IMPLEMENTATION
After building a transformation, we can now go to the RTLLR implementation. Building the estimate \({\hat{f}}_{\text{RTLLR}}\) of the unknown density \(f_{X}\) can be summarized into five steps \cite{2006} \cite{Wahbah_2019}:
  1. Binning. Electric load data is divided into \(T\approx n/10\) bins, where \(n\) is the number of data observations. Let \(Q_{1},\ldots,Q_{T}\) denote the positive integer corresponding to the number of observations in each bin, and \(x_{1},\ \ldots,x_{T}\) represent the centers of each of the \(T\) bins.
  2. Variance Stabilizing Root Transform. Calculate \(y_i\ =\sqrt{\frac{1}{10}}\cdot\sqrt{Q_i\ +\ \frac{1}{4}}\ \) , thus yielding a new paired data set with \(T\) observations: \((x_{1},\ y_{1}),\ \ldots,(x_{T},\ y_{T}).\)
  3. Nonparametric Regression. Any nonparametric regression can then be used on the new paired data \(\left(x_{1},\ y_{1}\right),\ \ldots,\left(x_{T},\ y_{T}\right)\).  We elect to use local linear regression. This will build a regression function  \(\hat{r}(x)\) where \({\hat{r}(x)}^{2}\) is an estimate of the PDF \(f_{X}.\) We have used local linear regression because of its efficiency and accuracy in regression modelling.
  4. Unroot. Reverse the root transform by squaring the function to obtain  \(\hat{f_{u}}(x)={\hat{r}(x)}^{2}\ \)
  5. Normalize. To ensure the estimator is a PDF (i.e. the estimator integrates to 1), we normalize \({\hat{f}}_{\text{RTLLR}}\) so that \({\hat{f}}_{\text{RTLLR}}=\ \frac{{\hat{f}}_{u}(x)}{\int_{0}^{1}{{\hat{f}}_{u}\left(x\right)\text{dx}}}\).

3.2 Methods for Assessing Model Performance

In this subsection, performance assessment methods are explored in order to evaluate the models presented in subsection 3.1.