Abstract :
[en] The purpose of this work is to provide models to compare the state of a region at different stages of its evolution. We formulate three mathematical models all based on several axioms that make sense in some geographic contexts. We start with a simple one, which doesn't take into account any geographic aspect. In the second one, we introduce one geographic aspect and then we generalize this model to take into account any finite number of geographic aspects.
In the field of land management, it's not uncommon to have a geographic map, representing a region under study, and divided into geographic units. Each unit is assessed on an ordinal scale describing its degree of suitability for some usage, for instance housing, or its state of degradation with respect to sustainable development criteria. We call such a map a decisional map. After a while, and for example, after the application of some policies aiming to improve the situation, the state of the units has evolved. For some units, the state has improved, and for some others, it has deteriorated. What we want to know is whether the global state of the map has improved or not. Of course, we must ask for an expert (decision maker) to answer this question. So, we propose several formal models aiming to compare two states of a same region, and taking into account the preferences of the decision maker who will face the decision problem.
In a first model due to Metchebon, we suppose that the only thing that matters is the distribution of the units in the different categories of the ordinal scale. We define the set of all possible decisional maps of a region as a mixture set so that we can apply Jensen's axioms, which initially come from the theory of decision under uncertainty. Thanks to these axioms, we can represent the preferences of the decision maker by a linear utility function. The model we obtain consists in giving a value to maps where all units are assigned to the same category, then weighting these values by the proportion of units actually assigned to each category, and finally summing these weighted values. For the elicitation of the parameters of the model, we can transpose a method used in the theory of decision under uncertainty, i.e. the comparison of lotteries. We formulate questions to the decision maker in terms of comparisons of well-chosen maps. The main drawback of this model is that it neglects all the geographic aspects.
The second model we have developed enables to take into account one ``geographic attribute'' that can have an influence on the preferences of the decision maker. Such geographic attributes can be the proximity to some habitations, some roads, some watercourses... For this model, we introduce what we have called an ``attribute map''. This is a fixed map divided in the same way as the decisional map considered, where all units are evaluated on an ordinal scale representing the levels of the geographic attribute considered, for instance the levels of proximity. Then, we define a partial map as a subregion of the attribute map consisting of all units that are assessed on a same level of the scale and we deal with each partial map individually. We define the set of all possible distributions of a partial map as a mixture set and we specify a preference relation on this set, which can be represented by a linear utility function. Finally, we aggregate the representations obtained by an additive value function.
The third model is a generalization of the previous one. It enables to take into account any finite number of geographic attributes. For this purpose, we introduce one attribute map for each geographic aspect we want to consider. The idea of the model is to superpose all these maps to obtain only one where all units are evaluated on the scale of each attribute map, so that we return to the previous form. The elicitation of the parameters of this model, as for the previous one, results of a dialog with the decision maker, who will have to compare specific partial maps.
