3.2.1 Error Metrics
In renewable energy research, some commonly used error metrics to assess
the performance of PDF models are the Root Mean Squared Error (RMSE),
Mean Absolute Error (MAE), Mean Absolute Percentage Error (MAPE) and
Mean Biased Error (MBE) \cite{Wahbah_2020} \cite{El_Dakkak_2019}. Each of these errors measures
different characteristics of fit, but all serve the same purpose. The
error metrics range from \([0,\infty)\), with the exception of
MBE which can be any real number, with 0 signifying that the model is a
perfect fit for the data. The error metrics formulas are presented down
below.
\[RMSE=\sqrt{\frac{1}{t} \sum_{i=1}^{t}\left(p_{i}-\hat{p_{i}}\right)^{2}\ }\nonumber \\
\]\[\begin{equation}
MAE=\frac{1}{t} \sum_{i=1}^{t}{|p_{i}-\hat{p_{i}}|}\nonumber \\
\end{equation}\]\[MAPE=\frac{1}{t} \sum_{i=1}^{t}{\left|\frac{p_{i}-\hat{p_{i}}}{y_{i}}\right|}\nonumber \\\]\[MBE=\ \frac{1}{t} \sum_{i=1}^{t}\left(p_{i}-\hat{p_{i}}\right)\nonumber \\\]
where \(t\) is the number of bins of data chosen using the
Freedman-Diaconis Rule \cite{Freedman_1981}, \(p_{i}\) is the probability of
electric load being within bin \(i\) calculated from the data set, and \(\hat{p_{i}}\) is the probability within the same bin calculated from
the estimated data set which found by integrating the model within bin \(i\).