Consider two sets with non-negative entries $\mathcal{S}=\{s_1,s_2,\cdots,s_N\}$ and $\mathcal{P}=\{p_1,p_2,\cdots,p_N\}$ , and consider an index set $\mathcal{T}$ of cardinality $K \leq N$ that contains indices of $K$ largest entries of set $P$. Then, let us define an asymmetric similarity metric
$D_{\mathcal{S}\rightarrow\mathcal{P}}=\frac{\sum_{t\in\mathcal{T}} \mathcal{S}(t)}{\sum_{t\in\mathcal{T}}\mathcal{P}(t)}$
It is clear that $0 \leq D_{\mathcal{S}\rightarrow\mathcal{P}}\leq 1$. But I haven't seen something similar in distance/similarity metrics literature, e.g.,
"Comprehensive Survey on Distance/Similarity Measures between Probability Density Functions"
I wonder if someone has seen something similar in academic literature? Thanks.