To investigate the efficiency of regression models obtained in the results section of the present study to estimate RWC, (Tc − Ta) and LWP, the correlation coefficient (R), root mean square error (RMSE), mean absolute error (MAE) and mean relative error (MRE) were used39.

In these equations, N is the number of measurements (samples), i index of each model, Oi and Pi measured and predicted values, and O¯i and Pi¯ mean measured and predicted values. However, the results from the above four criteria sometimes vary in different models40. The Ideal Point Error (IPE) index was used to select the best model with the highest accuracy. This index combines the effects of four error criteria and helps choosing the right model. The IPE index is based on identifying the ideal point in the n-dimensional space (n is the number of statistical criteria to evaluate the models), which every models seeks to approach. The ideal point coordinates should be RMSE = 0.0; MAE = 0.0; MRE = 0.0; R = 1.0. The IPE index (Eq. (13)) indicates how far the model is from the ideal point39.


A careful examination of the original IPEA equation shows that there is a contradiction in the standardization method applied to each component. In Eq. (13), the first three indices are standardized according to the worst model performance, while the last index (R) is standardized according to the best model performance. It should also be noted that standardization of R in the original IPEA equation is not designed for negative values41. Equation (14) or IPEB represents an improved variant of the original equation, which includes a more generalized and robust standardization procedure for R that can accommodate the full range [− 1, + 1]. IPEB is also consistent with the method of standardization with regard to the worst-case model performance. So, the new index eliminates the inconsistency (contradiction) of standardization of the original IPEA equation. This correction can lead to a significant difference between the output of the original IPEA and the IPEB, especially for the states containing medium or low R values41.