Abstract: Cellular automata (CA) can reproduce global patterns and behavior from local interactions of cells and they are used increasingly to simulate complex natural and human systems. Among their attributes are their computational simplicity and their explicit representation of space and time. However, the classic definition of CA limits their application to problems that involve a discrete space, and similar rules and neighborhoods for all cells. In addition, the standard raster-based CA model is sensitive to spatial scale. This paper presents a new vector-based geographic cellular automata model, called the VecGCA model, which defines space as a collection of irregular geographic objects. Each object has a geometric representation (a polygon) that evolves through time according to a transition function that depends on the influence of neighboring polygons. In this model, the neighborhood is defined as the region of influence on each geographic object, and the neighbors are all geographic objects located within the region of influence. An innovative aspect of the VecGCA model is that the procedure allows geometric transformation of objects. The area of a polygon (representing an object) is reduced in the region that is nearest to the neighbor that exerts an influence on it, and the area of that neighbor is increased accordingly. The proposed model was tested with real data and compared with a raster-based CA model to simulate land-use changes in an agroforested area in southern Quebec, Canada. The model was validated using two land-use maps, produced from satellite Landsat Thematic Mapper imagery, which were acquired in 1999 and 2002. The results obtained show that VecGCA can represent well the dynamics in the study area through an adequate evolution of the geometry of the geographic objects which are independent of the cell size, whereas, to generate similar outcomes in the raster-based CA model, a sensitivity analysis must be conducted to determine which cell size is needed. The geometric transformation procedure introduced in the VecGCA model executes the change of shape of a geographic object by changing its state in a portion of its surface, allowing a more realistic representation of the evolution of the landscape.

• Synthesis
  1. Cellular Automata → dynamic systems defined by a large tesselation of finite state cells whose states are updated at discrete time steps according to deterministic or probabilistic rules, such determines how the state of a cell can change on the basis of the state of its neighbors.
     • Constitute a powerfull tool to model complex natural and human systems, because is computationally simple and have explicit representation of space and time.
  2. Problem → assumption of regularity, uniformity and homogeneity in the classic definition of CA makes their application difficult for simulating real-world phenomena.
  3. Previous proposed CA extentions, trying to overcome these limitation:
     • Couclelis (1985) presented a model allowing the separation of the neighborhood set and the transition rules for each cell.
     • Geo-Algebra → mathematical framework to integrate CA and GIS, expressing the modelling paradigms of CA in the form of map equations. In addition, generalizes the structure of standard CA to accept arbitrary, spatially variant neighborhoods and transition rules.
     • White and Engelen (2000) and O'Sullivan (2001b) proposed an extention of the neighborhoods difinition to the set of all cells that influence the state of a particular cell, adjacent or not, referred to as a raduis of influence and a graph of influence.
  4. Most GIS integrated CA models have been used a discrete space representation and a regular tesselation of cells of the same size, and shape similar to the GIS raster model, but these raster-based CA models are sensitive to spatial scale. Several studies have demostrated that the choice of a particular cell size and neighborhood configuration impacts on results of the models developed using these CA models.
  5. Some models use solutions as vector-based or object-based models where the space is defined as a collection of irregular polygons that correspond to real entities in the study area, to mitigate scale sensitivity.
  6. Some implementations of irregular space in CA models:
     • Using Voronoi polygons → not necessarily correspond to real-world entities, because a Voronoi polygon represents a region grouping together the set of points closest to a spatial object, but it does not represent the spatial object itself.
     • Using GIS vector format to define space (Vector cellular automata model) → a geographic object is the conceptual representation of a real entity such as a city, a farm, and others. Each geographic object has a spatial representation under the cartesian coordinate system, and neighborhood relationships are defined using Voronoi diagrams. Two disadvantages:
       1. The lack of an explicit definition of the neighborhood relationships, because the Voronoi diagram is automatically generated from the vector map that represents the space.
       2. The model does not allow a change of shape or size of the objects, but only change of state.
     • Defining the space as a collection of irregular cadastral land parcels, with the neighbors composed of all adjacent parcels, parcels accessible from a road, and parcels within a buffer zone. The appearance of a new parcel is based on a set of predefined parcels that changes from one state to another. This model does not allow changes of shape or size of the land parcels.
  7. Changes of size and shape occurs continuously in the real-world, and should to be taken into account, so this inability have to be considered.
  8. Objectives:
     • To present a new vector-based CA model that overcomes the problem of cell-size sensitive by allowing an irregular space tesselation where the neighborhood definition and the transition rules are connected to the real properties of each geographic object within the study area, and that allows geometric transformations of the geometric objects as a result of the transition functions.
     • To test the vector-based geographic cellular automata (VecGCA) model with real data and compare the results with those obtained using a classical raster-based CA model under the same conditions.
       
  9. The proposed VecGCA model:
     • Conceptual Model → Three components that correspond to modification to the classical CA model:
       1. Space Definition → Space is composed of a collection of georeferenced geographic objects of irregular shape that represents the real-world entities. Each poligon is in a specific state and can define its propper transition function and neighborhood. Each geographic object is connected to others through adjacent sides and together they compose the whole space of a study area.
       2. Neighborhood Definition → The neighborhood defines which objects determine the change of shape of a central geographic object. Usually, in geographic applications, a geographic entity is influenced by adjacent or nonadjacent entities separated by a distance d. In VecGCA model, the neighborhood is defined as an external buffer (region of influence) around each geographic object, and all the objects (adjacent or not) that are partially or totally within the region of influence defined by d are the neighbors of the central object. The size of the buffer that delineates the region of influence is expressed in lenght units and is selected by the user.
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