Abstract: While cellular automata (CA) models have been increasingly used over the last decades to simulate a wide range of spatial phenomena, recent studies have illustrated that they are sensitive to cell size and neighborhood configuration. In this paper, a new vector-based cellular automata (VecGCA) model is described to overcome the scale sensitivity of the raster-based CA models. VecGCA represents space as a collection of geographic objects of irregular shape and size corresponding to real-world entities. The neighborhood includes the whole geographic space; it is dynamic and specific to each geographic object. Two objects are neighbors if they are separated by objects whose states favor the land-use transition between them. The shape and area of the geographic objects change through time according to a transition function that incorporates the influence of the neighbors on the specific geographic object. The model was used to simulate land-use/land cover changes in two regions of different landscape complexity, in Quebec and Alberta, Canada. The results revealed that VecGCA produces realistic spatial patterns similar to reference land-use maps. The space definition removes the dependency of the model to cell size while the dynamic neighborhood removes the rigid, arbitrarily defined zone of influence around each geographic object.

• Synthesis

1. A basic CA consists of five components:
   1. a grid space on which the model acts;
   2. cell states in the grid space;
   3. transition rules that determine the spatial dynamic process;
   4. a neighborhood that influences the central cell;
   5. time steps.
2. VecGCA Model: A vector-based CA model, where the space is represented as a collection of irregular structures that represents the geographic objects.
3. Problem: The CA models are sensitive to different cell resolutios and neighborhood configuration, in another words, they are sensitive to the scale of the model.
4. Proposed structure: In the dynamic neighborhood proposed, two objects are neighbors if they are adjacents or separated by other objects which states are favorable to the transition of states between them. There is no distance or fixed area that delineates it. The neighborhood includes the whole geographic space and is specific to each geographic object. A binary matrix as the following describes if a state X is favorable to the transition from the state Y to Z. The 1 value indicates that the state is favorable to the transition, and 0 indicates the opposite.

The influence of each neighbor on the central objet is defined by an influence value that varies between 0 and 1, and is variable on the surface of the central object, having the maximum value in the object's border and decreasing inside the object. If this value is higher than a threshold value that represents the resistance of the geographic object to change it's state, a geometric transformation which produce a change of shape of the object is performed.

Abstract: One of the main challenges for the development of spatial information theory is the formalization of the concepts of space and spatial relations. Currently, most spatial data structures and spatial analytical methods used in GIS embody the notion of space as a set of absolute locations in a Cartesian coordinate system, thus failing to incorporate spatial relations, which are dependent on topological connections and fluxes between physical or virtual networks. To answer this challenge, we introduce the idea of a generalized proximity matrix (GPM), an extension of the spatial weights matrix where the weights are computed taking into account both absolute space relations such as Euclidean distance or adjacency and relative space relations such as network connection. Using the GPM, two geographic objects (e.g. municipalities) are “near” each other if they are connected through a transportation or telecommunication network, even if thousands of kilometers apart or, using even more abstract concepts, if they are part of the same productive chain in a given economical activity. The generalized proximity matrix allows the extension of spatial analysis formalisms and techniques such as spatial autocorrelation indicators and spatial regression models to incorporate relations on relative space, providing a new way for exploring complex spatial patterns and non-local relationships in spatial statistics. The GPM can also be used as a support for map algebra operations and cellular automata models.

• Synthesis

1. Relative space → the relation of a spatial object to other objects.
2. Most spatial data structures and spatial analytical methods used in GIS embody the concept of space as a set of absolute locations in a cartesian coordinate system.
3. According to Castells geographical space is a combination of “spaces of fixed locations and spaces of fluxes”, where: spaces of fixed locations → spatial arrangements based on absolute space; spaces of fluxes → spacial arrangements based on relative space.
4. Absolute space relations are, for example, Euclidian distance or adjacence.
5. Relative space relations are, for example, topological connection on a network.
6. Given two objects o_i and o_j belongin to a set O, the neighborhood relation between o_i and o_j is denoted w_ij. The set of all relations w_ij defines a spatial weights matrix W that represents the neighborhood relationships between the objects in the set O.
7. Problem: Representation of relative space in a GIS uses arc-node data structures, which are usually completely unrelated to the analysis methods that use representations of absolute space, leading to a limited conception of space in GIS.
8. Proposed Structure: GPM is an extension of the spatial weights matrix, where the spatial relations are computed taking into account both absolute space and relative space relations. To compute the GPM, a graph G over the connected subset S provides the connectivity informations. G includes different types of networks, including physical links and logical links. Thus, the graph G represents the relative space and S represents the absolute space.
9. How to build a GPM?
   1. Functions proxabs and proxrel: associated to the proximity measures in the absolute space and in the relative space, respectively. There are several ways to define these functions, such as indicator functions or distance-based functions.
   2. Closed networks → objects connected only at network nodes. Examples are railroads and telecommunication networks.
   3. Openned networks → objects connect at any node or arc coordinate. Examples are transportation networks, such as rivers and roads. It is necessary to make use of the actual line coordinates that correspond to each arc, in order to be able to compute the closest entrance/exit points from any arbitrary position.
   4. The criterion to the construction depends on the type of the neighborhood.
   5. open networks
      • Two parameters are considered: (1) maximum distance from an object to the network; (2) maximum value of the shortest path between the two connection of two objects to the network. And the following functions are used:
        • cent → calculates the centroid of an object o_i;
        • dist → calculates the distance between two points;
        • shpath → calculates the shortest path between two locations;
        • clspt → determines the closest point to a given location in the graph.
      • Computing proxrel: Given two objects (o_i and o_j), the closest location (p_i and p_j) in the graph to its are computed, and the shortest path between them is calculated. Therefore, the objects are neighbors if the distance from their centroids to the network is smaller than a threshold value or if the poits p_i and p_j in the graph is smaller than a threshold value.
   6. closed networks
      • Two parameters considered: (1) maximum distance from each object to the closest node; (2) maximum limit to the shortest path between the two nodes chosen as connection points.
      • Having these two parameters, the proxrel function can be calculated in a similar way as using open networks.

Abstract: This paper investigates which, how and to what extent land-use related neighbourhood effects play a role in urban dynamics. To justify the use of cellular automata land-use models for spatial policy support, existing neighbourhood rules need to be better founded. This research eliminates a number of uncertainties in the land-use model outcomes by introducing improved empirically founded and regionalspecific neighbourhood rules. This allows for a better evaluation and justification of spatial policy scenarios and their effects on future land-use dynamics.

• Synthesis:
  1. CA has a considerable potential for urban land-use modelling.
  2. The validity of the neighborhood rules specify the 'behaviour' of land-uses and can be considered as being at the heart of the CA model.
  3. The neighborhood rules specify how the combined effects of spatial externalities, in the sense of mutual attraction and repulsion of land-use types, work out over distance. The main limitation to which these rules are subjected, as applied in practice, is their lack of theoretical foundation and empirical validation. It is often defined and calibrated on an ad hoc basis by trial and error methods.
  4. Before CA models can justifiably be applied to support spatial policy, their neighborhood rules in particular require a better empirical foundation.
  5. There are only few CA models being used in order to develop or assess land-use patterns for policy support. Usually CA models are suited to specific applications, being not very suitable to assessments in another scale which is not that for what it was created. For example, a model created to be applied in a national level to long-term analyses can not be appropriated to short-term assessments in a regional scale.
  6. There is a need for further developments to make present land-use models suitable as policy instruments in specific cases. The most crucial point of improvement is a sound foundation of the neighborhood rules for regional applications for relatively short time periods.
  7. Goals:
     1. Find out which are the key neighborhood effects that influence the processes of land-use change and how their net intensity, the aggregate influence of one land use on another, is shaped over distance. It is because the empirical justification of the neighborhood rules used in CA is often insufficient for policy intervention assessments.
     2. Determine to what extent land-use related net neighborhood influences explain past land-use changes and can be used to simulate future changes. Neighborhood rules derived from the first part is assessed by calibration and validation of the CA land-use model for a number of urban regions. Compare, based on it, the survey-based rules and the calibration-based rules, making it possible to assess their values for future CA modelling.
  8. Spatial externalities and urban dynamics:
     • Urban dynamics are closely related to the widely accepted to the first law of geography, formulated by Tobler in 1970: “Everything is related to everythig else, but near things are more related than distant things”. Many works have stressed the importace of the concept of “situation”, the relative location of a site.
     • The externalities, as defined in the paper (see page 41) are seen as the organising forces of urban patterns. Every activity of an element in an urban system might generate unpriced and perhaps non-monetary effects upon other elements.
     • Most external effects are constrained to a spatially limited area, “spatial externalities”. The definition of spatial externalities used is: the radiated priced or unpriced effects of one land-use on another land-use. It can be envisioned as a form of <font color=“green”>situational factors</font>.