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Modern Geomatics Technologies and Applications

indicates change and value 0 indicates no change in user interval. The solution of the logistic regression model is a function that
calculates the probability of each object belonging to predetermined classes according to a new set of dependent variables [14].
The following equation represents the logistic regression formula [15]:
1
( ) = (1)
1 + −( +∑ )
where ( ) indicates the probability of changing the use of each pixel and is the independent parameters of the model. α and

constant relation = ( . . … . ) are the regression coefficients that α and should be estimated [16]. The function of

0
1

logistic regression is a nonlinear function that must be logarithmic to be linear. Equation (2) and (3) show the stages of the linear
regression of logistic regression [12].
( )
( ) = ln ( ) (2)
1 − ( )
( )
ln ( ) = + + ⋯ + + (3)

1 1
1 − ( )
Logistic regression uses the maximum correction method to estimate the best combination of model fit coefficients
( . ) [16] .Equation (4) indicates the maximum correction function used in logistic regression [15].

= ∏ × (1 − ) (1− ) (4)

=1
where represents the probability, represents the observed value of the dependent variable for sample , and finally

indicates the predicted value for the dependent variable .
1
= (5)

1 + −(∑ =0 )
To maximize Equation (4), the following nonlinear equation must be resolved [16].

∑ ( − ) × = 0 (6)

=1
where indicates the observed value of factor for sample . The Newton-Raphson algorithm is used to solve equation 6 [15].

3.4 Markov chain
The Markov chain is used to determine the amount of change that will occur in the future. A Markovian process is a
method of determining the state of a system by knowing its previous state and the probability of its transition.
The Markov chain is a discrete random process with the Markov property, which determines the probability distribution
for the system in the next step, and all future steps depend only on the current state of the system and do not depend on the state
of the system in the previous steps [17]. Markov chain analysis is a good tool for modeling land use change and serves as an
indicator of the direction and magnitude of future change because it has descriptive capabilities and predicts the course of simple
land changes [18], [19].

The model used in this study uses Markov chain analysis to predict the transfer zone matrix to manage land use change
rates. The transfer zone matrix is performed by the following equation [20]:

11 12 ⋯ 1
⋯
= [ 21 22 2 ] (7)
⋮ ⋮ ⋱ ⋮
1 2 ⋯

3