Let us denote the area of the domain as A. Consider two different regionalizations of the domain. To make a further discussion more lucid, we will refer to the first one as a regionalization and to the second one as a partition. The regionalization R divides the domain into n regions ri∣i=1,…,n. The partition Z divides the domain into m zones zj∣j=1,…,n. Both R and Z are essentially integer-type vectors with equal elements.
h=1−m∑j=1AjASRjSR
where SR=−n∑i=1AiAlogAiA, SRj=−n∑i=1ai,jAjlogai,jAj, and ai,j represents the count of elements where R==i and Z==j. Ai is the number of elements in the vector where R==i, and Aj is the number of elements in the vector where Z==j.
By swapping R and Z, c can be calculated. Finally, the v-measure can be calculated useing the below formula:
Vβ=(1+β)hc(βh)+c