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D. Taniar et al. (Eds.): ICCSA 2010, Part I, LNCS 6016, pp. 105–115, 2010.

© Springer-Verlag Berlin Heidelberg 2010

Design of a Dynamic Land-Use Change Probability

Model Using Spatio-Temporal Transition Matrix

Yongjin Joo1, Chulmin Jun2, and Soohong Park3

1 Institute of Urban Sciences, University of Seoul, Seoul, Korea

yjjoo75@uos.ac.kr

2 Dept. of Geoinformatics, University of Seoul, Seoul, Korea

cmjun@uos.ac.kr

3 Dept. of Geoinformatic Engineering, Inha University, Incheon, Korea

shpark@inha.ac.kr

Abstract. This study aims to analyze land use patterns using time-series satel-

lite images of Seoul Metropolitan Area for the past 30 years, and present a mac-

roscopic model for predicting future land use patterns using Markov Chain

based probability model, and finally examine its applicability to Korea. Several

Landsat MSS and TM images were used to acquire land-use change patterns

and dynamic land-use change patterns were categorized from the classified im-

ages. Finally, spatio-temporal transition matrices were constructed from the

classified images and applied them into a Markov Chain based model to predict

land-use changes for the study area.

Keywords: land-use change prediction, spatio-temporal transition matrix,

Markov Chain, urban growth model.

1 Introduction

Urban economist and planners have consistently studied how urban areas have devel-

oped and what primary factors have affected. However, those research efforts have

not sufficiently presented theoretical models. That’s because the aspect of urbaniza-

tion is different between countries and varies with time. In addition, the process of

urbanization is so complicated that proposing theoretical validity is difficult through

feasible verification [10].

Detecting an urban spatial structure and predicting changing trend is very impor-

tant information in establishing the efficient urban policies. In Korea, however, there

have been minimal research efforts regarding analysis and prediction of the character-

istics of dynamic changes of land use. In order to predict land use change, models

represented in terms of space-time is needed and a variety of variables and data sup-

porting the model are also required [11]. The models in the previous studies, with

insufficient time-series data, show limitations in incorporating the past tendencies of

urbanization and explaining the past land use changes. Investigating the current state

of land use and comparing with the past ones requires significant time and efforts. In

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106 Y. Joo, C. Jun, and S. Park

the areas as Seoul Metropolitan Area (SMA), which shows fast population growth and

development, detecting the land-use variations happened in the past is very difficult.

In this situation, utilizing remote sensing data is a practical alternative for monitoring

visible change of urban spaces.

First of all, in this study, we examine the characteristics of land use transitions

through the time-series images of Landsat in SMA. Then, we develop a prediction

model for land-use change based on Markov chain methods and apply it to the simula-

tion of the land-use transition processes. Finally, we examine the validity of prediction

result using the actual data of1984, 1992. Before the implementation of the model, we

set up the prototype of the model through the consideration of the spatial characteris-

tics of the study area, the definition of model components, establishing input variable

data, and designing basic algorithm for the model. Satellite images (MSS, TM), digital

maps and data of the limited development district were used to establish the prediction

model. In other words, input data which are relevant to land-use changes (topography

and social phenomenon) and land cover data are developed. Then, by calculating the

land use conversion rate, a transition matrix was composed on two periods--

1972~1984, 1984~1992. The suitability of the model was evaluated by using a valida-

tion method comparing the derived results with the actual data (1984, 1992).

2 Markov Transition Model

2.1 Model Framework

Analysis of Markov Chain, a statistical method was used for predicting how topog-

raphical and social variations affect on the land use changes in the future through

examining dynamic characteristics of the past. It is based on the process of probability

called Markov Chain, which assumes that present state is determined only by the

immediate previous state. It is composed of the system state and transition probabil-

ity. The changes of states are called transitions, and the probabilities associated with

various state-changes are called transition probabilities.

When a probability analysis can be performed on matrix of random events accom-

panied by time, {Xt}, the row of random variables of each event is referred to as sto-

chastic process. If random variables Xt(t=1,2,…) change into one of state sets(S1, S2,

… Sk) at a certain moment, transforming from state Si to Sj is called a step. And when

the transition probability from Si to Sj (Pij) is only related to the immediate previous

state Si, such probability process is Markov Process. The following formula (1) indi-

cates the transition probability.

Pij = P {Xn = Sj | Xn-1 = Si ∧ P} = P {Xn = Sj | Xn-1 = Si}

(1)

The transition matrix of Pij is defined as follows;

⎥

⎦

⎥

⎥

⎤

⎢

⎣

⎢

⎢

⎡

=

kkkk

k

k

PPP

PPP

PPP

p

...

...

...

11

22121

1 12 11

for

10

<=<=

ijP

∑

=1

ij P

(2)

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Design of a Dynamic Land-Use Change Probability Model 107

In its simplest form, the state vector X(t) can be described if types of land use are

categorized into urban, water, forest, and agriculture. The formula is defined as

follows;

⎥

⎦

⎥

⎥

⎥

⎤

⎢

⎣

⎢

⎢

⎢

⎡

=

) ( 4

) ( 3

) ( 2

) ( 1

)(

tX

tX

tX

tX

tX

(3)

In the above formula, vector x(t) is land use/cover, transition probability P is land use

and land-use change from time t to t+1 is defined as. X(t+1) = P * x(t). Each element

of transition matrix Pij is the probability to move from i type of land use in time t to j

type of land use in time t+1. For example, let’s suppose there are 100 pixels of forests

in t time. If, after 20 years, there still remain 78 pixels of forests, 12 pixels change

into agricultural area and 10 pixels to urban area, then Pij is described as;

? P31 = 10 / 100 = 0.10

? P32 = 0 / 100 = 0.00

? P33 = 78 / 100 = 0.78

? P34 = 12 / 100 = 0.12

With the transition matrix being described as;

⎥

⎦

⎥

⎥

⎥

⎤

⎢

⎣

⎢

⎢

⎢

⎡

=

⎥

⎦

⎥

⎥

⎥

⎤

⎢

⎣

⎢

⎢

⎢

⎡

=

............

12. 0 78 . 000 . 010 . 0

......... ...

...... ......

4443 42 41

34 33 32 31

2423 2221

1413 1211

PPPP

PPPP

PPPP

PPPP

p

(4)

2.2 Problems of Spatial Influence Algorithm

Markov model has been used to predict aspects of land-use changes by human activi-

ties and understand the changing processes of natural forms [8]. The probability of a

land cover change in this model is based on spatial influence algorithm with neighbor-

hood effects having influence on adjacent land cover [12]. On the whole, this algorithm

supposes that change of a cell is carried out by transition probabilities. The range of

influence that adjacent cells reach can be set up with 4 or 8 neighbor cells. The values

are set in two-dimensional square cells and transition state of land use class from pre-

vious time (t1) to next time (t2) is calculated. After that, change of cell in the space can

be simulated by the time according to calculated transition probability.

Markov model is easily computed by using digital image or raster-based GIS data

and has an advantage to reflect transition tendency of current land use effectively.

Even though time passes along, transition matrix is always constant and applied

equally to all locations [7]. However, actual land use doesn’t change exactly accord-

ing to the assumption of Markov and obtaining the transition probability through

independent measurement is difficult. Also, the factors for land-use transition are

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108 Y. Joo, C. Jun, and S. Park

more affected by political and economic factors rather than biophysical ones [1].

Therefore, in case we apply previous algorithm to our study area, the Seoul metropoli-

tan area where urbanization has been on the rise at an extremely rapid rate for a short

period in Korea, some anticipated problems are as follows;

First of all, limited development districts (also called green-belt areas) have been set

in South Korea since 1972. It has been playing an important role in controlling spread

of urbanization and preserving green spaces. Because the change of center pixel is

affected only by the land uses of adjacent cells, it leads to a spread of habitation inside

the limited development areas. Secondly, one of the most important factors relevant to

land-use change is the slope. The high slope prevents the regions from being devel-

oped and populated by the development permit system in Korea. Unless the physical

properties of the land are considered, habitation cells located in hill sides and low

mountains will spread into neighboring cells.

In this paper, we thus improved previous model into a more practical land-use

model engrossed in urban structural change, which can incorporate the concept of

multi-dimensional spatial filter. In other words, political factor of land use regulation

(green-belt policy) is considered to prevent urbanized cells in green area and green-

belts from spreading. More importantly, we developed the methodology for dynamic

probabilities of transition matrix with the help of practical multi-temporal satellite

images accumulated for long periods.

3 Design of Land-Use Change Model

Land-use change model that we suggest is based on Cellular automata (CA), which

are both a body of knowledge and set of techniques for solving complex dynamic-

systems problems [9]. The model includes four components: a grid space, local states,

neighborhoods and a transition rule. Though these components, the model evolves in

discrete time steps by updating their local state according to a universal rule that is

applied to each cell synchronously at each time step. The value of each cell is deter-

mined by a geometrical configuration of neighbor cells, and is specified in the transi-

tion rule. Updated values of individual cells then become the inputs for the next

iteration. In this chapter, we specify each element for design of land-use change

model in detail.

Grid space: The first element of this model is grid spaces, which mean regular grid

of cells where interactions for urban sprawl are carried out. Theoretically, there is no

restriction to the tessellation of a grid space and it could be various forms of shape. In

general, however, square cells are the most common form in CA applications due to

their inherent convenience of implementation in computation. Therefore, grid space in

this study consists of regular square cells of 2 dimensions. Besides, because grid

spaces are determined by the spatial resolution of satellite images, input data (land

cover, green-belt, and slop) are determined as 60m grids according to Landsat MSS,

the lowest size of the sensor resolution.

Local states: The second element of the model is the local state. The local states

mean the status of each cell encoded by numerical values at a given time step. The

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Design of a Dynamic Land-Use Change Probability Model 109

range of its values is defined in the transition rules and depends on the actual imple-

mentation of model. In raster GISs, these local states are directly analogous to the

values of each grid cell in a layer. The local states in our study represent the land-use

characteristics of cells, that is, values assigned in all cells. Table 1 shows the cell state

of input data that we used in our experiment.

Table 1. The cell states of input data

Thematic Data State

Grid

Space(m)

Data type

Urban 1

60 ? 60

Water 2

60 ? 60

Forest 3

60 ? 60

Land cover

Agriculture 4

60 ? 60

Integer

Exclusive region 1

60 ? 60

Green-belt

Non-Exclusive region 0

60 ? 60

Byte

Gradient map Slope

0 ? 90 60 ? 60

Integer

Neighborhood cell: Neighborhood is a set of cells located adjacent to focus cell.

Such a neighborhood concept is very similar to the mask or moving window of spatial

filters in digital image processing and GIS. In theory, there is no limitation to the size

of a neighborhood and usually the configuration of a neighborhood can be extended to

the temporal dimension as well as the spatial dimensions. In this study, the Neighbor-

hood of Moor was used as the neighborhood definition. In fig 1, the cell in the 3 x 3

window (i.e, the neighborhood) is changed in a discrete time step according to the

transition rule.

Transition rule: GIS data such as land cover data of time series built from satellite

images, digital elevation models (DEM), and green-belt data are considered as input

variables. The local transition rule and constraints are applied to grid spaces repeatedly

resulting in state transitions from time t to time t +1. This transition rule defines how

each cell changes every time step, and models the process that a state of a cell changes

constantly in accordance with the effects of neighboring cells. As the process of this

algorithm, transition matrix is calculated by using time-periodic transition probability in

the study area. Transition index is calculated through examining the state of the focus

cell and the adjacent state of 8 cells representing land use followed by the computation

of the transition index. The transition index is the maximum value j of Nj × Pij (where Nj

is the number of land use elements in the current window size and Pij is the element in

the transition matrix from i to j). If the returned value of transition index is urban, then

model checks for such constraints as green-belt and slope. In case agricultural cell in

green-belt is changed into urban, then its state is maintained. Lastly, transition index is

assigned to the cell and move on to next cell. Process and algorithm for model execution

are as shown in figure 1.