rMAE

While scaled errors do indeed solve the issues of more traditional metrics, they have other associated problems that make them not unsuitable in the context of EPF:

  1. As MASE depends on the in-sample dataset, forecasting methods with different calibration windows will naturally have to consider different in-sample datasets. As a result, the MASE of each model will be based on a different scaling factor and comparisons between models cannot be drawn.
  2. The same argument applies to models with and without rolling windows. The latter will use a different in-sample dataset at every time point while the former will keep the in-sample dataset constant.
  3. In ensembles of models with different calibration windows, the MASE cannot be defined as the calibration window of the ensemble is undefined.
  4. Drawing comparisons across different time series is problematic as electricity prices are not stationary. For example, an in-sample dataset with spikes and an out-of-sample dataset without spikes will lead to a smaller MASE than if we consider the same market but with the in-sample/out-sample datasets reversed.

To solve these issues, an arguably better metric is the relative MAE (rMAE). Similar to MASE, rMAE normalizes the MAE by the MAE of a naive forecast. However, instead of considering the in-sample dataset, the naive forecast is built based on the out-of-sample dataset. In the context In the context of one-step ahead forecasting is defined as:

\[\begin{equation} \mathrm{rMAE} = \frac{1}{N}\sum_{k=1}^{N}\frac{|p_k-\hat{p}_k|}{\frac{1}{N-1}\sum_{i=2}^{N} |p_i - p_{i-1} |}. \end{equation}\]

For seasonal time series, the rMAE may be defined using the MAE of a seasonal naive model in the denominator:

\[\begin{equation} \mathrm{rMAE}_{m} = \frac{1}{N}\sum_{k=1}^{N}\frac{|p_k-\hat{p}_k|}{\frac{1}{N-m}\sum_{i=m+1}^{N} |p_i - p_{i-m} |} \end{equation}\]

where \(m\) represents the seasonal length (in the case of day-ahead prices that could be either 24 or 168 representing the daily and weekly seasonalities). As an alternative, the naive forecast can also be defined on the standard naive forecast for price forecasting (using daily seasonality for Tuesday to Friday and weekly seasonality for Saturday to Monday).

epftoolbox.evaluation.rMAE

epftoolbox.evaluation.rMAE(p_real, p_pred, m=None, freq='1H')[source]

Function that computes the relative mean absolute error (rMAE) between two forecasts:

\[\mathrm{rMAE}_\mathrm{m} = \frac{1}{N}\sum_{i=1}^N \frac{\bigl|p_\mathrm{real}[i]−p_\mathrm{pred}[i]\bigr|} {\mathrm{MAE}(p_\mathrm{real}, p_\mathrm{naive})}.\]

The numerator is the MAE of a naive forecast p_naive that is built using the dataset p_real and the naive_forecast function with a seasonality index m.

If the datasets provided are numpy.ndarray objects, the function requires a freq argument specifying the data frequency. The freq argument must take one of the following four values '1H' for 1 hour, '30T' for 30 minutes, '15T' for 15 minutes, or '5T' for 5 minutes, (these are the four standard values in day-ahead electricity markets).

Also, if the datasets provided are numpy.ndarray objects, m has to be 'D' or 'W', i.e. the naive_forecast cannot be the standard in electricity price forecasting because the input data does not have associated a day of the week.

p_real, p_pred, and p_real_in` can either be of shape \((n_\mathrm{days}, n_\mathrm{prices/day})\), \((n_\mathrm{prices}, 1)\), or \((n_\mathrm{prices}, )\) where \(n_\mathrm{prices} = n_\mathrm{days} \cdot n_\mathrm{prices/day}\)

Parameters:
  • p_real (numpy.ndarray, pandas.DataFrame) – Array/dataframe containing the real prices.
  • p_pred (numpy.ndarray, pandas.DataFrame) – Array/dataframe containing the predicted prices.
  • m (int, optional) – Index that specifies the seasonality in the naive_forecast used to compute the normalizing insample MAE. It can be be 'D' for daily seasonality, 'W' for weekly seasonality, or None for the standard naive forecast in electricity price forecasting, i.e. daily seasonality for Tuesday to Friday and weekly seasonality for Saturday to Monday.
  • freq (str, optional) – Frequency of the data if p_real, p_pred, and p_real_in are numpy.ndarray objects. It must take one of the following four values '1H' for 1 hour, '30T' for 30 minutes, '15T' for 15 minutes, or '5T' for 5 minutes, (these are the four standard values in day-ahead electricity markets). If the shape of p_real is (ndays, n_prices_day), freq should be the frequency of the columns not the daily frequency of the rows.
Returns:

The mean absolute scaled error (MASE).
Return type:

float

Example

>>> from epftoolbox.evaluation import rMAE
>>> from epftoolbox.data import read_data
>>> import pandas as pd
>>>
>>> # Download available forecast of the NP market available in the library repository
>>> # These forecasts accompany the original paper
>>> forecast = pd.read_csv('https://raw.githubusercontent.com/jeslago/epftoolbox/master/' +
...                       'forecasts/Forecasts_NP_DNN_LEAR_ensembles.csv', index_col=0)
>>>
>>> # Transforming indices to datetime format
>>> forecast.index = pd.to_datetime(forecast.index)
>>>
>>> # Reading data from the NP market
>>> _, df_test = read_data(path='.', dataset='NP', begin_test_date=forecast.index[0],
...                        end_test_date=forecast.index[-1])
Test datasets: 2016-12-27 00:00:00 - 2018-12-24 23:00:00
>>>
>>> # Extracting forecast of DNN ensemble and display
>>> fc_DNN_ensemble = forecast.loc[:, ['DNN Ensemble']]
>>>
>>> # Extracting real price and display
>>> real_price = df_test.loc[:, ['Price']]
>>>
>>> # Building the same datasets with shape (ndays, n_prices/day) instead
>>> # of shape (nprices, 1) and display
>>> fc_DNN_ensemble_2D = pd.DataFrame(fc_DNN_ensemble.values.reshape(-1, 24),
...                                   index=fc_DNN_ensemble.index[::24],
...                                   columns=['h' + str(hour) for hour in range(24)])
>>> real_price_2D = pd.DataFrame(real_price.values.reshape(-1, 24),
...                              index=real_price.index[::24],
...                              columns=['h' + str(hour) for hour in range(24)])
>>> fc_DNN_ensemble_2D.head()
                   h0         h1         h2  ...        h21        h22        h23
2016-12-27  24.349676  23.127774  22.208617  ...  27.686771  27.045763  25.724071
2016-12-28  25.453866  24.707317  24.452384  ...  29.424558  28.627130  27.321902
2016-12-29  28.209516  27.715400  27.182692  ...  28.473288  27.926241  27.153401
2016-12-30  28.002935  27.467572  27.028558  ...  29.086532  28.518688  27.738548
2016-12-31  25.732282  24.668331  23.951569  ...  26.965008  26.450995  25.637346
>>>

According to the paper, the rMAE of the DNN ensemble for the NP market is 0.403 when m='W'. Let’s test the metric for different conditions

>>> # Evaluating rMAE when real price and forecasts are both dataframes
>>> rMAE(p_pred=fc_DNN_ensemble, p_real=real_price)
0.5265639198107801
>>>
>>> # Evaluating rMAE when real price and forecasts are both numpy arrays
>>> rMAE(p_pred=fc_DNN_ensemble.values, p_real=real_price.values, m='W', freq='1H')
0.4031805447246898
>>>
>>> # Evaluating rMAE when input values are of shape (ndays, n_prices/day) instead
>>> # of shape (nprices, 1)
>>> # Dataframes
>>> rMAE(p_pred=fc_DNN_ensemble_2D, p_real=real_price_2D, m='W')
0.4031805447246898
>>> # Numpy arrays
>>> rMAE(p_pred=fc_DNN_ensemble_2D.values, p_real=real_price_2D.values, m='W', freq='1H')
0.4031805447246898
>>>
>>> # Evaluating rMAE when input values are of shape (nprices,)
>>> # instead of shape (nprices, 1)
>>> # Pandas Series
>>> rMAE(p_pred=fc_DNN_ensemble.loc[:, 'DNN Ensemble'],
...      p_real=real_price.loc[:, 'Price'], m='W')
0.4031805447246898
>>> # Numpy arrays
>>> rMAE(p_pred=fc_DNN_ensemble.values.squeeze(),
...      p_real=real_price.values.squeeze(), m='W', freq='1H')
0.4031805447246898