Tools for Assessment of Multiple Scale Land Change Models

Kristina Helle, Pedro Andrade, and Edzer Pebesma

The complex relations between biophysical and anthropological factors generate the land change patterns of our environment. In order to study this complex phenomena, we have to rely on simulation models, for example cellular automata or agent-based models. LUCC simulation models usually generate a new map given a real world map of land cover classes. In the figure below, the left map shows the real data and the right one the simulated results.

Figure 1: Example of real world data (left) and a simulation (right). Source: Pontius (2002)

Considering that “all models are wrong, but some are useful” (Box 1999), model assessment should address the most feasible requirements, such as testing how well the model fits the data and if it is useful for certain purposes. The fit to data is addressed by goodness-of-fit tests, whereas usability should guide the choice of tests. Goodness-of-fit tests compare the predicted map with the reality at the new time.

Some authors have been proposed ways to calculate metrics trying to inform the quality of the results to the scientist. With these metrics, it is possible to calibrate or to validate the model. In fact, the final objective of these goodness-of-fit methods is to point out how to improve the model.

Costanza (1989) was the first author in the literature to propose a model to compare the results of a simulation with real world data in a non-exclusively pixel-by-pixel way. The author proposes a multi-resolution approach, comparing growing sets of cells, trying to measure the spatial patterns of land use variables. Given the difference on different resolutions, the method calculate a weighted average called F, and a p-value-like metric (Figure 2).

Figure 2: Multi-scale goodness-of-fit test. Source: (Costanza 1989)

Pontius (2002) realized that we only need to take into account the cells that have changed, instead of comparing the whole maps. His more flexible approach allows to explicitly separate errors of quantity and of location and to use fuzzy classification. Later (Pontius et al. 2008) he refines his technique to compare real changes and predicted changes. Others like Jantz and Goetz (2005) did also address geometric porperties of the land use patterns like number and shape of clusters and length of edges. Some of these metrices are already implemented in TerraME but have not been used for testing.

Scale (here in terms of resolution and extend) is an important property of LUCC models. Li (2000) investigated the fractal properties which are typical to many land use (change) patterns. Another approach by Jantz and Goetz (2005) compared different goodness-of-fit measures on several resolutions for an urban growth model as land use changes may show varying behaviour on different scales. But also extend can change models a lot as Kok and Veldkamp (2001) showed for national vs. multinational models.

Calibration of cellular automata or agent-based models is not a trivial task as parameters influence is in most cases non-linear and often the number of parameters is high, making comprehensive evaluation of all combinations unfeasible. Simple approaches like by Clarke et al. (1998) generate lots of simulations to be evaluated by the user, they consider interactive visualization as an important tool. Multi-resolution search of the parameter space as described in Candau (2002) or Hakan et.al (2007) may help to detect important parameter combinations and subsequently to adjust them with feasible computational effort. Still users may not find the most influential parameter combinations. This task was addressed by Miller (1998) who used several robust optimization algorithms to investigate the parameter space. Wu (2002) uses the data to fit a prior distribution to the parameters and updates it according to the results of Monte Carlo simulations for calibrating a cellular automata model. Whereas calibration of agent based models is still a domain of econometrics (e.g. Rogers & von Tessin 2004).

The diversity of LUCC models may require different calibration and validation methods. An overview over current models is given by Agarwal et al., a comparison of several models by Pontius et al. (2008). Parker et al. (2003) focus on multi-agent models only.

The following open questions can be investigated by a PhD and a Master theses (Supervisors: Prof. Dr. Edzer Pebesma, Prof. Dr.Gilberto Câmara - not confirmed).

“Qualitatively, these goodness-of-fit measures range from completely non-spatial (the amount of change) to non-spatial metrics (edges and clusters), to explicitly spatial” (Jantz & Goetz 2005). A first task is to compare those tests and to investigate which of them fits best the needs of agent-based models of deforestation in Amazonia. The best test may vary depending on the aim of modelling: a good fit on the averaged deforestation rate needs only a global goodness-of-fit test whereas typical patterns need comparing geometric properties, and prediction about the actual areas affected besides needs comparison on pixel level.

The models in the literature compare two static maps for goodness-of-fit testing. They do not address the errors among more than two time steps and the propagation of errors in time.

The models of the literature do not take into account the intrinsic errors of classification in the data in each of the times. Errors can also emerge from uncertainty in the results, due to random procedures. How to separate this uncertainty from the errors of the model itself? Pontius (2002) approach using fuzzy classification could be a first attempt towards this task.

A current challenge in LUCC is to develop multi-scale models. Human behaviour can only be captured at different levels. Jantz and Goetz (2005) compared goodness-of-fit tests on different resolutions. But they did not address multi-scale models like the partially hierarchical model of Moreira et al. (2009), where the scale below is a finer grid of only a sub-area of the upper scale.

Figure 3: Multi-scale model. Source: (Moreira 2009)

The test results depend strongly on the resolution of the model (Jantz & Goetz 2005) therefore a multi-scale model (hierarchically nested or with resolution not constant in space) could be tested in many different ways. The ideas of multi resolution tests from Costanza (1989) may provide a possibility to aggregate test results on different resolutions.

To complete the design of a model calibration and final validation is needed. Both procedures require goodness-of-fit tests. Calibration should improve the most sensitive and important parameters. Validation finally tests if the calibrated model is overfitting the data or to which extent it is valid. Up to now, models in TerraME are not calibrated statistically but by expert advice. This expertise could be used together with Monte Carlo simulations to find the sensitive parameters.

Top-down models are easier to calibrate than bottom-up models as in the former demand (amount of change) and allocation can be separated. Top-down approaches for LUCC modelling have a demand separated to the allocation. Therefore, we have to adjust the quantity and location errors separately. In the case of agent-based models, each agent decide its own demand and its location independently, making much more difficult to calibrate the model. Jantz and Goetz (2005) and Manson (2000) have studied calibration of bottom-up approaches.

Traditionally, models use a set of data to calibrate the model and another set to validate the method. In the case of LUCC models we can have two configurations:

1. Validate the model with a future time not used in the calibration procedure. Then if there is a good prediction of the new time, the model can be used for building scenarios for future speculations, such as “always-as-usual” or increasing demand.
2. We calibrate the model with a given data set and then use the model with the same parameters with another data set of another region. The only question this procedure can answer is whether the two populations have similar behaviour in these areas, checking if there are significant differences.

TerraME does not include tools for model calibration nor goodness-of-fit tests yet. The latter are required before calibration and validation can be addressed. The first step is to implement Costanza's and Pontiu's models within the TerraME framework. There are some LUCC models developed by INPE's researches that can be used for testing the goodness of fit models. One example is the adaptation of CLUE, a multi-scale model, to a region in the Brazilian Amazonia (Moreira et. al 2009). Another option is the model proposed by Almeida et al. (2005), which is a cellular automata model of city growth in Sao Paulo state. Both models have been developed by INPE researches, although only the first model was already implemented in TerraME framework.

The results of goodness of fit tests may be simple numbers but often are curves or maps and probability distributions. These results must be communicated e.g. by visualization in a way that supports the usability of the models. The most important properties should have an easy interpretation and access. On the other hand, experts should be able to improve the model by thoroughly investigating the errors.

The thesis could be expanded in several directions.

1. The first goal of change models is usually prediction of the future but for to understand the process itself it may be useful to research the other direction in time, predicting the past or the intermediate states. This might require different calibration methods.
2. The validation methods suggested above are only goodness-of-fit measures. The validity of the rules used in the models and the fitness for certain purposes is not addressed. For a more profound understanding of the processes, this might be an important task.
3. The validation problem of calibration and validation rises questions about stationarity of the processes. Comparison of locally fitted parameters may help to understand the processes, typical trajectories and areas of similar development.
4. Providing tools for goodness-of-fit or even calibration and validation by a web service will involve more users and help to adjust the tools to their needs.

The topic is at the overlap of the research at INPE (agent-based and cellular automata models of LUCC) and IFGI (statistics, calibration / validation of models). Therefore the theses should take place as sandwich (exchange: PhD 6-12 months, MSc 2-3 months), starting either at INPE or IFGI.

AGARWAL, CH.; GREEN, G. M.; GROVE, J. M.; EVANS, T. P. & SCHWEIK, CH. M. A Review and Assessment of Land-Use Change Models Dynamics of Space, Time, and Human Choice. CIPEC Collaborative Report Series No. 1.

C. M. ALMEIDA, A. M. V. MONTEIRO, G. CAMARA, B. S. SOARES-FILHO, G. C. CERQUEIRA, C. L.pENNACHIN, M. BATTY. GIS and remote sensing as tools for the simulation of urban land-use change International Journal of Remote Sensing Vol. 26, No. 4, 20 February 2005, 759–774

BOX, G. E. P. Statistics as a Catalyst to Learning by Scientific Methods Part II - A Discussion. XLII Annual Fall Technical Conference of the Chemical and Process Industries Division and Statistics Division of the American Society for Quality and the Section on Physical & Engineering Sciences of the American Statistical Association. 1998.

CANDAU, J., 2002, Temporal calibration sensitivity of the SLEUTH urban growth model. Masters thesis, Department of Geography, University of California.

CLARKE, K.; HOPPEN, S. & GAYDOS, L. (1998): Methods And Techniques for Rigorous Calibration of a Cellular Automaton Model of Urban Growth. (accessed 25.03.09)

COSTANZA, R. Model Goodness of Fit: A Multiple Resolution Procedure. Ecological Modelling 47: 199-215. 1989.

HAKAN, O.; KLEIN, A.G.; SRINIVASAN, R. (2007): Calibration of the Sleuth Model Based on the Historic Growth of Huston. Journal of Applied Sciences 7 (14): 1843 - 1853.

JANTZ, C. A.; GOETZ, S. J. Analysis of scale dependencies in an urban land-use-change model. International Journal of Geographical Information Science. Vol. 19, No. 2, February 2005, 217–241.

KOK, K. & VELDKAMP, A. Evaluating impact of spatial scales on land use pattern analysis in Central America. Agriculture, Ecosystems and Environment 85 (2001) 205–221.

LI, B.-L. 2000. Fractal geometry applications in description and analysis of patch patterns and patch dynamics. . Ecological Modelling 132 (1/2): 33–50.

MANSON, S. M. Agent-based dynamic spatial simulation of land-use/cover change in the Yucatán peninsula, Mexico. 4th International Conference on Integrating GIS and Environmental Modeling (GIS/EM4): Problems, Prospects and Research Needs. Banff, Alberta, Canada, September 2 - 8, 2000.

MILLER, J. H. 1998. Active nonlinear tests (ANTs) of complex simulation models. Management Science 44 (6): 820–30.

E. MOREIRA, S. COSTA, A. P. AGUIAR, G. CAMARA, T. CARNEIRO Dynamic coupling of multiscale land change models: interactions and feedbacks across regional and local deforestation models in the Brazilian Amazonia, Ecological Modelling (submitted). 2009

PARKER, D. C.; Manson, S. M.; JANSSEN, M. A.; HOFFMANN, M. J. & DEADMAN, P. 2003 Multi-Agent Systems for the Simulation of Land-Use and Land-Cover Change: A Review. Annals of the Association of American Geographers 93 (2): 314–337.

PONTIUS, R. G. Statistical Methods to Partition Effects of Quantity and Location During Comparison of Categorical Maps at Multiple Resolutions. Photogrammetric Engineering & Remote Sensing Vol. 68, No. 10, October 2002, pp. 1041–1049

PONTIUS, R.G.; BOERSMA, W.; CASTELLA, J.-CH.; CLARKE, K.; DE NIJS, T.; DIETZEL, CH.; DUAN, Z.; FOTSING, E.; GOLDSTEIN, N.; KOK, K.; KOOMEN, E.; LIPPIT, CH. D.; MCCONNELL, W.; SOOD, A. M.; PIJANOWSKI, B.; PITHADIA, S.; SWEENEY, S.; TRUNG, T. N.; VELDKAMP, A. T. & VERBURG, P. H. Comparing the input, output, and validation maps for several models of land change. Ann Reg Sci (2008) 42: 11-37.

ROGERS, A.; VON TESSIN, P. (2004): Multi-Objective Calibration for Agent-Based Models. Proceedings 5th Workshop on Agent-Based Simulation.

WU, F., 2002, Calibration of stochastic cellular automata: the application to rural-urban land conversions. International Journal of Geographical Information Science, 16, pp. 795–818.

GILES, R. H., Jr. & TRANI, M. K..1999. Key elements of landscape pattern measures. Environmental Management 23 (4):477–81.)