The applications of bit population count extend further than might be obvious at first glance. As another example, consider computation of the correlation coefficient r between a pair of genetic variants, where some data may be missing. Formally, let n be the number of samples in the dataset, and {x1,x2,…,xn} and {y1,y2,…,yn} contain genotype data for the two variants, where each xi and yi has a value in {0,1,2,ϕ}. In addition, define \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $${\fontsize{9}{6}\begin{aligned} I_{x,y} & := \{i: x_{i}\ne \phi\ \text{and}\ y_{i}\ne \phi \}, \\ v_{i} & := \{0\mathrm{ if }x_{i}=\phi, (x_{i}-1)\text{otherwise}\}, \\ w_{i} & := \{0\mathrm{ if }y_{i}=\phi, (y_{i}-1)\text{otherwise}\}, \\ \overline{v} & := |I_{x,y}|^{-1}\sum_{i\in I_{x,y}}v_{i}, \\ \overline{w} & := |I_{x,y}|^{-1}\sum_{i\in I_{x,y}}w_{i}, \\ \end{aligned}} $$ \end{document}Ix,y:={i:xi≠ϕandyi≠ϕ},vi:={0ifxi=ϕ,(xi−1)otherwise},wi:={0ifyi=ϕ,(yi−1)otherwise},v¯:=|Ix,y|−1∑i∈Ix,yvi,w¯:=|Ix,y|−1∑i∈Ix,ywi,
