integrating cellular automata and GIS through Geo

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International Journal of Geographical Information Science

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Map dynamics: integrating cellular automata and GIS through GeoAlgebra Masanao Takeyama; Helen Couclelis

Online publication date: 12 January 2010

To cite this Article Takeyama, Masanao and Couclelis, Helen(1997) 'Map dynamics: integrating cellular automata and GIS

through Geo-Algebra', International Journal of Geographical Information Science, 11: 1, 73 — 91 To link to this Article: DOI: 10.1080/136588197242509 URL: http://dx.doi.org/10.1080/136588197242509

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int. j. geographical information science, 1997 , vol. 11 , no. 1 , 73± 91

Research Article M ap dynamics: integrating cellular automata and GIS through Geo-A lgebra MASANAO TAKEYAMA

Downloaded By: [University of California, Santa Barbara] At: 18:16 29 November 2010

Faculty of Environmental Information, Keio University, 5322 Endo, Fujisawa 252, Japan email: [email protected]

and HELEN COUCLELIS Department of Geography, University of California, Santa Barbara CA 93106, U.S.A. email: [email protected] ( Received 28 September 1994; accepted 17 April 1995 ) Abstract. In this paper the modelling formalism of cellular automata (CA) is generalized and extended within Geo-Algebra, a mathematical generalization of map algebra capable of expressing a variety of dynamic spatial models and spatial data manipulations within a common framework. Map dynamics, that is, the integration of the spatial dynamics re¯ ected in CA and the spatial data handling capabilities of map algebra, constitutes a critical element within a wider project which sets out to formulate a general framework for simultaneously supporting spatial database manipulations and static and dynamic modelling within GIS. Map dynamics can also allow the modelling of additional dynamic behaviours and phenomena such as adaptation, design, learning and gaming not currently expressible as GIS models.

1.

Introduction

For several years now, cellular automata (CA) have attracted the attention of researchers in geography, ecology and other environmental sciences because of their ability to model and visualize complex spatially distributed processes (Couclelis 1985, 1988, 1989, Phipps 1989, White and Engelen 1992, Hogeweg 1988, Green et al. 1990, Smith 1991 ). Despite their theoretical interest, cellular automata have not yet been widely used in applications, primarily because of the di culties of adapting a rather rigid formalism to the demands of modelling real-world phenomena. The assumptions of regularity, homogeneity, universality, closure, etc. of standard CA are indeed di cult to reconcile with real life. Also, though less theoretically challenging, interfacing CA with real databases has proved messy in practice. This paper describes how both these impediments can be removed through the integration of cellular automata with geographical information systems (GIS) by means of GeoAlgebra, a novel mathematical framework. The approach proposed will not only make cellular automata more widely applicable, but will also signi® cantly enhance the general modelling capabilities of GIS technology itself. The work described in this paper is an aspect of a wider e ort to integrate mathematical modelling and GIS. An e cient integration of this sort would bene® t 1365 ± 8816/97 $12´00

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1997 Taylo r & Francis Ltd.


74

M. Takeyam a and H. Couclelis

signi® cantly both modellers and GIS users. Modellers would be able to take advantage of the databases and data manipulation capabilities of GIS for building models, and to make better use of its graphical representation functions for visualizing their results. At the same time, the full integration of modelling concepts and functions into GIS would considerably enhance its current analytical capabilities, helping support a wider range of applications from scienti® c analysis to practical planning and decision making. Thus far, the lack of a consistent framework capable of tying together spatial data modelling, spatial data manipulation, and spatial process modelling, has forced modellers using GIS to switch back and forth between di erent, often incompatible structures. This process is time-consuming and prone to inconsistencies and errors. This is why integrating spatial modelling and GIS has become a major objective for researchers in many areas where GIS is used (Goodchild et al. 1993, Fischer and Nijkamp 1993 ). Our approach, called Geo-Algebra, is an extension and generalization of map algebra, as developed by Tomlin ( 1990). Tomlin’s map algebra is a sophisticated framework for integrating spatial data modelling and analytic GIS functions. In particular, it organizes the operations for transforming and synthesizing rasterencoded spatial data represented in the form of map layers. The digital map overlap manipulations are classi® ed into four di erent classes of operations on and between map layers: local, neighbourhood (incremental and focal ), zonal, and global. Any complex manipulation of map layers is then represented as an algebraic composition of such operations. Thus, the structure of map algebra is similar to that of traditional algebra in the sense that map layers are treated as if they were numbers in an equation ( Berry 1987 ). In spite of its quasi-algebraic structure, however, map algebra is not expressed mathematically. Being based on map analysis operations, it is also not appropriate for modelling spatial processes. From a mathematical point of view, cellular automata are usually de® ned as dynamical systems that are the discrete analogs of partial di erential equations ( Wolfram 1986 ). At the same time, they have many structural similarities with map algebra. The states of elementary automata arranged over a regular array form a cellular automaton con® guration, a high-resolution spatial data structure similar to a raster-encoded map layer. Dynamic transitions of CA con® gurations are determined by a local transition rule analogous to the neighbourhood operator de® ned in map algebra. There are also important di erences: while in map algebra, as applied to map analysis, the focus is the sequence of static transformations on map layers, the central interest of CA modellers is the dynamic sequence of CA con® gurations. Because CA models can exhibit highly complex space-time dynamics, they are especially suggestive when that behaviour is visualized on a computer screen. Needless to say, the visualization of spatial information is also an essential strength of GIS, though still primarily for static patterns. Realizing process modelling within GIS in a general and consistent manner is di cult largely because of the very di erent modes of representing information in GIS on the one hand, and expressing a mathematical model on the other. While the basic form of high-level information representation in GIS is usually the map layer (or object), mathematical models take the form of equations of numerical, categorical, or symbolic variables. Certainly the map layer is a natural metaphor for representing static spatial patterns and structures, but it cannot express rules, regularities and processes which are more appropriately represented by equations. The structural similarities between CA and map algebra leads us to explore the possibility of


Integrating cellular automata and GIS

75

combining the best of both worlds by integrating these two approaches within a common mathematical framework. This paper describes how this problem is solved within the formalism of Geo-Algebra, developed to support both dynamic and static spatial modelling in conjunction with GIS-based spatial data manipulation capabilities. The other related work ( Takeyama 1996) shows how static geographic models can be represented as mathematical equations involving maps. The present paper focuses on extending the argument to dynamic models and to the notion of map dynamics. The paper is structured as follows: Section 2 brie¯ y summarizes the basic ideas of Tomlin’s map algebra. Section 3 introduces the basic concepts of Geo-Algebra such as maps, multi-variate maps, relational maps, and operations on and between these maps. In particular, a common mathematical formulation is provided for representing arbitrary non-local operations such as neighbourhood, zonal, and global operations. In Section 4, the notion of map dynamics is developed and used to generalize cellular automata by relaxing several of the constraining assumptions of uniformity and regularity that makes these models hard to apply in practice. Finally map dynamics are further generalized ® rst into multi-layer map dynamics, supporting interactions among multiple attributes, and then also into dynamic models open to external inputs. 2.

M ap algebra

Since Geo-algebra is a formalization and generalization of map algebra, we begin with a brief description of the latter. Map algebra was ® rst introduced by Tomlin ( 1983 ) as à starting point for coordination of the development of digital overlay mapping techniques’. The structure of map algebra consists of a set of map layers, primitive operations on and between map layers, and sequences of these operations ( Berry 1987, 1993). A map layer is an element of a cartographic model, de® ned as a collection of map layers each of which represents the spatial distribution of a particular attribute over a common study area. The elementary unit of a map layer is the location, usually represented as a cell in a grid space. Each location on a map layer is associated with a numerical value representing the value of the corresponding attribute at that location. A collection of locations which have the same value on a map layer is referred to as a zone. The map algebra operations are used to transform map layers, location by location, into new map layers in order to extract information useful to the user. The operations are classi® ed into local, zonal, incremental, and focal operations ( Tomlin 1990): Local operations are used to compute a new value for every location as a function of one or more existing values associated with that location. Examples are: LocalSum and LocalMaximum. LocalSum ( LocalMaximum) computes, as a new value for every location, the sum (the maximum) of values at the location on di erent map layers. Zonal operations compute a new value for each location as a function of the values from a speci® ed layer that are associated not just with that location itself but with all locations that occur within its zone on another speci® ed layer. Examples are ZonalMean and ZonalPercentage. ZonalMean computes, as a new value for every location, the mean value of all the values in the zone that includes the location.


76


ZonalPercentage assigns a new value to each location within a second layer zone indicating what percent of that zone shares the location’s ® rst layer value. Incremental operations characterize each location as an increment of one-, twoor three-dimensional cartographic form. The size and shape of these increments are inferred from the value(s) of each location relative to those of its adjacent neighbours on one or more speci® ed layers. Examples are IncrementalVolume and IncrementalAspect. The IncrementalVolume operation computes a new value for each location indicating the sur® cial volume associated with whatever portion of an areal condition is represented by that location. IncrementalAspect measures sur® cial aspect, the compass direction toward which an inclined surface descends most rapidly. Focal operations are those that compute each location’s new value as a function of the existing values, distances, and/or directions of neighbouring ( but not necessarily adjacent) locations on a speci® ed map layer. Examples are FocalSum and FocalMean. FocalSum ( FocalMean) computes, as a new value for every location, the sum (the mean) of the values within its neighbourhood. In addition to the three standard types of map algebra operations, recent implementations of map algebra in GIS often include global ( per-layer) operations (Menon et al. 1992 ). For example, global operations are used for the generation of Euclidean distance and weighted cost distance maps, shortest path maps, nearest neighbour allocation maps, for the grouping of zones into connected regions, for geometric transformations, raster-vector interconversion, and interpolation. In these extensions as in the original map algebra, statements and operations are normally expressed in an English-like syntax stimulating an intuitive programming language. The following is an example of that syntax expressing a neighbourhood operation which computes the average of the elevations of all locations within a speci® ed radius ( Tomlin 1991, p. 367). AltitudeNearby = Focal Mean of Altitude within 1000

Because the output of each operation, i.e., a new map can be used as input to another operation, multiple operations can be organized into a sequence called a procedure. A procedure for analysing maps thus functions just like an equation in traditional algebra. This quasi-algebraic structure of map algebra provides a formal foundation for map analysis. Figure 1 shows the procedure of map analysis where each box represents a raster

Figure 1.

Map analysis (Tomlin 1990).



77

map. The equations in the ® gure are de® ned on the basis of operations on these maps. Operations on maps in turn are induced by applying conventional algebraic operations to the values at each location. While map algebra is a well organized and simple framework for the analysis of spatial data in GIS, it is not appropriate for expressing spatial models. First, operands, operations, and procedures are not described in a mathematically precise manner. The English-like statements used in map algebra do facilitate programming, and it is actually possible to represent some spatial models with existing primitive operations in that language (Gao et al. 1996). However, it is di cult to ® nd out how one model mathematically relates to another unless both models happen to be represented with the same operands and operations. This makes comparing and combining models rather di cult. Second, map algebra does not deal explicitly with spatial relations and interactions among locations, which are central concepts in spatial modelling. Even though spatial interaction in the sense of in¯ uence can be included in the computation of a neighbourhood operation, map algebra does not have the structure to represent interactions explicitly or to deal with operations involving spatial interactions. Third, map algebra is designed for static map analysis. It can model spatial patterns and structures but not dynamic spatial processes. As discussed in the following, Geo-Algebra builds on the strengths of map algebra just as it transcends its limitations. 3.

Geo-Algebra

Geo-Algebra aims to provide a rigorous common formalism for map analysis and spatial modelling. This chapter summarizes some basic concepts of Geo-Algebra. Geo-Algebra is designed through a mathematical generalization of map algebra. It formalizes the ideas and structures de® ned in map algebra in the language of modern (abstract) algebra such that it can express arbitrary operations and functions. Modern algebra provides a general framework for the study of arbitrary operations de® ned on arbitrary operands, whereas traditional algebra only deals with operands that are quantities. Thus the relation of Geo-Algebra to modern algebra is not an intuitive analogy as in the case of map algebra relative to traditional algebra, but GeoAlgebra is characterized mathematically as a subalgebra of modern algebra. As with every other formal algebra, the structure of Geo-Algebra consists of several sets of operands and the operations de® ned on these sets. The operand sets here are: locations, values, maps, relational maps, and meta-relational maps. The basic operations de® ned in Geo-Algebra are unary and binary local operations (i.e., the operations on and between maps), and non-local operations which are compositions of meta-relational maps, operations between maps and meta-relational maps, and local in¯ uence functions. Just as map algebra was in¯ uenced by ideas and techniques of image processing, the formalism of Geo-Algebra generally follows that of image algebra developed by Ritter ( Ritter et al. 1990 ). Image algebra is a high algebraic language for describing image processing algorithms. With the concept of generalized templates, a generalization of the concepts of templates, masks, windows, and structuring elements of mathematical morphology (Serra 1982), image algebra formalizes widely-used imageto-image transformations such as addition, multiplication, and convolution into a common mathematical framework. Although Geo-Algebra owes a large part of its formalization to image algebra with regard to the expression of simple map-to-map transformation, it goes beyond the idea of image algebra in that it can express a

78



variety of static and dynamic spatial process models. In particular, Geo-Algebra introduces new concepts such as the relational and meta-relational maps to represent arbitrary situational spatial information (Couclelis 1991 ) explicitly in forms of maps. Moreover it de® nes arbitrary operations on and between these di erent types of maps, so that we can process simultaneously site-based and situation-based spatial information. As a result, such a framework for complex map processing integrates representations of spatial data models and spatial process models. A map consists of pairs of locations and associated values, such pairs being the fundamental units of geographical information (Goodchild 1992). They are called geo-units in Geo-Algebra. Mathematically, a map is a function from a set of locations to a set of values f : L V de® ned by: m = {( l, m ( l )) | l × L }

where l × L is a location and m ( l ) × V is a value. The set of all V valued maps on L L is denoted by V following the usual mathematical convention of denoting the set A of all functions from A to a set B by B . Although it is possible to de® ne any arbitrary structure on a set of locations ( Takeyama 1996), in this paper we assume a regular grid structure for simplicity. A regular grid structure can be generalized into a continuous structure where the location set is mathematically denoted as L = R Ö R where R is the set of reals. Similarly a set of values can be any set such as a set of integers, of binary numbers, of real numbers, or of symbols. Note that the term map used here must be distinguished from its mathematical meaning (i.e., mapping) as well as the cartographic usage (mapping from places on the Earth to points on a sheet of paper). The idea of a map as a function can be generalized into the multi-variate case in which every location takes a multiple number of values simultaneously corresponding to a multiple number of attributes. Mathematically a multi-variate map is a V valued map on L such that the set of values is the set product of the same or di erent sets of values, i.e., n

V = a Vi . i

=1

This de® nition of a multi-variate map corresponds to the idea of map layers in map algebra. Then the i th map layer is derived from a multi-variate map with i th coordinate projection, i.e., m i = p i ( m ) = {( l, m i ( l )) | m i ( l ) = p i ( m ( l ))}

where p i is the projection function, n

p i : a Vj j

=1

Vi

de® ned by: p i (v 1 , . . . , v i , . . . , v n ) = v i .

Thus m = ( m 1 , m 2 , . . ., m n ). The simplest operation on maps is a unary local operation induced by simultaneous application of a unary variable function at each location of a map. Let f be a L L unary function. Then a unary local operation on maps is a function f : V V


79

de® ned by: f ( m ) = {( l, n ( l )) | n ( l ) = f ( m ( l )), l × L }.

The most often used unary local operation is a m ap characteristic fu nction L L {0, 1} induced by a characteristic function for a value x : V {0, 1} . This function computes a binary map according to whether the value at each location is included in a predetermined set S. For example, let S = {3, 4, 5}. Then a map m as in ® gure 2 is transformed into a new map x S ( m ) by the map characteristic function x S . Note that a map characteristic function corresponds to the reclassi® cation of a map in map algebra. In a similar manner, any binary local operation between maps, i.e., L L L f : V Ö V V is induced by a binary operation on a value set V, i.e., f : V Ö V V. For example, the simple binary operations of addition, multiplication, and maximum L on the set of real values maps R on L are de® ned as follows: L Let m , n × R


x: V

m +n m 1n m n

8

{( l, h ( l )) | h ( l ) = m ( l ) + n ( l ), l × L } {( l, h ( l )) | h ( l ) = m ( l ) n ( l ), l × L } {( l, h ( l )) | h ( l ) = m ( l ) n ( l ), l × L }.

8

Compared to local operations, the mathematical de® nition of arbitrary nonlocal operations becomes more complex because it must consider the relation among locations. In Geo-Algebra, any nonlocal operation is computed in the following three steps. First, for each location of a map, all locations in¯ uencing that location are de® ned. Secondly, the values at these in¯ uencing locations are computed. Finally a new value for each location is computed as a function of the values at the in¯ uencing locations. In other words, a computation of a nonlocal operation is decomposed into the computations of (i ) where are the interacting locations, (ii ) how much is there to interact, and (iii ) what is the result of the interaction. A key concept of Geo-algebra, the meta-relational map, plays a central role in the initial step of de® ning the in¯ uencing locations. A m eta-relational m ap represented here by R is the extension of the idea of map as a function where, instead of a value or a vector, an entire map called a relational m ap is assigned to each location. A relational map associated with the location l, denoted as R l , represents all the locations in¯ uencing that particular location in the form of a map in which all the in¯ uencing locations have the value 1 and all others have the value 0 (® gure 3). Therefore a relational map represents the situational information for each location rather than the site information speci® c to the location itself (Couclelis 1991). By de® nition, then, a relational map can express any arbitrary neighbourhood or zone associated with a location. Operations on and between maps, relational maps and meta-relational maps can be consistently de® ned ( Takeyama 1996).

Figure 2.

Application of a map characteristic function.

80



Figure 3.

A meta-relational map.

In the second step, the values at each of the in¯ uencing locations are computed by a multiplication operation between the map m and the meta-relational map R . This multiplication between two di erent types of maps is performed such that the map m is multiplied in parallel, location by location, by each relational map R l associated with each location of a meta-relational map R ( ® gure 4). The result is a new meta-relational map m # 1 R representing the values of the in¯ uencing locations for each location.

Figure 4.

Multiplication between a map and a meta-relational map.

81



new

Finally, a value at each location of a new map m is computed as a function of the values at the in¯ uencing locations. This function which transforms each map associated with each location of a relational map R l into a new value at the location v l is called a local in¯ uence function and is represented by i l . One example of a local in¯ uence function is the sum of a map which computes the sum of all the values associated with the map ( ® gure 5). By applying such a local in¯ uence function to each map at each location of a relational map simultaneously, the relational map is ® nally transformed into a new map in which a new value is assigned to each location (® gure 6). The parallel application of local in¯ uence functions to all the locations of a relational map can be spatially homogeneous or heterogeneous and is collectively denoted as I and is called a globa l in¯ uence f unction. A global in¯ uence function itself can be de® ned by a map being called a f unction assignm ent m ap and denoted as i . The computation of a nonlocal operation is summarized in the following map equation: m

new

= I ( m# 1 R)

( 1)

A meta-relational map is mathematically equivalent to the generalized template of image algebra. In image algebra, some basic template operations such as generalized convolution, additive maximum, and multiplicative maximum are de® ned as operations between images and templates. For example, a generalized convolution is given by:

G

aCt

K

( y, c ( y)) c ( y) = x

×

a (x) t y (x), y × Y

X

H

where a and t are an image and a template, respectively ( Ritter 1990 ). However, arbitrary nonlocal operations in Geo-Algebra, while including the equivalents of these basic image algebra operations, extend them to functions of an intermediate meta-relational map m 1#R representing explicitly the values of in¯ uencing locations. new Thus in Geo-Algebra a nonlocal operation of the form m = I( m # 1 R ) generalizes the template operations of image algebra.

Figure 5.

Figure 6.

An example of a local in¯ uence function.

A parallel applications of local in¯ uence functions.

82 4.

M. Takeyam a and H. Couclelis Cellular automata generalized within Geo-Algebra

In this section we examine how the modelling paradigm of CA is generalized and extended within the framework of Geo-Algebra. Following the review of the conventional formalism of CA, we ® rst show that any arbitrary CA in the conventional formalism is homomorphic to a subalgebra of Geo-Algebra. Then the new concepts of a multi-variate map dynamics and an open map dynamics are introduced as further extensions of CA. A cellular automaton is usually de® ned as a tuple, (X, S, N, f ).


where the de® nitions of the elements are in order: d

(i ) X is a subset of the d dimensional coordinate space Z called a cellular space in which each element called a cell is of the form x = (x 1 , x 2 , . . . , x d ), where each coordinate x i (i = 1, 2, . . . , d ) is an integer. (ii ) S is a nonempty ® nite set of automaton states. An automaton state is denoted by s . (iii ) N is a neighb ourhoo d template N = {v1 , v 2 , . . . , v k } where each coordinate v j ( j = 1, 2, . . . , k) is a vector of d integers. k (iv) f is a state transition fu nction f : S S de® ned by sx + = f (s x t

t

where s x

1

t

t

+v 1 , . . . , s x +v k )

+v j is the automaton state at the location x + v j at time t .

Well-known neighbourhood templates of two-dimensional CA are the von Neumann neighbourhood and the Moore neighbourhood de® ned by: N = {( 0, 0 ), ( 1, 0 ), ( 0, 1 ), (Õ N = {( 0, 0 ), ( 1, 0 ), ( 1, 1 ), ( 0, 1 ), (Õ

1, 1 ), (Õ

1, 0 ), ( 0, Õ 1, 0 ), (Õ

and

1 )}, 1, Õ

1 ), ( 0, Õ

1 ), ( 1, Õ

1 )}.

respectively. Those neighbourhood templates are represented pictorially in ® gure 7. More generally these neighbourhood templates can be expanded into n th order neighbourhood templates. The n th order von Neumann neighbourhood template is given by: NNeumann = {v i = (v i1 , v i2 ) | | v i1 | + | v i2 |

2

n, (v i1 , v i2 ) ×Z }

and the n th order Moore neighbourhood template is given by: NMoore = {v i = (v i1 , v i2 ) | | v i1 |

n and | v i2 |

2

n, (v i1 , v i2 ) ×Z }

where v i1 and v i2 are the ® rst and the second coordinates of v i which is the i th element of the neighbourhood template. From these ideas, a few additional important concepts follow. A con® guration of a cellular automaton is a function c: X S de® ned by: c = {(x, c(x )) | x ×X}

Figure 7.

The von Neumann ( left) and the Moore (right) neighbourhoods.

83


where c (x) ×S thus c (x) = sx . A con® guration of a cellular automaton de® nes the assignment of an automaton state to every cell of the cellular space and represents a globa l state of the cellular automaton. x x A globa l transition function is a function F: S S given by: F (c) (x) = f (c(x + v 1 ), . . . , c (x + v k ))


x

d

where S is the set of all con® gurations for given X, L and x ×Z . This de® nition indicates that a global transition function is given by the simultaneous application of the local transition function f to all cells of the cellular space. We can now set up a correspondence between CA and some notions de® ned in Geo-Algebra to prove that an arbitrary CA is homomorphic to a subalgebra of GeoAlgebra. Consider any arbitrary cellular automaton M = (X, S, N, f ). We ® rst de® ne the location set L and the value set V such that the two functions g: X L and h: S V are injective (one-to-one). Then we can de® ne a mapping from the set of all X L con® gurations to a set of all maps g : S V given by: g (c) = {( g (x), h (c(x))) | x ×X}

This mapping ensures that any con® guration of a cellular automaton c corresponds uniquely to a map m preserving the functional relation from cells to automaton states in the functional relation from locations to values (® gure 8). With these mappings g and h, any neighbourhood template N = {v 1 , . . . , v k } is L L also uniquely mapped to the meta-relational map R ×({0, 1} ) de® ned by: R l i ( lj ) = 1

=0

when for some x ×X, l i = g (x) and l j × {g(x + v 1 ), . . . , g (x + v k )} otherwise

l i , lj× L

which preserves the neighbourhood spatial relation in the spatial relation de® ned by a relational map. Also an arbitrary transition function f is mapped to a local in¯ uence function i l i given by: il i ( m 1R l i ) =

=

h( f (c(x + v 1 ), . . . , c (x + v k )))

if for some x, l i = g(x ) otherwise

0

l i× L

Then obviously I ( m1#R ) = g (F (c)) and therefore I simulates exactly the same behaviour as that of the cellular automaton. As a result, we can conclude that for any arbitrary cellular automaton M = (X, S, N, f ), we can always construct an algebra L L L L L (A, O ), where A = {V , ( V ) , ({0, 1} ) } and O = {1#, I } which is homomorphic to M. The schematic view of the correspondence is as follows (® gure 9 ). As examples of mapping CA into Geo-Algebra, a couple of well-known CA models are represented in that formalism in the following. First, consider the voting model, one of the simplest two-dimensional cellular automaton models, as de® ned

Figure 8.

84



Figure 9.

in the usual notation, c (i, j, t + 1 ) = 1

=0

if

(p ,q )

×

c( p, q, t)

3

( 2)

n ( i,j)

otherwise

In this model, each cell (representing a voter) can assume either state 1 (say, Democrat) or state 0 (say, Republican). The voting rule updates the state of each cell c (i, j ) into 1 if the state 1 is in the majority (i.e., the neighbourhood sum is larger than three) in its ® ve (® rst order) von Neumann neighbourhood cells (including itself ) n(i, j ) given by: n(i, j ) = ((i, j ), (i + 1, j ), (i, j + 1 ), (i Õ

1, j ), (i, j Õ

1 ))

Otherwise, the new cell state becomes 0. The same model is now expressed in the following dynamic global in¯ uence function: n = x 3 ( I ( m 1#R )).

( 3)

Let us look at the procedure for computing this function with a small ( 3 by 3) cellular space as an example. First, in the Geo-Algebra notation, the initial con® guration and the neighbourhood within unbounded torus-like space are represented by a map m (® gure 10 ) and a meta-relational map R (® gure 11) respectively. The multiplication between m and R computes a new relational map m # 1R representing the values at the locations in the neighbourhood ( ® gure 12). I is a global in¯ uence function which is the parallel application of the same local in¯ uence function S for all of the locations of a meta-relational map R de® ned by: I ( R ) = {( l, S R l ) | l × L }.

Applying this global in¯ uence function to m # 1 R , we compute another intermediate map I ( m1#R ) representing the neighbourhood sum at each location of m (® gure 13). The map characteristic function x 3 transforms this map into a new map n by

Figure 10.

Initial map m .

85



Figure 11.

Relational map R representing the von Neumann neighbourhood (circular boundary).

Intermediate map m # 1 R.

Figure 12.

Intermediate map I ( m# 1 R ).

Figure 13.

assigning 1 to locations with a state value larger than or equal to 3 and 0 to locations with a state value less than 3 (® gure 14). As the second example, the popular rule of the Game of Life usually de® ned in the form, c (i, j, t + 1 ) = 1

if

(p ,q )

×n ( i,j)

or

(p ,q )

=0

×

c ( p, q, t) = 2

and

c( p, q, t) = 3

n ( i,j)

otherwise

Figure 14.

Final map n = x

H

c ( p, q, t) = 1

1 R )). µ3 ( I (m#

( 4)

86


is equivalent to the map equation,


n=x

1 R )) 1m + x =3 ( I ( m 1#R )) =2 ( I ( m #

( 5)

where I ( R ) = {( l, S R l ) | l × L } . Here the logical operations such as ànd’ and òr’ in the regular notation correspond to the Geo-Algebraic operations of `1’ and `+’ respectively. As the above examples show, the expression of a cellular automaton model in Geo-Algebra has two important characteristics. First, because the model is expressed in the form of a map equation, the algorithm for computing a map dynamics equation becomes exactly the same as the procedure for analysing maps with map algebra. As a result, it becomes possible to use the same set of spatial data both for map analysis and for process modelling. This also enables one to combine map analysis and map-equation-based model building. Thus Geo-Algebra drastically reduces the gap between spatial data manipulation and mathematical process modelling. Secondly, the expression of models themselves becomes simpler. In particular, logical statements such as ìf ’, `then’, and èlse’ and logical operations like ànd’ and òr’ often occurring in the regular formulation of cellular automata, are translated into simple functions and operations such as a characteristic function, addition, and multiplication on and between maps. This compressed expression of Geo-Algebra is more advantageous when the model has a more complex expression in the regular CA notation. For example, the following is a CA-like ® re spread model described in the Arc-Info GRID model language (Gao et al. 1993 ): if (old_® re> 15) new_® re = 0 elseif (old_® re> 10 ) new_® re = old_® re+ 1 elseif (old_® re ( 1, 0 ) > 10 or old_® re ( 1, 1) > 10 or old_® re ( 0, 1) > 10 new_® re = 10 else new_® re = 0 endif.

( 6)

That same model is described in Geo-Algebra in just one equation plus the de® nition of a meta-relational map R , which is easily constructed from the neighbourhood of the CA model, as follows: n=m Õ

16 x 16 m + x 10

m t(l) 14 m

+ 10 x 0 m 1x 1 ( I ( m1#R ))

( 7)

where I ( R ) = {( l, S R l ) | l × L }. To describe dynamic geographical processes more precisely within Geo-Algebra, it is necessary to extend the framework such that it can express the idea of time explicitly. Mathematically this requires us to add another set of operands to represent time to the operand sets of Geo-Algebra. The idea of time is considered as a set of coordinates in a one-dimensional Euclidean space. A tim e coordinate or simply tim e is denoted by t whereas the tim e coordinate set or tim e set is denoted by T . A time set T can be either a subset of integers Z or a subset of reals R. The times in a


87

time set may or may not be evenly spaced. As with location sets and value sets, the time set T can be assumed to correspond systematically to real world time by means of a mapping from a set of points or periods in real world time to a time set. Introducing a time coordinate set, a geo-unit becomes a tuple of a location, a time, and a value associated with the location and the time: ( l, t, v).

A geo-unit with time incorporated may be used to represent any information at a certain location at a certain time in the real world. Given L , T and V, all possible geo-units are given by the Cartesian product: Downloaded By: [University of California, Santa Barbara] At: 18:16 29 November 2010

LÖ

TÖ

V.

Then the idea of a map as a function from a location set to a value set is extended into the idea of a temporal map. A V valued temporal m ap on L Ö T is a function from the Cartesian product between a location set and a time set to a value set m : L Ö T V de® ned by: m = {(( l, t), m ( l, t)) | ( l, t) × L Ö

V}

where m ( l, t) × V. Although we do not elaborate on this here, it is obvious that the notions of relational and meta-relational maps as well as the operations de® ned on and between them are analogously extended into those incorporating time by replacing L by L Ö T . A temporal map can always be viewed as a series of map slices. Mathematically a m ap slice of a temporal map m at a time t i× T is given by the restriction of m : L Ö T V to L Ö {t i } , i.e., m |L

Ö

{t i}

: LÖ

{t i }

V

such that m | L {t i} ( l, t) = m ( l, t) for ( l, t) × L Ö {t i }. For notational convenience, we Ö denote a time slice m | L {t i} : L Ö {t i } V by m t i : L V. Ö With the idea of a map slice we can introduce the idea of a map dynamics which is a mathematical analog of a CA within Geo-Algebra. A m ap dyna m ics is an algebra L

L L

L L

({T , V , ( V ) , ({0, 1} ) }, {1#, i }) where i is a dyna m ic globa l in¯ uence fu nction de® ned by: m t l = i ( m t 1#R )

+

( 8)

Then the map slice at time t 0 + k is computed recursively from the initial map slice at time t 0 by: m t0

+k = i (. . . ( i ( i ( m t 01#R )1#R ) . . . )1#R ) ~ k times~

~ k times ~

Such a recursive operation generates a series of map slices: m t0 m t0

+1 m t 0+2 . . . m t 0+k

which is called a dyna m ic behaviour of the map dynamics. 5.

M ulti-variate and open map dynamics L

L L

L L

The map dynamics {( T , V , ( V ) , ({0, 1} ) }, 1#, i }) not only represents any arbitrary CA model but also helps to generalize the formalism into a more ¯ exible


88


framework for modelling dynamic spatial processes. First, whereas a CA model normally assumes a regular rectangular cellular structure over a space, locations de® ned in a map dynamics are neither limited to a rectangular structure nor spaced regularly. Second, the structure of space and variables of the conventional formalism are intrinsically discrete but both a location set and a value set in a map dynamics can be continuous. Third, we can de® ne any arbitrary size and shape of translation variant or invariant neighbourhood by a meta-relational map. Fourth, the assignment of transition functions over cellular space can be spatially homogeneous or heterogeneous using a global in¯ uence function with a function assignment map. These extensions of the conventional CA formalism allow us to apply the framework for modelling a dynamic spatial process in which locations are irregularly spaced and the pattern of interaction and the rule to generate the dynamic process are di erent from region to region. In addition to the generalization of neighbourhoods and transition rules, GeoAlgebra can generalize the conventional formalism of CA even further. Since a cellular space in the conventional formalism is usually considered to be a single plane or ® eld corresponding to a single variable, it cannot deal with interactions among a multiple number of attribute variables. For example, the dynamic behaviour of ® re spread taking into account vegetation and wind direction, cannot be modelled easily as a CA except by fairly complex transition rules with many if-then type conditions. Because operations on multi-variate maps are de® ned consistently within Geo-Algebra, we can extend the concept of a map dynamics into a multi-layer (or multi-variate) map dynamics which deals in a very straightforward manner with dynamic interactions among multiple map layers corresponding to multiple variables and attributes. Recall that a multi-variate map is de® ned by: n

a Vi

m: L

=1

i

and this can be viewed as layers of single variate maps, i.e., m = ( m 1 , m 2 , .. . , m n ) where m i = p i ( m ). The idea of a multi-variate map is easily extended into a temporal multi-variate map given by: n

m: L Ö

a Vi

T i

which

viewed as a series of multi-variate map slices m = + +mÕ 1 ) where m t j = m | LÖ {t j} . Because each map slice is seen as layers of maps, a temporal multi-variate map can be represented by the matrix (table 1 ), where m it 0 j is the i th layer of the multi-variate map slice at time t 0 + j. ( m t0 , m t 0

can

=1

1

be

, . . . , mt0

+

Table 1 Temporal multi-variate map. mt m1 m2 e mn

0

mt

0+

1

m 1t m 2t 0 e0

m 1t 1 + m 2t 0 1 + e0

mn t

mn t

0

0+

1

,

mt

, , , ,

m 1t k + m 2t 0 k + e0

0+

mn t

k

0+

k

89



Figure 15.

Multi-variate relational map.

Suppose a value at every location on m it 0 j is computed as a function of the values + on map layers of the map slice of previous time m 1 t 0 j 1 , m 2 t 0 j 1 , . . . , m nt 0 j 1 , we +Õ +Õ +Õ can represent any spatial relation among locations on di erent layers mathematically by means of a set of layers of meta-relational maps (® gure 15) or equivalently by a multi-variate meta-relational map. A m ulti-variate m eta-relational m ap is a function: n

a

R: L

i

de® ned by:

CA B D L

n

aV i

i

R = ( R 1 , R2 , . . . , Rn )

= (( R 11 , . . . , R 1 n ), . . . , ( R n 1 , . . . , R nn )), where R i is a multi-variate meta-relational map representing spatial relations between all locations on i th map layer and all locations on all map layers, and R ij is a single variable meta-relational map representing spatial relations between all locations on the i th map layer and all locations on the j th map layer. A global in¯ uence function is similarly extended into a m ulti-variate globa l in¯ uence function, which transforms a multi-variate meta-relational map into a multi-variate map: I:

given by:

G C A B DH A B n

n

a

aV

i

i

L

L

n

i

L

aV i

I ( R ) = ( I 1 ( R 1 ), . . . , I n ( R n ))

= ( I 1 ( R 11 , . . . , R 1 n ), . . . , I n ( R n 1 , . . . , R nn )). Then the equation m t 1 = I ( m t1#R ) de® nes a m ulti-variate m ap dyna m ics where + every i th map layer of the map slice at time t + 1 is given by the n ary map transformation: m it

+1 = I i ( m t1#R i ) = I 1 i ( m t 1 ( t)1#R i1 , . . . , m tn ( t) # 1 R in ).

While the regular cellular automaton is a closed system, external input data into c map dynamics is consistently represented as a map m . If the input is a simple

90



Figure 16.

An interactive map dynamics.

addition to the initial map m , then map dynamics open to external input is denoted as: m it

c

+1 = I i (( m t + m t )1#R i )

( 9)

This open m ap dyna m ics provides a mathematically consistent description of interactive CA (® gure 16 ). The structure of such an interactive map dynamics can be viewed as a generalization of the structure of a board game like chess or even a geographical game, such as, the Community Land Use Game (CLUG) ( Feldt 1972) and SimCity ( Wilson 1990 ). By representing an interactive spatial modelling as open map dynamics within Geo-Algebra, new behaviours such as design, learning and gaming can be also integrated into the same framework along with modelling and analysing spatial information. 5.

Conclusion

We have shown how Geo-Algebra can express and generalize, in the form of map dynamics, the modelling paradigm of CA, which thus becomes a paradigm for a very wide range of spatio-temporal modelling approaches. Because the models of map dynamics are formulated as map equations within Geo-Algebra, dynamic spatial modelling is implemented in exactly the same way as the analysis of map data within map algebra. Such a mathematical integration of CA and map algebra removes the signi® cant gap between map-based spatial manipulation and equation-based spatial modelling. In addition, the structure of standard CA was generalized to accept arbitrary, spatially variant neighbourhoods and transition functions, making the CA formalism suitable for modelling a variety of real-world spatial phenomena. As a result of such integration and generalization, models with various structures can be compared and combined within a common mathematical environment. Furthermore, the Geo-Algebra expression of CA models is fairly simple compared with the standard notation. Finally, additional features such as multi-layer interactions and the inclusion of external input map layers can be consistently incorporated into map dynamics. An interactive map dynamics opens up the possibility of integrating into GIS new kinds of phenomena and behaviours such as design, learning and gaming. Referen ces B erry, J . K ., 1987, A mathematical structure for analyzing maps. Journal of Environm ental Management, 11, 317± 325. B erry, J . K ., 1993 , Cartographic modeling: The analytical capabilities of GIS. In Environm ental Modeling with GIS, edited by M. F. Goodchild, B. O. Parks, and L. T. Steyaert (New

York: Oxford University Press), pp. 58± 74.


91

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