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 \(i\).