Quantity, exchange, and shift components of difference in a square contingency table

A common task is to measure the difference between two maps that show the same spatial extent for the same categorical variable, such as land-cover type. One popular technique is to express the overall difference as the sum of two components called quantity and allocation. This article shows how to take an additional step to express allocation difference as the sum of two components called exchange and shift. Exchange exists for a pair of pixels when one pixel is classified as category A in the first map and as category B in the second map, while simultaneously the paired pixel is classified as category B in the first map and as category A in the second map. If there are more than two categories, then it is possible to have a component called shift, which is allocation difference that is not exchange. Our article shows how to compute all three components of overall difference: quantity, exchange, and shift. We show also how to compute the three components for each category and to reveal the category pairs that account for the largest exchanges. Our article applies the principles to characterize both temporal changes and classification errors using land-cover maps from suburban Massachusetts, USA.