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Table 1 An overview of definitions for proposed measures and concepts in sections that follow with the same name

From: A new concordant partial AUC and partial c statistic for imbalanced data in the evaluation of machine learning algorithms

1. The horizontal partial area under the curve (a section that follows)
This partial area denoted pAUCx, was suggested by Walter [2] and is defined for part or an ROC curve r(·) defined by TPR = [y1, y2] with inverse function r−1(·):
\( {pAUC}_x:={\int}_{y_1}^{y_2}1-{r}^{-1}(y) dy \)
2. The concordant partial area under the curve (a section that follows)
This partial area denoted pAUCc (Fig 1b) is defined for part of an ROC curve r(·) defined by FPR = [× 1, × 2] and TPR = [y1, y2], with inverse function r−1(·):
\( {\displaystyle \begin{array}{c} pAU{C}_c\triangleq \frac{1}{2} pAU C+\frac{1}{2} pAU{C}_x\\ {}=\frac{1}{2}{\int}_{x_1}^{x_2}r(x) dx+\frac{1}{2}{\int}_{y_1}^{y_2}1-{r}^{-1}(y) dy\end{array}} \)
3. The concordance matrix for ROC data (a section that follows)
A matrix that depicts the exact relationship between the unique scores of positives and negatives in data and their corresponding points along a matrix border that exactly matches the (empirical) ROC curve. It geometrically and procedurally equates area measures AUC and pAUCc to the statistics c and c.
4. The partial c statistic for ROC data (a section that follows)
This statistic denoted c is defined for ROC data with P actual positives {p1…P} and N actual negatives {n1…N} and a partial curve specified by a subset of J positives and K negatives, i.e., \( \left\{{p}_{1\dots J}^{\prime}\right\} \) and \( \left\{{n}_{1\dots K}^{\prime}\right\} \); with Heaviside function H(·) and classification scores g(·). We present simple c (the non-interpolated version) here:
\( {\displaystyle \begin{array}{c}\mathrm{simple}\ {\mathrm{c}}_{\varDelta}\triangleq \frac{1}{2 JN}\sum \limits_{j=1}^J\sum \limits_{k=1}^NH\left(g\left({p}_j^{\prime}\right)-g\left({n}_k\right)\right)\\ {}+\frac{1}{2 PK}{\sum}_{j=1}^P{\sum}_{k=1}^KH\left(g\left({p}_j\right)-g\left({n}_k^{\prime}\right)\right)\end{array}} \)