# 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:

$$$\mathrm{MASE} = \frac{1}{N}\sum_{k=1}^{N}\frac{|p_k-\hat{p}_k|}{\frac{1}{n-1}\sum_{i=2}^{n} |p^\mathrm{in}_i - p^\mathrm{in}_{i-1} |},$$$

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:

$$$\mathrm{MASE}_{m} = \frac{1}{N}\sum_{k=1}^{N}\frac{|p_k-\hat{p}_k|}{\frac{1}{n-m}\sum_{i=m+1}^{n} |p^\mathrm{in}_i - p^\mathrm{in}_{i-m} |}$$$

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 forecast Ynaive_in that is built using the insample dataset p_real_in 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 24 or 168, 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. p_real_in (numpy.ndarray, pandas.DataFrame) – Insample dataset that is used to compute build a naive_forecast and compute its MAE 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). The mean absolute scaled error (MASE). float

Example

>>> from epftoolbox.evaluation import MASE
>>> import pandas as pd
>>>
>>> # Download available forecast of the NP market available in the library repository
>>> # These forecasts accompany the original paper
...                       '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)])
>>>
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
`