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:
 As MASE depends on the insample dataset, forecasting methods with different calibration windows will naturally have to consider different insample datasets. As a result, the MASE of each model will be based on a different scaling factor and comparisons between models cannot be drawn.
 The same argument applies to models with and without rolling windows. The latter will use a different insample dataset at every time point while the former will keep the insample dataset constant.
 In ensembles of models with different calibration windows, the MASE cannot be defined as the calibration window of the ensemble is undefined.
 Drawing comparisons across different time series is problematic as electricity prices are not stationary. For example, an insample dataset with spikes and an outofsample dataset without spikes will lead to a smaller MASE than if we consider the same market but with the insample/outsample 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 insample dataset, the naive forecast is built based on the outofsample dataset. In the context In the context of onestep ahead forecasting is defined as:
For seasonal time series, the rMAE may be defined using the MAE of a seasonal naive model in the denominator:
where \(m\) represents the seasonal length (in the case of dayahead 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:
\[\mathrm{rMAE}_\mathrm{m} = \frac{1}{N}\sum_{i=1}^N \frac{\biglp_\mathrm{real}[i]−p_\mathrm{pred}[i]\bigr} {\mathrm{MAE}(p_\mathrm{real}, p_\mathrm{naive})}.\]The numerator is the
MAE
of a naive forecastp_naive
that is built using the datasetp_real
and thenaive_forecast
function with a seasonality indexm
.If the datasets provided are numpy.ndarray objects, the function requires a
freq
argument specifying the data frequency. Thefreq
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 dayahead electricity markets).Also, if the datasets provided are numpy.ndarray objects,
m
has to be'D'
or'W'
, i.e. thenaive_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 insampleMAE
. 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
, andp_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 dayahead electricity markets). If the shape ofp_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: 20161227 00:00:00  20181224 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 20161227 24.349676 23.127774 22.208617 ... 27.686771 27.045763 25.724071 20161228 25.453866 24.707317 24.452384 ... 29.424558 28.627130 27.321902 20161229 28.209516 27.715400 27.182692 ... 28.473288 27.926241 27.153401 20161230 28.002935 27.467572 27.028558 ... 29.086532 28.518688 27.738548 20161231 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