The Flow Matrix Offers a Straightforward Alternative to the Problematic Markov Matrix
Abstract
:1. Introduction
2. Materials and Methods
2.1. Illustrative Example
2.2. Equations
2.2.1. Raw Matrix
Variable | Description |
---|---|
c_{ij} | Size of transition from category i at time t_{0} to category j at time t_{1} |
C | Raw matrix with J rows, J columns, and entries c_{ij} |
D(n)_{d} | Family of row vectors for time interval n where each member of the family corresponds to a specific d with d = 0, 1,…, n . Each row vector has J columns where each entry is a category’s size at t_{n} that derives from d incidents from t_{0} to t_{n}. |
d | Number of incidents which is an integer on the interval [0, n] defined as the number of times a pixel’s category has transitions from time t_{0} to t_{n} |
f_{ij} | Size per unit of time for the transition from category i at time t_{0} to a different category j at time t_{1} |
F | Flow matrix with J rows, J columns, and entries f_{ij} |
G | Gain matrix with J rows, J columns, and entries m_{ij} off the diagonal and zeroes on the diagonal |
H(n) | Matrix with J rows, J columns, and the same entries as M^{n} off the diagonal and zeroes on the diagonal |
i | Index for a category at the start time of a time interval |
I | Identity matrix, which has 1 for each diagonal entry and 0 elsewhere |
j | Index for a category at the end time of a time interval |
J | Number of categories >1 |
m_{ij} | Proportion of category i at time t_{0} that transitions to category j at time t_{1} |
M | Markov matrix with J rows, J columns, and entries m_{ij} |
n | Index for the time interval from time t_{n}_{−1} to time t_{n} where n = 1, 2, … ∞ |
P | Persistence matrix with J rows, J columns, and entries m_{ii} on the diagonal and zeroes off the diagonal |
R(n) | Matrix with J rows, J columns, and the same entries as M^{n} on the diagonal and zeroes off the diagonal |
s_{i}(0) | Size of category i at time t_{0} |
s_{i}(n) | Size of category i at time t_{n} |
S(0) | Row vector with J columns and entries s_{i}(0) |
S(n) | Row vector with J columns and entries s_{i}(n) |
t | Continuous time on the interval [t_{0}, T] |
t_{0} | Start time of the calibration time interval |
t_{1} | End time of the calibration time interval |
t_{n} | End time of time interval n, which is also the start time of interval n + 1 |
T_{i} | Time when category i reaches zero persistence for the Flow extrapolation |
T | Time when the Flow extrapolation stops |
w_{ij}(t) | Size of transition from category i at time t_{0} to category j at time t |
w_{jj}(t) | Size of persistence of category j from time t_{0} to time t |
W(t) | Matrix with J rows, J columns, and entries w_{ij}(t) for the Flow extrapolation |
2.2.2. Flow Matrix
2.2.3. Markov Matrix
2.3. Case Studies
3. Results
3.1. Wetland Case Study
3.2. Suburban Case Study
4. Discussion
4.1. Comparison of Characteristics
Characteristic | Flow | Markov |
---|---|---|
Necessary mathematical knowledge | Line | Matrix Algebra |
Extrapolates at most one incident of change | Yes | No |
Extrapolates through continuous time | Yes | No |
Assumes constant size change per time change | Yes | No |
Computes time point to reach zero persistence | Yes | No |
Extrapolates to any desired time point | Constrained | Maybe |
Extrapolates into the infinite future | No | Yes |
Category sizes approach a steady state | No | Frequently |
Category’s gain depends on sizes of other categories | No | Yes |
Can extrapolate acceleration of change | No | No |
Can extrapolate transitions that calibration lacks | No | Yes |
Matches true systems through a time series | Testable | Testable |
4.2. Extrapolation to Specific Time Points
4.3. Which Method to Choose?
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Strzempko, J.; Pontius, R.G., Jr. The Flow Matrix Offers a Straightforward Alternative to the Problematic Markov Matrix. Land 2023, 12, 1471. https://doi.org/10.3390/land12071471
Strzempko J, Pontius RG Jr. The Flow Matrix Offers a Straightforward Alternative to the Problematic Markov Matrix. Land. 2023; 12(7):1471. https://doi.org/10.3390/land12071471
Chicago/Turabian StyleStrzempko, Jessica, and Robert Gilmore Pontius, Jr. 2023. "The Flow Matrix Offers a Straightforward Alternative to the Problematic Markov Matrix" Land 12, no. 7: 1471. https://doi.org/10.3390/land12071471