MASE¶
Considering the errors of standard metrics described in the introduction, metrics based on scaled errors, where a scaled error is simply the MAE scaled by the in-sample MAE of a naive forecast
, are arguably better. A scaled error has the nice interpretation of being lower/larger than one if it is better/worse than the average naive forecast evaluated in-sample.
A metric based on this concept is the mean absolute scaled error (MASE), and in the context of one-step ahead forecasting is defined as:
where \(p^\mathrm{in}_i\) is the \(i^\mathrm{th}\) price in the in-sample dataset and \(n\) the size of the in-sample dataset. For seasonal time series, the MASE may be defined using the MAE of a seasonal naive model in the denominator:
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.MASE¶
-
epftoolbox.evaluation.
MASE
(p_real, p_pred, p_real_in, m=None, freq='1H')[source]¶ Function that computes the mean absolute scaled error (MASE) between two forecasts:
\[\mathrm{MASE}_\mathrm{m} = \frac{1}{N}\sum_{i=1}^N \frac{\bigl|p_\mathrm{real}[i]−p_\mathrm{pred}[i]\bigr|} {\mathrm{MAE}(p_\mathrm{real\_in}, p_\mathrm{naive\_in}, m)}.\]The numerator is the
MAE
of a naive forecastYnaive_in
that is built using the insample datasetp_real_in
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 day-ahead electricity markets).Also, if the datasets provided are numpy.ndarray objects,
m
has to be 24 or 168, 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.
- p_real_in (numpy.ndarray, pandas.DataFrame) – Insample dataset that is used to compute build a
naive_forecast
and compute itsMAE
- 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
, 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 day-ahead electricity markets).
Returns: The mean absolute scaled error (MASE).
Return type: float
Example
>>> from epftoolbox.evaluation import MASE >>> 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_train, 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']] >>> real_price_insample = df_train.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)]) >>> real_price_insample_2D = pd.DataFrame(real_price_insample.values.reshape(-1, 24), ... index=real_price_insample.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 >>>
Let’s test the metric for different conditions.
>>> # Evaluating MASE when real price and forecasts are both dataframes >>> MASE(p_pred=fc_DNN_ensemble, p_real=real_price, ... p_real_in=real_price_insample, m='W') 0.5217886515713188 >>> >>> # Evaluating MASE when real price and forecasts are both numpy arrays >>> MASE(p_pred=fc_DNN_ensemble.values, p_real=real_price.values, ... p_real_in=real_price_insample.values, m='W', freq='1H') 0.5217886515713188 >>> >>> # Evaluating MASE when input values are of shape (ndays, n_prices/day) instead >>> # of shape (nprices, 1) >>> # Dataframes >>> MASE(p_pred=fc_DNN_ensemble_2D, p_real=real_price_2D, ... p_real_in=real_price_insample_2D, m='W') 0.5217886515713188 >>> # Numpy arrays >>> MASE(p_pred=fc_DNN_ensemble_2D.values, p_real=real_price_2D.values, ... p_real_in=real_price_insample_2D.values, m='W', freq='1H') 0.5217886515713188 >>> >>> # Evaluating MASE when input values are of shape (nprices,) >>> # instead of shape (nprices, 1) >>> # Pandas Series >>> MASE(p_pred=fc_DNN_ensemble.loc[:, 'DNN Ensemble'], ... p_real=real_price.loc[:, 'Price'], ... p_real_in=real_price_insample.loc[:, 'Price'], m='W') 0.5217886515713188 >>> # Numpy arrays >>> MASE(p_pred=fc_DNN_ensemble.values.squeeze(), ... p_real=real_price.values.squeeze(), ... p_real_in=real_price_insample.values.squeeze(), m='W', freq='1H') 0.5217886515713188