Table 2 The mathematical equation of different statistical benchmark of the predictive potential for CORAL models

\({R}^{2}=1-\frac{\sum {({Y}_{obs}-{Y}_{prd})}^{2}}{\sum {({Y}_{obs}-{\overline{Y} }_{train})}^{2}}\)

Y_obs is the observed endpoint for the training set, and Y_pred is the predicted endpoint values for the training set of compounds

\({\overline{Y}}_{train}\) is the mean observed endpoint of the training set

\({Q}^{2}=1-\frac{\sum {({Y}_{prd(train)}-{Y}_{obs(train)})}^{2}}{\sum {({Y}_{obs(train)}-{\overline{Y}}_{train})}^{2}}\)

Y_obs(train) is the observed endpoint, and Y_pred(train) is the predicted response of the training set compounds

\(CCC=\frac{2\sum ({X}_{i}-\overline{X })({Y}_{i}-\overline{Y })}{\sum_{i=1}^{n}{({X}_{i}-\overline{X })}^{2}+\sum_{i=1}^{n}{({X}_{i}-\overline{X })}^{2}+n(\overline{X }-\overline{Y })}\)

n is the number of compounds, and x_i and y_i denote the mean of observed and predicted values, respectively

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\({\mathrm{c}}_{{R}_{p}^{2}}=R\sqrt{\left({R}^{2}-{R}_{r}^{2}\right)}\)

\({R}^{2}\) is squared correlation coefficient of models and \({R}_{r}^{2}\) is squared mean correlation coefficient of randomized models

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\(\overline{{r }_{m}^{2}}=\frac{{r}_{m}^{2}+{r}_{m}^{{\prime}2}}{2}\)

r² is the squared correlation coefficient value between observed and predicted endpoint values, and \({\mathrm{r}}_{0}^{2}\) and \({\mathrm{r{\prime}}}_{0}^{2}\) are the respective squared correlation coefficients when the regression line is passed through the origin by interchanging the axes

For the acceptable prediction, the value of all \(\Delta {\mathrm{r}}_{\mathrm{m}}^{2}\) metrics should preferably be lower than 0.2 provided that the value of r²m is more than 0.5

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\(\Delta {r}_{m}^{2}=\left|{r}_{m}^{2}-{r{\prime}}_{m}^{2}\right|\)

\({r}_{m}^{2}={r}^{2}\times \left(1-\sqrt{{r}^{2}-{r}_{0}^{2}}\right)\)

\({r{\prime}}_{m}^{2}={r}^{2}\times \left(1-\sqrt{{r}^{2}-{r{\prime}}_{0}^{2}}\right)\)

\(MAE=\frac{1}{n}\times \sum \left|{Y}_{obs}-{Y}_{prd}\right|\)