Flow matrix avoids problems of the popular Markov matrix

Authors: Jessica Strzempko*, Clark University, Robert Gilmore Pontius Jr., Clark University
Topics: Spatial Analysis & Modeling, Geographic Information Science and Systems
Keywords: Markov matrix, Flow matrix, transition matrix, extrapolation, LTER
Session Type: Virtual Paper
Day: 4/8/2021
Start / End Time: 1:30 PM / 2:45 PM
Room: Virtual 16
Presentation File: No File Uploaded


The Markov matrix gives the proportion of each initial category that transitions to each subsequent category. Scientists use Markov matrices for extrapolation in dynamic systems where a single observation can change during each time interval. Scientists routinely rely exclusively on the Markov matrix as a default modeling method in spite of substantial mathematical limitations. The Flow matrix offers an alternative to the Markov matrix. The Flow matrix gives a constant size per time for each transition assuming each observation changes at most once. We compare the Flow and Markov matrices in terms of mathematical properties and implications in application. The Markov matrix frequently extrapolates to a steady state with no categories reaching extinction. Each category must be present at the initial time. The Markov matrix extrapolates to future time points that are an integer multiple of the historical time interval with a new incident possible at each observation for each additional future time interval. In contrast, the Flow matrix extrapolates until a category becomes extinct. The initial size of a category can equal zero. The Flow matrix extrapolates on a time continuum to any future time point with a maximum possible incident of one per observation. We use case studies from Long Term Ecological Research (LTER) datasets to demonstrate these characteristics. Scientists can gain insight from both approaches and should select an extrapolation method that suits their needs and field of application.

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