Table 3 Description of the considered validation metrics present in the benchmark. \(\Omega ^\pm \triangleq \{(i,j), A_{i,j} = \pm 1 \mid i \le N_S, j \le N_P\}\) is the set of all positive (\(\Omega ^+\)) or negative (\(\Omega ^-\)) drug-disease associations, whereas \(\Omega ^+_j \triangleq \{i \mid A_{i,j} = +1\}\) is the set of drugs involved in positive associations with disease j and \(\widetilde{\Omega }_j \triangleq \{(i,i') \mid A_{i,j} > A_{i',j}\}\) for any \(j \le N_P\) is the set of correctly ordered pairs of drugs for the score ranking in disease j. In the benchmark, \(t=0\) and \(\mathbbm {1}(C)\) is equal to 1 if C is satisfied, 0 otherwise. \(\sigma _{V}\) is the permutation that sorts all coefficients of any vector V of length n in decreasing order, that is, \(V_{\sigma _V(1)} \ge V_{\sigma _V(2)} \ge \dots \ge V_{\sigma _V(n)}\). The true positive rate is formally defined as \(\texttt {TPR}(t; \hat{R}, A) = \sum _{(i,j),A_{i,j}=+1} \mathbbm {1}(\hat{R}_{i,j}>t)/\sum _{(i,j)} \mathbbm {1}(\hat{R}_{i,j}>t)\) and \(\texttt {FPR}(t; \hat{R}, A) = \sum _{(i,j),A_{i,j}=-1} \mathbbm {1}(\hat{R}_{i,j}>t)/\sum _{(i,j)} \mathbbm {1}(\hat{R}_{i,j}\le t)\) is the false positive rate. Finally, \(N^{+,j}_S\) is defined as \(\min (N_S,|\Omega ^+_j|)\).

From: Comprehensive evaluation of pure and hybrid collaborative filtering in drug repurposing