You need to get the data from different time periods and extract the rules from each. In recent times the rules change along with technology, transportation, and economics.

Thanks Jerry. In the field of urban planning, we generally have a series of land cover maps where we extracted the ca transition rules from. In some cities, these land cover maps may have been derived from remote sensing imagery way back in 1970s (using Landsat). Thus, we can implement your suggestion to extract the rules from different time periods. However, some other cities have short time-period of land use maps (let say less than 10 years or even only two periods). Deriving land use maps from rs imagery is not an option because of different classification criteria. How should we deal with this case? Thank you

I think that you only can check it with data or by your own knowledge (you also can use expert knowledge instead). Anyway, if you go for the first option, data comes with tons of uncertainty, especially when you are working with a long temporal series (because of lot of inconsistencies between the different years) so in the end you spend a lot of time trying to figure out something that you cannot.

Usually Ca models work for short calibration periods or, at least, a few decades, so it is supposed that the transition rules will not change. It also depends on your model and how complex it is. If being simple, it is quite likely that the rules will not change. I mean, the new urban areas will be always located near of previous ones and maybe the accesibility provided by highways or main roads will continue playing an essential role.

The key is in your observation about "extracting the transition rules". David is quite right that this is a complex process involving considerable uncertainty. The way of identifying transition rules depends on the modelling technique you apply. There are several variants of CA, the two key ones I think are the SLEUTH model approach (developed by Keith Clarke and colleagues) and the approach of Roger White, Guy Engelen and Inge Uljee, which is now usually known by its commercial software brand name "Metronamica". My collegaues and I have developed an approach for the R platform under the name SIMLANDER (and APoLUS for multiple land uses and land-based actors) based on this earlier work.

Under these approaches, the transition rules are input by the user and then modified through a back and forth trial and error approach involving simulation of a historic date for which a land use map already exists. Calibration usually involves setting values for neighborhood (attraction and repulsion of land use types to each other based on their distance), accessibility (proximity of communication networks, usually road and rail but also could be electricity networks or irrigation channels, depending on the land use to be modelled), suitability (the underlying biophysical capacity of a cell to recieve a particular land use, e.g. slope, soil type or other characteristic) and zoning (institutional preferences relating to spatial planning regulations, e.g. not inside protected areas). A random factor is also useful to allow for the natural uncertainty inherent in human-environment interaction processes.

The starting values for each of these parameters N, A, S, Z and R (or alpha) would be usually determined with reference to literature, consultation with knowledgeable stakeholders, and researchers' own knowledge of the territory.

By modifying these values and testing the goodness-of-fit of simuations made using these rules against historic maps using spatial statistical approaches (kappa simulation statistic, patch size, fractal dimension, clumpiness etc), it is possible to identify rules that provide the best spatial match to the calibration map.

So if you start with a map of land use in (e.g.) the year 2000, determine rules using the above described procedure to simulate the observed changes between this map and a reference map for (e.g.) the year 2010, you can then run the simulation for a future date, e.g. the year 2050.

The assumption, as you say, is that these transition rules are stationary for the duration of the simulation. The way to test if this assumption of stationarity holds true is to test the rules for a second reference map which you did not use while you were developing the transition rules. If you find, for example, that the transition rules you developed for the change period 2000-2010 also produce a good simulation for the change period 2010-2016, then you can say that your assumption of stationary transiton rules seems to be correct.

Often you will find that this is not the case. That's to say, in most situations, transition rules DO change over time. For this reason it is important to state clearly that future simulations have been developed under the assumption of stationary transition rules. For this reason, major socio-economic changes that were not present in the reference dataset, such as a major economic crisis, producing a sharp decline in urban development, do not produce realistic simulations. For example, in Europe, most of the simulation models developed before the economic crisis produced estimates of urban land expansion that we now know to be incorrect.

This is more to do with the way we use these tools than any actual failure in the modelling process. What we say is that, under certain assumptions, urban land is likely to grow. If these assumptions are violated (e.g. by an unpredictable even like an economic crisis), our predictions will no longer be reliable.