Example of CA using Markov Chain

To illustrate the technique of modelling landcover change by CA, we will present a relatively simple but effective model that combines cellular automata spatial rules with Markov chain transition rules. For example, this approach is proposed as a module called MARKOV / CA_MARKOV in the raster GIS IDRISI (Eastman 2008). The procedure models predictive Land Use Cover (LUC) changes within two stages

Stage 1

A Markov chain analysis (MARKOV) is performed in order to estimate the transition matrix between the two past and documented dates (date 1 and date 2) and to estimate probabilities of change for the third date (date 3) to be predicted. Input and output parameters for this analysis are the following:

• The LUC distribution for the two dates is provided as two images
• The interval of time between the two documented dates (date 1 and date 2) as well as the one between the second date and the date to be predicted (date 2 and date 3) are expressed as regular time steps (iterations)
• A mask image can be introduced in order to limit the development and change to another LUC category due to constraint rules. This will modify the transition probability matrix values
• A transition probability matrix is produced. It expresses the possibility that a cell of a given LUC category will change into any other category
• A transition area matrix is derived. It contains the total area (in cells) expected to change in the next time period
• A group of conditional probability images are generated, one image for each category. They express the probability that each cell will belong to the designated category in the next time period.

Stage 2

A Cellular automata predicting model (CA_MARKOV) estimates the spatial distribution of landcover at a later date (date 3):

• Using the output data produced by the Markov chain analysis, the predicting model will apply a contiguity filter to “grow out” landcover from date 2 to a later time (date 3).
• This CA filter develops a spatially explicit contiguity-weighting factor to change the state of a cell based on its neighbours.

The data used in this illustration are landcover map layers of 1952, 1974 and 1993 for the town of Bulle in the canton of Fribourg in Switzerland. As already previously commented, this dataset has been developed by K. Al-Ghamdi in the context on his PhD research work on LUCC modelling (Al-Ghamdi 2008). The objective of this illustration was to analyse the LUC change during the period from 1952 to 1974 and based on this modelling to predict the evolution for the date 1993. The ground truth 1993 LUC layer will then be used to assess the performance of the procedure. As already illustrated in figure 3.4 for the dates 1946 and 2001, six categories of landcover are identified: lake/pond, river, wetland/marshland, forest, agricultural/open field and urban. Linear features, such as the road network, were not included into the analysis as the 10m resolution of the LUC would overestimate its impact and development.

Several comments can be made about this spatial modelling of LUCC:

• The mask image limits the development of the land cover during the period 1974-1993 to only two categories: urban and agriculture. Remaining categories are considered as protected from the land planning politics. Therefore, the land competition is limited to urban and agricultural development.
• From last slideshow one can observe that urban growth mainly occurs at the expense of agricultural lands and open fields. The expansion of urban areas is modelled on the edges of already urbanised areas, due to the contiguity constraint assigned to the model rules. This type of growth is called “organic growth”.
• f) shows discrepancies between the modelled and the real landcover distribution in 1993. The lack of urban expansion corresponds to new urbanised areas (in blue) that are compensated by extra contiguous zones as part of the organic growth. The inability to model new seeds of development is one of the strong limitations of such a model.

Among other limitations one should identify the inability to take into account the influence of the road network that could potentially generate new “seeds” of development.