data
# ======================================================================================
data = fetch_dataset('fuel_consumption')
data = data.rename(columns={'Gasolinas': 'litters'})
data.index.name = 'date'
data = data.loc[:'1985-01-01 00:00:00', 'litters']
display(data.head(4))

# Train-test dates
# ======================================================================================
end_train = '1980-01-01 23:59:59'
data_train = data.loc[:end_train]
data_test  = data.loc[end_train:]

print(f"Train dates : {data_train.index.min()} --- {data_train.index.max()}  (n={len(data_train)})")
print(f"Test dates  : {data_test.index.min()} --- {data_test.index.max()}  (n={len(data_test)})")

# Plot
# ======================================================================================
set_dark_theme()
fig, ax = plt.subplots(figsize=(7, 3))
data_train.plot(ax=ax, label='train')
data_test.plot(ax=ax, label='test')
ax.set_title('Monthly fuel consumption in Spain')
ax.legend();

# Download data # ====================================================================================== data = fetch_dataset('fuel_consumption') data = data.rename(columns={'Gasolinas': 'litters'}) data.index.name = 'date' data = data.loc[:'1985-01-01 00:00:00', 'litters'] display(data.head(4)) # Train-test dates # ====================================================================================== end_train = '1980-01-01 23:59:59' data_train = data.loc[:end_train] data_test = data.loc[end_train:] print(f"Train dates : {data_train.index.min()} --- {data_train.index.max()} (n={len(data_train)})") print(f"Test dates : {data_test.index.min()} --- {data_test.index.max()} (n={len(data_test)})") # Plot # ====================================================================================== set_dark_theme() fig, ax = plt.subplots(figsize=(7, 3)) data_train.plot(ax=ax, label='train') data_test.plot(ax=ax, label='test') ax.set_title('Monthly fuel consumption in Spain') ax.legend();

╭──────────────────────────────── fuel_consumption ────────────────────────────────╮
│ Description:                                                                     │
│ Monthly fuel consumption in Spain from 1969-01-01 to 2022-08-01.                 │
│                                                                                  │
│ Source:                                                                          │
│ Obtained from Corporación de Reservas Estratégicas de Productos Petrolíferos and │
│ Corporación de Derecho Público tutelada por el Ministerio para la Transición     │
│ Ecológica y el Reto Demográfico. https://www.cores.es/es/estadisticas            │
│                                                                                  │
│ URL:                                                                             │
│ https://raw.githubusercontent.com/skforecast/skforecast-                         │
│ datasets/main/data/consumos-combustibles-mensual.csv                             │
│                                                                                  │
│ Shape: 644 rows x 5 columns                                                      │
╰──────────────────────────────────────────────────────────────────────────────────╯

date
1969-01-01    166875.2129
1969-02-01    155466.8105
1969-03-01    184983.6699
1969-04-01    202319.8164
Freq: MS, Name: litters, dtype: float64

Train dates : 1969-01-01 00:00:00 --- 1980-01-01 00:00:00  (n=133)
Test dates  : 1980-02-01 00:00:00 --- 1985-01-01 00:00:00  (n=60)

Create and train forecaster¶

In [3]:

# Create and fit forecaster
# ==============================================================================
forecaster = ForecasterEquivalentDate(
                 offset    = pd.DateOffset(months=12),
                 n_offsets = 2,
                 agg_func  = np.mean
             )

forecaster.fit(y=data_train)
forecaster

# Create and fit forecaster # ============================================================================== forecaster = ForecasterEquivalentDate( offset = pd.DateOffset(months=12), n_offsets = 2, agg_func = np.mean ) forecaster.fit(y=data_train) forecaster

Out[3]:

ForecasterEquivalentDate

General Information

• Estimator: NoneType
• Offset:
• Number of offsets: 2
• Aggregation function: mean
• Window size: 24
• Creation date: 2026-08-21 16:16:21
• Last fit date: 2026-08-21 16:16:21
• Skforecast version: 0.24.0
• Python version: 3.14.3
• Forecaster id: None

Training Information

• Training range: [Timestamp('1969-01-01 00:00:00'), Timestamp('1980-01-01 00:00:00')]
• Training index type: DatetimeIndex
• Training index frequency: MS

📖 API Reference    📝 User Guide

Prediction¶

In [4]:

# Predict
# ==============================================================================
predictions = forecaster.predict(steps=len(data_test))
predictions.head(3)

# Predict # ============================================================================== predictions = forecaster.predict(steps=len(data_test)) predictions.head(3)

Out[4]:

1980-02-01    385298.35315
1980-03-01    472815.89325
1980-04-01    462944.81705
Freq: MS, Name: pred, dtype: float64

In [5]:

# Plot predictions
# ==============================================================================
fig, ax = plt.subplots(figsize=(6, 3))
data_train.plot(ax=ax, label='train')
data_test.plot(ax=ax, label='test')
predictions.plot(ax=ax, label='predictions')
ax.legend();

# Plot predictions # ============================================================================== fig, ax = plt.subplots(figsize=(6, 3)) data_train.plot(ax=ax, label='train') data_test.plot(ax=ax, label='test') predictions.plot(ax=ax, label='predictions') ax.legend();

In [6]:

# Prediction error
# ==============================================================================
error_mae = mean_absolute_error(
                y_true = data_test,
                y_pred = predictions
            )

print(f"Test error (mae): {error_mae}")

# Prediction error # ============================================================================== error_mae = mean_absolute_error( y_true = data_test, y_pred = predictions ) print(f"Test error (mae): {error_mae}")

Test error (mae): 16785.29897833333

Probabilistic forecasting¶

Conformal prediction is a framework for constructing prediction intervals that are guaranteed to contain the true value with a specified probability (coverage probability). It works by combining the predictions of a point-forecasting model with its past residuals, differences between previous predictions and actual values. These residuals help estimate the uncertainty in the forecast and determine the width of the prediction interval that is then added to the point forecast.

To learn more about conformal predictions in skforecast, visit the Probabilistic Forecasting: Conformal Prediction user guide.

In [7]:

# Store in-sample residuals
# ==============================================================================
# Residuals are not stored during `fit` unless `store_in_sample_residuals=True`,
# so they are computed here to enable conformal prediction intervals.
forecaster.set_in_sample_residuals(y=data_train)

# Store in-sample residuals # ============================================================================== # Residuals are not stored during `fit` unless `store_in_sample_residuals=True`, # so they are computed here to enable conformal prediction intervals. forecaster.set_in_sample_residuals(y=data_train)

In [8]:

# Prediction intervals
# ==============================================================================
predictions = forecaster.predict_interval(
    steps                   = len(data_test),
    interval                = [0.1, 0.9],  # 80% prediction interval
    method                  = 'conformal',
    use_in_sample_residuals = True
)

predictions.head(4)

# Prediction intervals # ============================================================================== predictions = forecaster.predict_interval( steps = len(data_test), interval = [0.1, 0.9], # 80% prediction interval method = 'conformal', use_in_sample_residuals = True ) predictions.head(4)

Out[8]:

	pred	lower_bound	upper_bound
1980-02-01	385298.35315	334204.43875	436392.26755
1980-03-01	472815.89325	413163.96550	532467.82100
1980-04-01	462944.81705	403292.88930	522596.74480
1980-05-01	477889.17740	418237.24965	537541.10515

In [9]:

# Plot intervals
# ==============================================================================
plot_prediction_intervals(
    predictions         = predictions,
    y_true              = data_test,
    target_variable     = "litters",
    title               = "Predicted intervals",
    kwargs_fill_between = {'color': 'gray', 'alpha': 0.4, 'zorder': 1}
)

# Plot intervals # ============================================================================== plot_prediction_intervals( predictions = predictions, y_true = data_test, target_variable = "litters", title = "Predicted intervals", kwargs_fill_between = {'color': 'gray', 'alpha': 0.4, 'zorder': 1} )

Backtesting¶

In [10]:

# Backtesting
# ==============================================================================
cv = TimeSeriesFold(
         steps              = 15,
         initial_train_size = len(data_train),
         refit              = True
     )

metric, predictions = backtesting_forecaster(
                          forecaster = forecaster,
                          y          = data,
                          cv         = cv,
                          metric     = 'mean_absolute_error'
                      )

print("Backtest error:")
metric

# Backtesting # ============================================================================== cv = TimeSeriesFold( steps = 15, initial_train_size = len(data_train), refit = True ) metric, predictions = backtesting_forecaster( forecaster = forecaster, y = data, cv = cv, metric = 'mean_absolute_error' ) print("Backtest error:") metric

  0%|          | 0/4 [00:00<?, ?it/s]

Backtest error:

Out[10]:

	mean_absolute_error
0	18575.076906

In [11]:

# Backtesting predictions
# ==============================================================================
predictions.head(4)

# Backtesting predictions # ============================================================================== predictions.head(4)

Out[11]:

	fold	pred
1980-02-01	0	385298.35315
1980-03-01	0	472815.89325
1980-04-01	0	462944.81705
1980-05-01	0	477889.17740

In [12]:

# Plot backtesting predictions
# ==============================================================================
fig, ax = plt.subplots(figsize=(6, 3))
data_test.plot(ax=ax, label='test')
predictions['pred'].plot(ax=ax, label='predictions')
ax.legend();

# Plot backtesting predictions # ============================================================================== fig, ax = plt.subplots(figsize=(6, 3)) data_test.plot(ax=ax, label='test') predictions['pred'].plot(ax=ax, label='predictions') ax.legend();

Searching for the best configuration¶

Although ForecasterEquivalentDate has no trainable estimator, its behaviour depends on a few configuration parameters (offset, n_offsets, and agg_func) that jointly define which past values are used and how they are aggregated. Finding the best combination is itself a small search problem: the function grid_search_equivalent_date evaluates a set of candidate configurations using time series backtesting and ranks them by the selected metric.

In [16]:

# Search over baseline configurations
# ==============================================================================
cv = TimeSeriesFold(
         steps              = 15,
         initial_train_size = len(data_train),
         refit              = False
     )

# Each dictionary defines one explicit baseline configuration.
# `offset` can be an integer (number of steps back) or a pandas DateOffset.
param_grid = [
    # Integer offset: number of steps back (monthly data -> months)
    {'alias': 'Last month',             'offset': 1,  'n_offsets': 1, 'agg_func': np.mean},
    {'alias': '3-month moving average', 'offset': 1,  'n_offsets': 3, 'agg_func': np.mean},
    {'alias': 'Same month last year',   'offset': 12, 'n_offsets': 1, 'agg_func': np.mean},
    {'alias': 'Mean of last 2 years',   'offset': 12, 'n_offsets': 2, 'agg_func': np.mean},
    # pandas DateOffset: calendar-aware step (requires a DatetimeIndex with frequency)
    {'alias': 'Mean of last 3 years',   'offset': pd.DateOffset(months=12), 'n_offsets': 3, 'agg_func': np.mean},
    {'alias': 'Median of last 3 years', 'offset': pd.DateOffset(months=12), 'n_offsets': 3, 'agg_func': np.median},
]

# Alternatively, pass a dict to evaluate all combinations (Cartesian product).
# Use this form only when every combination is meaningful:
# param_grid = {
#     'offset':    [1, 12],
#     'n_offsets': [1, 2, 3],
#     'agg_func':  [np.mean, np.median],
# }  # -> 2 x 3 x 2 = 12 configurations

results = grid_search_equivalent_date(
              forecaster  = forecaster,
              y           = data,
              cv          = cv,
              param_grid  = param_grid,
              metric      = 'mean_absolute_error',
              return_best = True,
          )

results

# Search over baseline configurations # ============================================================================== cv = TimeSeriesFold( steps = 15, initial_train_size = len(data_train), refit = False ) # Each dictionary defines one explicit baseline configuration. # `offset` can be an integer (number of steps back) or a pandas DateOffset. param_grid = [ # Integer offset: number of steps back (monthly data -> months) {'alias': 'Last month', 'offset': 1, 'n_offsets': 1, 'agg_func': np.mean}, {'alias': '3-month moving average', 'offset': 1, 'n_offsets': 3, 'agg_func': np.mean}, {'alias': 'Same month last year', 'offset': 12, 'n_offsets': 1, 'agg_func': np.mean}, {'alias': 'Mean of last 2 years', 'offset': 12, 'n_offsets': 2, 'agg_func': np.mean}, # pandas DateOffset: calendar-aware step (requires a DatetimeIndex with frequency) {'alias': 'Mean of last 3 years', 'offset': pd.DateOffset(months=12), 'n_offsets': 3, 'agg_func': np.mean}, {'alias': 'Median of last 3 years', 'offset': pd.DateOffset(months=12), 'n_offsets': 3, 'agg_func': np.median}, ] # Alternatively, pass a dict to evaluate all combinations (Cartesian product). # Use this form only when every combination is meaningful: # param_grid = { # 'offset': [1, 12], # 'n_offsets': [1, 2, 3], # 'agg_func': [np.mean, np.median], # } # -> 2 x 3 x 2 = 12 configurations results = grid_search_equivalent_date( forecaster = forecaster, y = data, cv = cv, param_grid = param_grid, metric = 'mean_absolute_error', return_best = True, ) results

params grid:   0%|          | 0/6 [00:00<?, ?it/s]

Out[16]:

	alias	params	mean_absolute_error	offset	n_offsets	agg_func
0	Mean of last 3 years	{'offset': <DateOffset: months=12>, 'n_offsets...	17969.624487	<DateOffset: months=12>	3	mean
1	Mean of last 2 years	{'offset': 12, 'n_offsets': 2, 'agg_func': <fu...	18575.076906	12	2	mean
2	Median of last 3 years	{'offset': <DateOffset: months=12>, 'n_offsets...	20038.362477	<DateOffset: months=12>	3	median
3	Same month last year	{'offset': 12, 'n_offsets': 1, 'agg_func': <fu...	20099.924027	12	1	mean
4	3-month moving average	{'offset': 1, 'n_offsets': 3, 'agg_func': <fun...	55317.400922	1	3	mean
5	Last month	{'offset': 1, 'n_offsets': 1, 'agg_func': <fun...	63229.415890	1	1	mean