# A Regression-Based Procedure for Markov Transition Probability Estimation in Land Change Modeling

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## Abstract

**:**

## 1. Introduction

_{t}, the annual transition matrix P

_{1}can be determined as:

^{th}root of the base matrix. Currently, there are two mathematical approaches to solve this problem [6]. One is to use matrix logarithms and exponentials:

_{i}is the i

^{th}eigenvalue of matrix P

_{t}(the base matrix for t years) and U is a k × k eigenmatrix, given k land cover classes. However, as discussed by Takada et al. [6], these two comparable approaches can present problems. Using eigendecomposition, encoded into a computer program called TAM, they showed that mathematically there are multiple solutions for estimation of the annual rate of change on the order of t

^{k}, where t is the number of years between the two historical land covers examined (10 years in the above example), and k is the number of land cover classes (7 in this example). Thus, for the example with Florida, there are ten million (10

^{7}) annual solutions, all of which obey the power rule to yield the base matrix. Many of these are complex number solutions and of those that are real, some involve negative transition probabilities. This is not reasonable since Markov transition probability matrices are stochastic matrices, which are defined by being composed entirely of non-negative real numbers that sum to 1 across each row [7]. Takada et al. [6] also indicate that it is possible that more than one stochastic annual transition matrix may be found, or indeed that none will be found. In a follow-up study, Hasegawa et al. [8] found that only 11% of 62 cases examined produced stochastic matrices. However, they showed that by forcing small negative transitions to be 0, it was possible to increase this number to 87%. That said, the biggest issue is the computational burden of the procedure used. Hasegawa et al. [8] cite an example of solving a case with 10 classes over 24 years requiring over 1000 days for computation.

- Mathematical solution yields multiple results, many of which involve complex numbers and negative transition probabilities. Such matrices are unusable for land cover modeling of future transitions.
- It is possible that no non-negative real stochastic solution can be achieved. In these cases, an approximation can sometimes be achieved by forcing small negative transition probabilities to 0.
- Most importantly, for computational reasons, solutions are only feasible when either or both the historical time period and the number of land cover classes are very small.

## 2. Regression-Based Markov

_{ijt}represents the transition probability for land cover i transitioning to land cover j at time step t, where t assumes values of 0, 1 and 2 for the three matrices generated. Since there are three data points for each transition, the regression will yield a perfect fit. The fitted equation is then used to estimate the transition probability for any period of time between these control matrices. Since the transition has k × k entries, a total of k

^{2}regressions are computed. Figure 2 illustrates this concept.

## 3. Materials and Methods

#### 3.1. Study Area and Data

#### 3.2. Data Analysis

^{2}was then calculated to determine the goodness of fit. Since the TAM results are understood to be the true values, this allows a determination of the degree to which a quadratic regression fits the true evolution of the transition probabilities over the span of three control matrices.

## 4. Results

^{2}values were all above 99%, with the lowest value being from shrub/grassland to barren in Georgia with an Adjusted R

^{2}of 99.94%. Figure 4 illustrates this regression. As can be seen, the fit is exceptionally good.

## 5. Discussion

^{2}of 99.94%. Comparing the fitted transition probabilities to the correct values as computed by TAM (Table 1), half the errors in each of the five focus states (as measured by the median) were less than 0.00001 and no errors were as high as one percent (the maximum was 0.006).

_{1}using the following formula:

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**A transition probability matrix for land cover change in Florida, USA, from 2001 to 2011 based on the land cover maps depicted. Florida is one of the lower 48 conterminous states, highlighted in red in the inset. The state centroid (marked with a yellow dot) is at 28.68° N, 82.46° W.

**Figure 2.**An illustration of the logic of the Regression-Based Markov (RBM) algorithm. (

**a**) To estimate the transition probability from forest to shrub/grassland in 2035, the base matrix (Figure 1) is taken to the powers of 2, 3 and 4. In each matrix, the rows represent land cover in 2001 and the columns represent land cover at the later date specified. (

**b**) The values in the three powered matrices (an example is highlighted in yellow) are then regressed against time expressed in 10-year steps from 2031. The value for 2035 would use the fitted equation using a value of 0.4 for the value of t (since 2035 is 4/10 of the time from 2031 to 2051). The predicted transition probability is 0.2831.

**Figure 3.**Schematic of the RBM algorithm for cases where the future date is between: (

**a**) the present and one time step ahead, (

**b**) one and two time steps ahead, and (

**c**) x and x + 1 time steps ahead. The input land cover maps are shown in light blue. Analytical operations are symbolized in green. Dark blue symbols represent transition probability matrices. With this logic, the future transition probability matrix is always interpolated, using quadratic regression, between control matrices P0 and P1. Note that recursive matrix powering is required in (

**b**) and (

**c**) to generate control matrices.

**Figure 4.**Evolution of the probability of transition from shrub/grassland to barren in Georgia as determined from the TAM program for durations from 1 to 20 years and set to 0 for 0 years (markers) along with the quadratic fit determined by the RBM procedure.

**Figure 5.**Maximum, mean and median error in transition probability estimation for all 49 transitions in Georgia, USA from 2011 to 2021.

**Figure 6.**The error distribution in estimated transition probabilities over all 49 transitions for Florida from 2011 to 2071.

**Table 1.**Highest median, mean and maximum errors in transition probability across all 49 transitions, by state.

State | Median | Mean | Maximum |
---|---|---|---|

Alabama | 0.000005 | 0.000203 | 0.005277 |

Arizona | 0.000002 | 0.000016 | 0.000569 |

Delaware | 0.000002 | 0.000018 | 0.000408 |

Florida | 0.000004 | 0.000085 | 0.001367 |

Georgia | 0.000005 | 0.000196 | 0.006353 |

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Eastman, J.R.; He, J. A Regression-Based Procedure for Markov Transition Probability Estimation in Land Change Modeling. *Land* **2020**, *9*, 407.
https://doi.org/10.3390/land9110407

**AMA Style**

Eastman JR, He J. A Regression-Based Procedure for Markov Transition Probability Estimation in Land Change Modeling. *Land*. 2020; 9(11):407.
https://doi.org/10.3390/land9110407

**Chicago/Turabian Style**

Eastman, J. Ronald, and Jiena He. 2020. "A Regression-Based Procedure for Markov Transition Probability Estimation in Land Change Modeling" *Land* 9, no. 11: 407.
https://doi.org/10.3390/land9110407