sMAPE¶

Another popular metric in the field of electricity price forecasting is the symmetric mean absolute percentage error (sMAPE):

\[\begin{split}\begin{equation} \mathrm{sMAPE} = \frac{1}{N}\sum_{k=1}^{N}2\frac{|p_k-\hat{p}_k|}{|p_k| + |\hat{p}_k|},\\ \end{equation}\end{split}\]

This metric computes the MAE between the predicted prices and the real prices and normalizes it by the average of the absolute value of both quantities. Note, that there are multiple versions of sMAPE and here we consider the most sensible one.

Although the sMAPE provides a metric based on relative errors that would grant comparison between datasets and even though it solves some of the issues of MAE, RMSE, and MAPE and, it has a statistical distribution with undefined mean and infinite variance.

epftoolbox.evaluation.sMAPE¶

epftoolbox.evaluation.sMAPE(p_real, p_pred)[source]¶

Function that computes the symmetric mean absolute percentage error (sMAPE) between two forecasts

Note that there are multiple versions of sMAPE, here we con-sider the most sensible one :

\[\mathrm{sMAPE} = \frac{1}{N}\sum_{i=1}^N \frac{2\bigl|p_\mathrm{real}[i]−p_\mathrm{pred}[i]\bigr|}{ \bigl|P_\mathrm{real}[i]\bigr|+\bigl|P_\mathrm{pred}[i]\bigr|}\]

p_real and p_pred 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.
Returns:	The symmetric mean absolute percentage error (sMAPE).
Return type:	float

Example

>>> from epftoolbox.evaluation import sMAPE
>>> 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 sMAPE of the DNN ensemble for the NP market is 4.85%. Let’s test the metric for different conditions

>>> # Evaluating sMAPE when real price and forecasts are both dataframes
>>> sMAPE(p_pred=fc_DNN_ensemble, p_real=real_price) * 100
4.846295174735425
>>>
>>> # Evaluating sMAPE when real price and forecasts are both numpy arrays
>>> sMAPE(p_pred=fc_DNN_ensemble.values, p_real=real_price.values) * 100
4.846295174735425
>>>
>>> # Evaluating sMAPE when input values are of shape (ndays, n_prices/day) instead
>>> # of shape (nprices, 1)
>>> # Dataframes
>>> sMAPE(p_pred=fc_DNN_ensemble_2D, p_real=real_price_2D) * 100
4.846295174735425
>>> # Numpy arrays
>>> sMAPE(p_pred=fc_DNN_ensemble_2D.values, p_real=real_price_2D.values) * 100
4.846295174735425
>>>
>>> # Evaluating sMAPE when input values are of shape (nprices,)
>>> # instead of shape (nprices, 1)
>>> # Pandas Series
>>> sMAPE(p_pred=fc_DNN_ensemble.loc[:, 'DNN Ensemble'],
...       p_real=real_price.loc[:, 'Price']) * 100
4.846295174735425
>>> # Numpy arrays
>>> sMAPE(p_pred=fc_DNN_ensemble.values.squeeze(),
...       p_real=real_price.values.squeeze()) * 100
4.846295174735425