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This manuscript provides guidance concerning how to use the Total Operating Characteristic (TOC) when (1) analyzing change through time, (2) ranking a categorical independent variable, and (3) constraining the extent for a gaining category. The illustrative variable is the marsh land-cover category in the Plum Island Ecosystems of northeastern Massachusetts, USA. The data are an elevation map and maps showing the land categories of water, marsh, and upland in 1938, 1971, and 2013. There were losses and gains near the edge of the marsh between 1938 and 1972 and between 1972 and 2013. The TOC curves show that marsh gained most intensively at intermediate elevations during the first time interval and then had a weaker association with elevation during the second time interval. Marsh gains more intensively from water than from upland during both time intervals. The TOC curves also demonstrate that the marsh gains occurred where marsh was previously lost, a phenomenon called Alternation. Furthermore, eliminating far distances and extreme elevations from the spatial extent decreased the area under the curve (AUC) for distance and increased the AUC for elevation. We invite scientists to use the TOC because the TOC is easier to interpret and shows more information than the Relative Operative Characteristic. 

The Flow matrix is a novel method to describe and extrapolate transitions among categories. The Flow matrix extrapolates a constant transition size per unit of time on a time continuum with a maximum of one incident per observation during the extrapolation. The Flow matrix extrapolates linearly until the persistence of a category shrinks to zero. The Flow matrix has concepts and mathematics that are more straightforward than the Markov matrix. However, many scientists apply the Markov matrix by default because popular software packages offer no alternative to the Markov matrix, despite the conceptual and mathematical challenges that the Markov matrix poses. The Markov matrix extrapolates a constant transition proportion per time interval during whole-number multiples of the duration of the calibration time interval. The Markov extrapolation allows at most one incident per observation during each time interval but allows repeated incidents per observation through sequential time intervals. Many Markov extrapolations approach a steady state asymptotically through time as each category size approaches a constant. We use case studies concerning land change to illustrate the characteristics of the Flow and Markov matrices. The Flow and Markov extrapolations both deviate from the reference data during a validation time interval, implying there is no reason to prefer one matrix to the other in terms of correspondence with the processes that we analyzed. The two matrices differ substantially in terms of their underlying concepts and mathematical behaviors. Scientists should consider the ease of use and interpretation for each matrix when extrapolating transitions among categories.