Towards a General Theory of Representation in GIS

1. Towards a General Theory of Representation in GIS Michael F. Goodchild University of California Santa Barbara

2. Outline • A quick history of data modeling in GIS • A general theory

3. What is a data model? • A template for data • A template that matches the application • providing "slots" for everything relevant to the application • A round hole into which the square peg of an application must be forced • A data model for GIS • a template for describing some aspect of the planet's surface

4. Some basic data models • The Powerpoint template • fitting a presentation into a template that allows for: • titles • slides • bullets • pictures • animation • not a good template for GIS

5. The table data model • Fitting data into rows and columns • features and their attributes

6. The georelational data model • Derived from a mainstream model • Tables linked by common keys • storing topology • Early ARC/INFO • the Coverage model • coordinates stored separately • variable number of coordinates per arc

11. 1990 2004 1995

12. Disadvantages • No ways to handle: • change through time • overlapping areas • points • networks • hierarchies of objects • But great for: • soil maps, the cadaster • Census data

16. * 0..1 MINARD NAPOLEON MAP 0..2 * INTERACTION * 0..1 KARST FLOW ROUTES 0..2 * ORIGINAL USE CASE MODELS 0..1

18. * 0..1 0..2 * 0..1 Generic Flow Model

22. Advantages • Closer to how we think about phenomena • Generalization/specialization • Hierarchies of complex objects • Events in space and time • Beyond the map metaphor

23. Disadvantages • Dealing with continuous phenomena • surfaces • continuous linear features • must be discretized

24. An increasing confusion • Rasters and vectors (1960s) • Topological data structures (1977) • Relational databases (1980) • Object-oriented databases (1990) • Fields and objects (1990) • Time and the third spatial dimension • Object fields, metamaps, ... • Why does it have to be so complex? • science loves simplicity

25. Desiderata of a general theory (Galton, 2003) • To provide suitable forms of representation and manipulation to do justice to the rich network of interconnections between field-based and object-based views of the world • To extend the field-based and object-based views, and the forms of representation developed to handle them, into the temporal domain • To provide a means to develop different views of spatio-temporal extents and the phenomena that inhabit them, especially with reference to those phenomena which seem to represent dual aspects as both object-like and field-like

26. (discrete) Objects and (continuous) Fields • The geographic world is like a tabletop littered with discrete, countable objects • pieces of a jigsaw puzzle • may retain form when moved • The geographic world is like a series of continuous layers, each describing the variation of one theme • value is a function of location • z = f(x) • z can be qualitative or quantitative

31. Major questions • Can the complexity be reduced? • can some of these options be integrated under a single umbrella? • Can objects and fields be integrated? • and are they the only options?

32. Relevant literature • Berry’s geographical matrix • Sinton’s three-dimensional schema • control one dimension, hold a second constant, observe variation in the third • Kjenstad (IJGIS, inpress) • UML framework • schemata for objects and fields • merged into a single schema • Parameterized Geographic Object Model

33. Geographic information • Associates places with properties • Reducible to an atomic form • <x,Z,z(x)> • x is a location in space–time • Z is a property • z(x) is the value of that property at x • the geo-atom • point properties are sometimes measured and managed • other structures are aggregations of point sets

34. Geo-fields • Defined over a domain D • Aggregates all geo-atoms within the domain for some property Z • Scalar • a single property • qualitative or quantitative • nominal, ordinal, interval, ratio • Vector • directional • one property per dimension of space–time • Aliases • surface, coverage

35. Representation of geo-fields • Domain and variation within domain • Six familiar methods • discretization is a required and inherent property of the representation of any geo-field

36. raster points raster • polygons polygons polylines

37. Measurement of geo-field properties • Point measurements • weather data • Convolutions • population density • width of kernel

38. The result of applying a 150km-wide kernel to points distributed over California A typical kernel function

39. Geo-objects • Aggregations of geo-atoms • based on certain rules applied to values • all geo-atoms with elevation > 100 • all geo-atoms with elevation = 100 • all geo-atoms with county = “jefferson” • aliases • entity, feature • Tobler’s First Law • ensures geo-objects of non-trivial size • multipart polygons • multipart polylines

40. Criteria for geo-objects • Homogeneity • bona fide geo-objects • Complementarity • functional regions • Fiat • administrative decisions • Fuzzy membership • geo-objects with indeterminate boundaries

41. A general theory of bona-fide geo-objects • m dimensional "phase" space defined by field variables • partition into n regions • m fields locate x in phase space • Assign x to one of n classes • compare classifiers

42. z1(x) z2(x) z3(x) c=4 z2 c=2 c=1 c=3 z1 2 4 3 3 1 2 2

43. Tables and classes • Groups of similar geo-objects • same topological dimension • same set of attributes • Relational and object-oriented models • Two steps removed from geo-atoms • aggregated to geo-objects, aggregated to tables and classes • aggregated to geo-fields and then discretized

44. Geo-dipoles • Properties associated with pairs of points • <x1,x2,Z,z(x1,x2)> • Interaction properties • distance, interaction, flow, direction, ...

45. Visible areas • Z: “is visible from” • z(x1,x2) = 1 if x2 is visible from x1 else 0 • aggregate all such geo-atoms to form a geo-object • the “area visible from” x1 = O(x1) • the “area from which x2 is visible” • Object field (Cova and Goodchild, 2002) • a mapping from location to an object • an object for every point • can have indeterminate boundary • if Z denotes membership

46. Metamaps (Takeyama and Couclelis, 1997) • A raster of cells • {Oi, i = 1,n} • a pair of such cells {Oi,Oj} • a metamap is the set of such pairs for all i,j and their properties zij • the set of all interactions • Each <Oi,Oj,zij> is an aggregation of geo-dipoles

47. Association classes • The object pair (Goodchild, 1991) • a pair of geo-objects having properties that exist only for the pair • distance, interaction, direction, flow... • distance matrices, turntables, the W matrix • An association class • An aggregation of geo-dipoles

48. Dynamics and time dependence • Already included in the definition of a geo-atom • Dynamic geo-fields • a discretization of space–time • an ordered sequence of spatial fields • with a common discretization? • isolines, bona fide polygons, and TINs rediscretized • fiat polygons, sample points can form a constant discretization • Dynamic geo-objects

50. Concluding comments • Increasing complexity • mainstream database concepts • increasing stress • A single, unifying theory • integrating objects and fields • integrating all major concepts • Geo-atoms and geo-dipoles • aggregation of point sets and point pairs • indeterminate, fiat, and bona fide geo-objects • geo-fields