Pre-Conference Seminar

Michael F. Goodchild
National Center for Geographic Information and Analysis

University of California

Santa Barbara, CA 93106

805 893 8049 (phone)

805 893 3146 (FAX)

805 893 8224 (NCGIA)


July 8, 2001


Four sessions:

Sunday July 8:

8:30am - 10:00am

10:30am - 12:00pm


1:30pm - 3.00pm

3:30pm - 5:00pm

  Instructor profile

Michael F. Goodchild is Professor of Geography at the University of California, Santa Barbara; Chair of the Executive Committee, National Center for Geographic Information and Analysis (NCGIA); Associate Director of the Alexandria Digital Library Project; and Director of NCGIA’s Center for Spatially Integrated Social Science. He received his BA degree from Cambridge University in Physics in 1965 and his PhD in Geography from McMaster University in 1969. After 19 years at the University of Western Ontario, including three years as Chair, he moved to Santa Barbara in 1988. He was Director of NCGIA from 1991 to 1997. He has been awarded honorary doctorates by Laval University (1999) and the University of Keele (2001). In 1990 he was given the Canadian Association of Geographers Award for Scholarly Distinction, in 1996 the Association of American Geographers award for Outstanding Scholarship, and in 1999 the Canadian Cartographic Association’s Award of Distinction for Exceptional Contributions to Cartography; he has won the American Society of Photogrammetry and Remote Sensing Intergraph Award and twice won the Horwood Critique Prize of the Urban and Regional Information Systems Association. He was Editor of Geographical Analysis between 1987 and 1990, and serves on the editorial boards of ten other journals and book series. In 2000 he was appointed Editor of the Methods, Models, and Geographic Information Sciences section of the Annals of the Association of American Geographers. His major publications include Geographical Information Systems: Principles and Applications (1991); Environmental Modeling with GIS (1993); Accuracy of Spatial Databases (1989); GIS and Environmental Modeling: Progress and Research Issues (1996); Scale in Remote Sensing and GIS (1997); Interoperating Geographic Information Systems (1999); Geographical Information Systems: Principles, Techniques, Management and Applications (1999); and Geographic Information Systems and Science (2001); in addition he is author of some 300 scientific papers. He was Chair of the National Research Council’s Mapping Science Committee from 1997 to 1999. His current research interests center on geographic information science, spatial analysis, the future of the library, and uncertainty in geographic data.

For a complete CV see the NCGIA web site www.ncgia.ucsb.edu under Personnel
Other related web sites: UCSB Geography www.geog.ucsb.edu, Alexandria Digital Library www.alexandria.ucsb.edu



1. What is Spatial Analysis?

        Basic GIS data models

        GIS function descriptions

2. Spatial Statistics

        Spatial interpolation

3. Spatial Interaction Models

4. Spatial Dependence

5. Spatial Decision Support

        Spatial search


What is Spatial Analysis?

GIS is designed to support a range of different kinds of analysis of geographic information: techniques to examine and explore data from a geographic perspective, to develop and test models, and to present data in ways that lead to greater insight and understanding. All of these techniques fall under the general umbrella of "spatial analysis". The statistical packages like SAS, SPSS, S, or Systat allow the user to analyze numerical data using statistical techniques—GIS packages like ArcInfo give access to a powerful array of methods of spatial analysis.

Purpose of the Course

The course will introduce participants with some knowledge of GIS to the capabilities of spatial analysis. Each of the five major sections will cover a major application area and review the techniques available, as well as some of the more fundamental issues encountered in doing spatial analysis with a GIS.


Section 1 - What is spatial analysis? - Basic GIS concepts for spatial analysis - GIS functionality - Integrating GIS and spatial analysis - Issues of error and uncertainty:

  • Definition of spatial analysis, major types and areas for application.
  • How should an analyst view a spatial database? Fields and discrete objects, attributes, relationships
  • How to organize the functions of a GIS into a coherent scheme.
  • Levels of integration of GIS and spatial analysis - loose and tight coupling, and full integration. Scripts and macros, lineage and analytical toolboxes.
  • The uncertainty problem - why is it such an issue in spatial analysis? What can we do now about data quality?
Section 2 - Spatial statistics - Simple measures for exploring geographic information - The value of the spatial perspective on data - Intuition and where it fails - Applications in crime analysis, emergencies, incidence of disease:
  • Measures of spatial form - centrality, dispersion, shape.
  • Spatial interpolation - intelligent spatial guesswork - spatial outliers.
  • Exploratory spatial analysis - moving windows, linking spatial and other perspectives.
  • Hypothesis tests - randomness, the null hypothesis, and how intuition can be misleading.
 Section 3 - Spatial interaction models - What they are and where they're used - Calibration and "what-if" - Trade area analysis and market penetration:
  • The Huff model and variations.
  • Site modeling for retail applications - regression, analog, spatial interaction.
  • Modeling the impact of changes in a retail system.
  • Calibrating spatial interaction models in a GIS environment.
Section 4 - Spatial dependence - Looking at causes and effects in a geographical context:
  • Spatial autocorrelation - what is it, how to measure it with a GIS.
  • The independence assumption and what it means for modeling spatial data.
  • Applying models that incorporate spatial dependence - tools and applications.
Section 5 - Site selection - Locational analysis and location/allocation - Other forms of operations research in spatial analysis - Spatial decision support systems - Linking spatial analysis with GIS to support spatial decision-making:
  • Shortest path, traveling salesman, traffic assignment.
  • What is location/allocation, and where can it be applied?
  • Modeling the process of retail site selection. Criteria.
  • Electoral districting and sales territories.
  • What is an SDSS? What are its component parts? How does it compare to a GIS or a DSS? Why would you want one? Building SDSS.
  • Examples of SDSS use - site selection, districting.





Section 1 - What is spatial analysis? - Basic GIS concepts for spatial analysis - GIS functionality - Integrating GIS and spatial analysis - Issues of error and uncertainty:

  • Definition of spatial analysis, major types and areas for application.
  • How should an analyst view a spatial database? Objects, layers, relationships, attributes, object pairs, data models.
  • How to organize the functions of a GIS into a coherent scheme.
  • Levels of integration of GIS and spatial analysis - loose and tight coupling, and full integration. Scripts and macros, lineage and analytical toolboxes.
  • The uncertainty problem - why is it such an issue in spatial analysis? What can we do now about data quality?
What is spatial analysis?

A set of techniques for analyzing spatial data

  • used to gain insight as well as to test models
  • ranging from inductive to deductive
  • finding new theories as well as testing old ones
  • can be highly technical, mathematical
    • can also be very simple and intuitive

    "A set of techniques whose results are dependent on the locations of the objects being analyzed"

  • move the objects, and the results change
  • e.g. move the people, and the US Center of Population moves
  • e.g. move the people, and average income does not change
  • most statistical techniques are invariant under changes of location
  • compare the techniques in SAS, SPSS, Systat etc.
    "A set of techniques requiring access both to the locations of objects and also to their attributes"
  • requires methods for describing locations (i.e. a GIS)
  • some techniques do not look at attributes
  • mapping is a form of spatial analysis?
    Is spatial analysis the ultimate objective of GIS?

    Some books on spatial analysis:

    • Anselin L (1988) Spatial Econometrics: Methods and Models. Kluwer
    • Bailey T C, Gatrell A C (1995) Interactive Spatial Data Analysis. Harlow: Longman Scientific & Technical
    • Berry B J L, Marble D F (1968) Spatial Analysis: A Reader in Statistical Geography. Prentice-Hall
    • Boots B N, Getis A (1988) Point Pattern Analysis. Sage
    • Burrough P A, McDonnell R A (1998) Principles of Geographical Information Systems. New York: Oxford University Press
    • Cliff A D, Ord J K (1973) Spatial Autocorrelation. Pion
    • Cliff A D, Ord J K (1981) Spatial Processes: Models and Applications. Pion
    • Fischer M, Scholten H J, Unwin D J, editors (1996) Spatial Analytical Perspectives on GIS. London: Taylor & Francis
    • Fotheringham A S, O'Kelly M E (1989) Spatial Interaction Models: Formulations and Applications. Kluwer
    • Fotheringham A S, Rogerson P A (1994) Spatial Analysis and GIS. Taylor and Francis
    • Fotheringham A S, Wegener M (2000) Spatial Models and GIS: New Potential and New Models. London: Taylor and Francis
    • Fotheringham A S, Brundson C, Charlton M (2000) Quantitative Geography: Perspectives on Spatial Data Analysis. London: SAGE
    • Getis A, Boots B N (1978) Models of Spatial Processes: An Approach to the Study of Point, Line and Area Patterns. Cambridge University Press
    • Ghosh A, Imgene C A (1991) Spatial Analysis in Marketing: Theory, Methods, and Applications. JAI Press
    • Ghosh A, Rushton G (1987) Spatial Analysis and Location-Allocation Models. Van Nostrand Reinhold
    • Goodchild M F (1986) Spatial Autocorrelation. CATMOG 47, GeoBooks
    • Griffith D A (1987) Spatial Autocorrelation: A Primer. Association of American Geographers
    • Griffith D A (1988) Advanced Spatial Statistics. Special Topics in the Exploration of Quantitative Spatial Data Series. Kluwer
    • Haggett P, Chorley R J (1970) Network Analysis in Geography. St Martin's Press
    • Haggett P, Cliff A D, Frey A (1977) Locational Methods. Wiley
    • Haggett P, Cliff A D, Frey A (1978) Locational Models. Wiley
    • Haining R P (1990) Spatial Data Analysis in the Social and Environmental Sciences. Cambridge University Press
    • Harries K (1999) Mapping Crime: Principle and Practice. Washington, DC: Crime Mapping Research Center, Department of Justice
    • Haynes K E, Fotheringham A S (1984) Gravity and Spatial Interaction Models. Sage
    • Hodder I, Orton C (1979) Spatial Analysis in Archaeology. Cambridge: Cambridge University Press
    • Leung Y (1988) Spatial Analysis and Planning under Imprecision. Amsterdam: North Holland
    • Longley P A, Batty M, editors (1996) Spatial Analysis: Modelling in a GIS Environment. Cambridge: GeoInformation International
    • Mitchell, A (1999) The ESRI Guide to GIS Analysis, Volume 1: Geographic Patterns and Relationships. ESRI Press
    • Odland J (1988) Spatial Autocorrelation. Sage
    • Raskin R G (1994) Spatial Analysis on the Sphere: A Review. Santa Barbara, CA: National Center for Geographic Information and Analysis
    • Ripley B D (1981) Spatial Statistics. Wiley
    • Ripley B D (1988) Statistical Inference for Spatial Processes. Cambridge University Press
    • Taylor P J (1977) Quantitative Methods in Geography: An Introduction to Spatial Analysis. Houghton Mifflin
    • Unwin D (1981) Introductory Spatial Analysis. Methuen
    • Upton G J G, Fingleton B (1985) Spatial Data Analysis by Example. Wiley

    Geographic Information Systems and Science

    Paul Longley, Mike Goodchild, David Maguire, and David Rhind
    Wiley, 2001
    Some background slides:
    Landsat image of New York area

    Indianapolis database

    Snow map of Soho, 1854

    the pump
    Openshaw GAM map of NE England

    Atlantic Monthly mystery map

    Northridge earthquake epicenters

    Environmental justice in LA

    World map

    England and Wales demography

    South Wales demography

    Vandenberg service station

    Service station subsurface

    Service station plume

    How does an analyst/modeler/decision-maker work with a GIS?

    What tools exist for helping/conceptualizing/problem-solving?

    Assumption: these (analysis, modeling, decision-making) are the primary purposes of GIS technology.

    The cost of input to a GIS is high, and can only be justified by the benefits of analysis/modeling/decision-making performed with the data.

  • 60 polygons per hour = $1 per polygon
  • estimates as high as $40 per polygon
  • 500,000 polygon database costs $500,000 to create using the low estimate
  • $20m using the high estimate
  • What types of analysis can justify these costs?
    • Query (if it is faster than manual lookup)
      • very repetitive
      • highly trained user
    • Analyses which are simple in nature but difficult to execute manually
      • overlay (topological)
      • map measurement, particularly area
      • buffer zone generation
    • Analyses which can take advantage of GIS capabilities for data integration
      • global science
    • Browsing/plotting independently of map boundaries and with zoom/scale-change
      • seamless database
      • need for automatic generalization
      • editing
    • Complex modeling/analysis (based on the above and extensions)
    The list of possibilities is endless
    • List of generic GIS functions has 75 entries
  • ESRI's ARC/INFO has over 1000 commands/functions
    How can we organize/conceptualize the possibilities?
    • A taxonomy/classification of GIS functions
    • A customized view of a spatial database designed for the needs of the analyst/modeler
    • A set of tools to support analysis and database manipulation
    • Associated tools for defining needs in the analysis/modeling area, and testing systems against those needs
    • Methods for dealing with problems associated with analysis/modeling of spatial databases, particularly error/inaccurac

    A geographical data model consists of the set of entities and relationships used to create a represention of the geographical world. The choices made when the world is modeled determine how the database is structured, and what kinds of analysis can be done with it. These choices occur when the data are captured in the field, recorded, mapped, digitized, and processed.

    There are two distinct ways of conceiving of the geographical world.

    In the field view, the world is conceived as a finite set of variables, each having a single value at every point on the Earth's surface (or every point in a three-dimensional space; or a four-dimensional space if time is included).

      Examples of fields: elevation, temperature, soil type, vegetation cover type, land ownership

      Some field-like phenomena: elevation, spectral response

    To be represented digitally, a field must be constructed out of primitive one, two, three, or four-dimensional objects. There are six ways of representing fields in common use in GIS:
      raster (a rectangular array of homogeneous cells)

      grid (a rectangular array of sample points)

      irregular points

      digitized contours



    Other methods can be found in environmental modeling, but not commonly in GIS.
    finite element methods

    The field view underlies the following ESRI implementation models:




    but not shapefiles
    in the Arc8 Geodatabase the distinction can be implemented in object behaviors

    In the discrete object view an otherwise empty space is littered with objects, each of which has a series of attributes. Any point in space (two, three, or four dimensional) can lie in any number of discrete objects, including zero, and objects can therefore overlap, and need not exhaust the space.

    objects can be counted
    how many mountains are there in Scotland?
    what's a mountain?
    objects can be manipulated
    they maintain integrity as they move
    objects are homogeneous
    the whole thing is the object
    parts can inherit properties from the whole
    Field and discrete object views can be implemented in either raster or vector forms
    compare manipulation of shapefiles (objects) and coverages (fields)
    the distinction concerns how the world is conceived, and the rules governing object behavior

    a field can be represented as raster cells, points (e.g., spot heights), triangles (TIN), lines (contours), or areas (land ownership)

    in many of these cases the primitive elements are not real (cannot be located on the ground), but are artifacts of the representation
    If we ignore the field/discrete object distinction we may easily apply meaningless forms of analysis
    buffer makes sense only for discrete objects

    interpolation makes sense only for fields

    Attributes can be of several types:




    Spatial objects are distinguished by their dimensions or topological properties:

            points (0-cells)
            lines (1-cells)
            areas (2-cells)
            volumes (3-cells)

    A class of objects is a set with the same topological properties (e.g. all points) and with the same set of attributes (e.g. a set of wells or quarter sections or roads). In the Arc8 Geodatabase a class also has the same behaviors, and may inherit behaviors from other classes. A class is associated with an attribute table.

    Geodatabase introduces a consistent set of terms for primitive geometric objects

    When a class represents a field, certain rules apply to the component objects. The objects belonging to one class of area or volume objects will fill the area and will not overlap (they are space-exhausting, they partition or tesselate the space, they are planar enforced).

             the layer provides one value at every point (recall the definition of a field)

      • e.g. soil type
      • e.g. elevation
      • e.g. zoning
    Slide: Planar enforcement

    Spatial objects are abstractions of reality. Some objects are well-defined (e.g. road, bridge) but others are not. Objects representing a discrete entity view tend to be well-defined; objects representing a field are not.

    • A TIN or DEM is an approximation to a topographic surface, with an accuracy which is usually undetermined. Even if accuracy is known at the sampled points, it is unknown between them.
    • We assume that all of the points within an area object have the attributes ascribed to the object. In reality the area inside the object is not homogeneous, and the boundaries are zones of transition rather than sharp discontinuities (e.g. soil maps, climatic zones, geological maps).
    A topographic surface can be represented as either a TIN or a DEM.

    Slides: Elevation model options

    digital elevation model (raster)

    digitized contours

    triangular mesh


    Advantages of TIN:
    • sampling intensity can adapt to local variability
    • many landforms are approximated well by triangular mosaics
    • triangles can be rendered quickly by graphics processors
    Advantages of DEM:
    • uniform sampling intensity is suited to automatic data collection via e.g. analytical stereoplotter
    • many applications require uniform-sized spatial objects.
    A spatial database consists of a number of classes of spatial objects with associated attribute tables.

    The methods used to store the attribute and locational information about the objects are not of immediate concern to the analyst/modeler.

  • In fact this object/attribute view of the database may have little in common with the actual data structures/models used by the system designer.

    A database encodes and represents the complex relationships which exist between objects.

    • spatial relationships
    • functional relationships
  • A GIS must be capable of computing these relationships through such geometrical operations as intersection.

     Spatial relationships include:

    • Relationships between objects of different classes
    • Relationships between objects of the same class

    The potential set of relationships within a complex spatial database is enormous. No system can afford to compute and store all of them in the database.

    A cartographic data structure stores no spatial relationships among objects.

  • Since it must compute any relationship as and when needed it is inefficient for complex spatial analyses.
    A topological data structure stores certain spatial relationships among objects. Common stored relationships are:
    • ID of incident links stored as attributes of nodes in line networks

    UML relationship types

    a functional linkage between objects in different classes
    aggregation and composition
    linkage between an object and its component objects
    type inheritance
    classes inherit properties from more general classes
    Relations between objects

    An object pair is a combination of objects of the same or different types/classes which may have its own attributes.

  • e.g. the hydrologic relationship between a spring and a sink may have attributes (direction, volume of flow, flow through time) but may not exist as a spatial object itself.
  • The ability to generate object pairs, give them attributes and include them in analysis is an important component of a full GIS.
    giving attributes to associations
    Examples of object pairs:
    • Matrix of distances between pairs of objects
    • Hydrologic flows
    • Traffic flows between origin/destination pairs

    Object pairs in ESRI products

    turntable (link-link pairs)

    distance matrix (first object, second object, distance)

    association class in UML

    attributed relationship class in Geodatabase

    Visio example

    Example: Data Model for Traffic Routing

    What are the essential components of a data model for route planning in a complex street network?

      1. Links - attributes: length, street name, traffic count, terminal nodes. A street can be represented by a single link with attributes which include <one-way?> or by a pair of links with associated directions - one of the pair being omitted in the case of one-way streets.

      2. Nodes or intersections - attributes: incident links, presence of traffic light.

    •  Turn prohibitions - are attributes not of nodes or links but of link/link object pairs ("turntable")
    •   Stop signs - are attributes of link/node object pairs.



      Visio example

    Data modeling examples

    1. Design a database to capture and analyze data on recreational fishing in the Scottish Highlands, to support decision-making by the tourist industry and regulatory agencies. The database should be able to represent the following:

    • locations of fishing (rivers, lakes)
    • locations of accommodation (hotels, guest houses)
    • preferences and rights (fishing locations owned by hotels, locations accessible to hotels)
    2. Design a database to support analysis and modeling of shoreline erosion on the Great Lakes. It is necessary to represent conditions and processes transverse to the shoreline in much more detail than variation parallel to the shoreline.

    3. Design a database to support water resource analysis and planning for complex hydrographic networks that include streams, rivers, lakes and reservoirs.



    A1 Digitizing (di)

    Digitizing is the process of converting point and line data from source documents to a machine-readable format.

    A2 Edgematching (ed)

    Edgematching is the process of joining lines and polygons across map boundaries in creation of a "seamless" database.† The join should be topological as well as graphic, that is, a polygon so joined should become a single polygon in the data base, a line so joined should become a single line segment.

    A3 Polygonization (po)

    Polygonizing is the process of connecting together arcs ("spaghetti") to form polygons.

    A4 Labelling (la)

    This process transfers labels describing the contents (attributes) of polygons, and the characteristics of lines and points, to the digital system.† This input of labels must not be confused with the process of symbolizing and labelling output described below.

    A5 Reformatting digital data for input from other systems (rf)

    Data previously digitized are made accessible through an interface or converted by software to the system format, and made to be topologically useful as well as graphically compatible.

    A6 Reformatting for output to other systems (ro)

    This function is the inverse of the previous one. Internal data is reformatted to meet the requirements of other systems or standards.

    A7 Data base creation and management (db)

    Data is typically digitized from map-sheets, and may be edgematched. The creation of a true "seamless" database requires the establishment of a map sheet directory, and may include tiling to partition the database.

    A8 Raster/vector conversion (rv)

    The ability to convert data between vector and raster forms with grid cell size, position and orientation selected by the user.

    A9 Edit and display on input (ei)

    This function allows continuous display and editing of input data, usually in conjunction with digitizing.

    A10 Edit and display on output (eo)

    The ability to preview and edit displays before creation of hard copy maps.

    A11 Symbolizing (sy)

    To create high quality output from a GIS, it is necessary to be able to generate a wide variety of symbols to replace the primitive point, line and area objects stored in the database.

    A12 Plotting (pl)

    Creation of hard copy map output.

    A13 Updating (up)

    Updating of the digital data base with new points, lines, polygons and attributes.

    A14 Browsing (br)

    Browse is used to search the data base to answer simple locational queries, and includes pan and zoom.


    B1 Create lists and reports (cl)

    This is the ability to create lists and reports on objects and their attributes in user-defined formats, and to include totals and subtotals.

    B2 Reclassify attributes (ra)

    Reclassification is the change in value of a set of existing attributes based on a set of user specified rules.

    B3 Dissolve lines and merge attributes (dm)

    Boundaries between adjacent polygons with identical attributes are dissolved to form larger polygons.

    B4 Line thinning and weeding (lt)

    This process is used to reduce the number of points defining a line or set of lines to a user defined tolerance.

    B5 Line smoothing (ls)

    Automatically smooth lines to a user-defined tolerance, creating a new set of points (compare B4).

    B6 Complex generalization (cg)

    Generalization which may require change in the type of an object, or relocation in response to cartographic rules.

    B7 Windowing (wi)

    The ability to clip features in the database to some defined polygon.

    B8 Centroid calculation and sequential numbering (cn)

    Calculate a contained, representative point in a polygon and assign a unique number to the new object.

    B9 Spot heights (sh)

    Given a digital elevation model, interpolate the height at any point.

    B10 Heights along streams (hs)

    Given a digital elevation model and a hydrology net, interpolate points along streams at fixed increments of height.

    B11 Contours (isolines) (ci)

    Given a set of regularly or irregularly spaced point values, interpolate contours at user-specified intervals.

    B12 Elevation polygons (ep)

    Given a digital elevation model, interpolate contours of height at user-specified intervals.

    B13 Watershed boundaries (wb)

    Given a digital elevation model and a hydrology net, interpolate the position of the watershed between basins.

    B14 Scale change (sc)

    Perform the operations associated with change of scale, which may include line thinning and generalization.

    B15 Rubber sheet stretching (rs)

    The ability to stretch one map image to fit over another, given common points of known locations.

    B16 Distortion elimination (de)

    The ability to remove various types of systematic distortion generated by different input methods.

    B17 Projection change (pc)

    The ability to transform maps from one map projection to another.

    B18 Generate points (gp)

    The ability to generate points and insert them in the database.

    B19 Generate lines (gl)

    The ability to generate lines and insert them in the database.

    B20 Generate polygons (ga)

    The ability to generate polygons and insert them in the database.

    B21 Generate circles (gc)

    The ability to generate circles defined by center point and radius.

    B22 Generate grid cell nets (gg)

    The ability to generate a network of grid cells given a point of origin, grid cell dimension and orientation.

    B23 Generate latitude/longitude nets (gn)

    The ability to generate graticules for a variety of map projections.

    B24 Generate corridors (gb)

    This process generates corridors of given width around existing points, lines or areas.

    B25 Generate graphs (gr)

    Create a graph illustrating attribute data by symbols, bars or fitted trend line.

    B26 Generate viewshed maps (gv)

    Given a digital elevation model and the locations of one or more viewpoints, generate polygons enclosing the area visible from at least one viewpoint.

    B27 Generate perspective views (ge)

    From a digital elevation model, generate a three-dimensional block diagram.

    B28 Generate cross sections (cs)

    Given a digital elevation model, show the cross-section along a user-specified line.

    B29 Search by attribute (sa)

    The ability to search the data base for objects with certain attributes.

    B30 Search by region (sr)

    The ability to search the data base within any region defined to the system.

    B31 Suppress (su)

    The ability to exclude objects by attribute (the converse of selecting by attribute).

    B32 Measure number of items (mi)

    The ability to count the number of objects in a class.

    B33 Measure distances along straight and convoluted lines (md)

    The ability to measure distances along a prescribed line.

    B34 Measure length of perimeter of areas (mp)

    The ability to measure the length of the perimeter of a polygon.

    B35 Measure size of areas (ma)

    The ability to measure the area of a polygon.

    B36 Measure volume (mv)

    The ability to compute the volume under a digital representation of a surface.

    B37 Calculate - arithmetic (ca)

    The ability to perform arithmetic, algebraic and Boolean calculations separately and in combination.

    B38 Calculate bearings between points (cb)

    The ability to calculate the bearing (with respect to True North) from a given point to another point.

    B39 Calculate vertical distance or height (ch)

    Given a digital elevation model, calculate the vertical distance (height) between two points.

    B40 Calculate slopes along lines (gradients) (al)

    The ability to measure the slope between two points of known height and location or to calculate the gradient between any two points along a convoluted line which contains two or more points of known elevation.

    B41 Calculate slopes of areas (sl)

    Given a digital elevation model and the boundary of a specified region (e.g., a part of a watershed), calculate the average slope of the region.

    B42 Calculate aspect of areas (aa)

    Given a digital elevation model and the boundary of a specified region, calculate the average aspect of the region.

    B43 Calculate angles and distances along linear features (ad)

    Given a prescribed linear feature, generalize its shape into a set of angles and distances from a start point, at user-set angular increments, and constrained to any known points along the linear feature.

    B44 Subdivide area according to a set of rules (sb)

    Given the corner points of a rectangular area, topologically subdivide the area into four quarters.

    B45 Locations from traverses (lo)

    Given a direction (one of eight radial directions) and distance from a given point, calculate the end point of the traverse.

    B46 Statistical functions (sf)

    The ability to carry out simple statistical analyses and tests on the database.

    B47 Graphic overlay (go)

    The ability to superimpose graphically one map on another and display the result on a screen or on a plot.

    B48 Point in polygon (pp)

    The ability to superimpose a set of points on a set of polygons and determine which polygon (if any) contains each point.

    B49 Line on polygon overlay (lp)

    The ability to superimpose a set of lines on a set of polygons, breaking the lines at intersections with polygon boundaries.

    B50 Polygon overlay (op)

    The ability to overlay digitally one set of polygons on another and form a topological intersection of the two, concatenating the attributes.

    B51 Sliver polygon removal (sp)

    The ability to delete automatically the small sliver polygons which result from a polygon overlay operation when certain polygon lines on the two maps represent different versions of the same physical line.

    B52 Line of sight (ln)

    The ability to determine the intervisibility of two points, or to determine those parts of pairs of lines or polygons which are intervisible.

    B53 Nearest neighbor search (nn)

    The ability to identify points, lines or polygons that are nearest to points, lines or polygons specified by location or attribute.

    B54 Shortest route (ps)

    The ability to determine the shortest or minimum cost route between two points or specified sets of points.

    B55 Contiguity analysis (co)

    The ability to identify areas that have a common boundary or node.

    B56 Connectivity analysis (cy)

    The ability to identify areas or points that are (or are not) connected to other areas or points by linear features.

    B57 Complex correlation (cx)

    The ability to compare maps representing different time periods, extracting differences or computing indices of change.

    B58 Weighted modelling (wm)

    The ability to assign weighting factors to individual data sets according to a set of rules and to overlay those data sets and carry out reclassify, dissolve and merge operations on the resulting concatenated data set.

    B59 Scene generation (sg)

    The ability to simulate an image of the appearance of an area from map data. The image would normally consist of an oblique view, with perspective.

    B60 Network analysis (na)

    Simple forms of network analysis are covered in Shortest route and Connectivity. More complex analyses are frequently carried out on network data by electrical and gas utilities, communications companies etc. These include the simulation of flows in complex networks, load balancing in electrical distribution, traffic analysis, and computation of pressure loss in gas pipes. In many cases these capabilities can be found in existing packages which can be interfaced to the GIS database.

    Other groupings of GIS functions:

    Berry, J.K., 1987, "Fundamental operations in computer-assisted map analysis". International Journal of GIS 1 119-36.

    • Reclassifying maps
    • Overlaying maps
    • Measuring distance and connectivity
    • Characterizing neighborhoods
    Goodchild, M.F., 1988, "Towards an enumeration and classification of GIS functions". Proceedings, IGIS '87

    Tomlin, Dana, 1990. Geographic Information Systems and Cartographic Modeling. Prentice Hall.
    based on a standard, semi-formal taxonomy of analytic functions for raster data

    • Focal: operations that process a single cell
    • Local: operations that process a cell and a fixed neighborhood
    • Zonal: operations that process an area of homogeneous characteristics
    • Global: operations that process the entire map
    Maguire, David, 1991. Chapter 21: The Functionality of GIS. In D.J. Maguire, M.F. Goodchild and D.W. Rhind, editors, Geographical Information Systems: Principles and Applications. Longman, London.
    • Capture
    • Transfer
    • Validate and Edit
    • Store and Structure
    • Restructure
    • Generalize
    • Transform
    • Query
    • Analyze
    • Present

    A Six-way Classification of Spatial Analysis

    1. Query and reasoning

    based on database views





    linked views

    2. Measurement

    simple geometric measurements associated with objects
    area, distance, length, perimeter, shape
    3. Transformation

    point in polygon

    polygon overlay


    density estimation

    4. Descriptive summaries


    spatial dependence


    5. Optimization
    best routes
    raster version

    network version

    Paul's ride

    best locations
    6. Hypothesis testing
    inference from sample to population

    Integration of GIS and Spatial Analysis

    1. Full integration (embedding)

    • spatial analysis as GIS commands
    • requires modification of source code
      • difficult with proprietary packages
      • analysis is not the strongest commercial motivation
    • third party macros, scripting languages
    • COM components and VBA
    2. Loose coupling
    • what we have now
    • unsatisfactory
    • hooks too awkward
    • loss of higher structures in data
    • transfer of simple tables
    3. Close coupling
    • better hooks
    • common data models
    • discretization problem
      • discretization often not explicit in models
      • e.g. slope, length
    • user interface design
      • models easy to use?
      • the user-friendly grand piano
      • user community is already frustrated






    Section 2 - Spatial statistics - Simple measures for exploring geographic information - The value of the spatial perspective on data - Intuition and where it fails - Applications in crime analysis, emergencies, incidence of disease:

    • Measures of spatial form - centrality, dispersion, shape.
    • Spatial interpolation - intelligent spatial guesswork - spatial outliers.
    • Exploratory spatial analysis - moving windows, linking spatial and other perspectives.
    • Hypothesis tests - randomness, the null hypothesis, and how intuition can be misleading.
    Measures of spatial form:

    How to sum up a geographical distribution in a simple measure?

    Two concepts of space are relevant:


    • travel can occur anywhere
    • best for small scales, or where a network is too complex or too costly to capture or represent
  • an infinite number of locations exist
  • a means must exist to calculate distances between any pair of locations, e.g. using straight lines
  • Discrete:
    • travel can occur only on a network
    • only certain locations (on the network) are feasible
    • all distances (between all possible pairs of locations) can be evaluated using any measure (travel time, cost of transportation etc.)
    In discrete space places are identified as objects; in continuous space, places are identified by coordinates

    A metric is a means of measuring distance between pairs of places (in continuous space)

    • e.g. straight lines (the Pythagorean metric)
  • e.g. by moves in N-S and E-W directions (the Manhattan or city-block metric)
  • simple metrics can be improved using barriers or routes of lower travel cost (freeways)
  • The most useful single measure of a geographical distribution of objects is its center

    Definitions of center:

    The centroid

    • the average position
    • computed by taking a weighted average of coordinates
    • the point about which the distribution would balance
    • the basis for the US Center of Population (now in MO and still moving west)
    The centroid is not the point for which half of the distribution is to the left, half to the right, half above and half below
    • this is the bivariate median
    The centroid is not the point that minimizes aggregate distance (if the objects were people and they all traveled to the centroid, the total distance traveled would be minimum)
    • this is the point of minimum aggregate travel (MAT), sometimes called the median (very confusingly)
    • for many years the US Bureau of the Census calculated the Center of Population as the centroid, but gave the MAT definition
    • there is a long history of confusion over the MAT
    • no ready means exist to calculate its location
    • the MAT must be found by an iterative process
    • an interesting way of finding the MAT makes use of a valid physical analogy to the resolution of forces - the Varignon frame
    • on a network, the MAT is always at a node (junction or point where there is weight)
     The definition of centrality becomes more difficult on the sphere
  • e.g. the centroid is below the surface
  • the centroid of the Canadian population in 1981 was about 90km below White River, Ontario
  • the bivariate median (defined by latitude and longitude) was at the intersection of the meridian passing through Toronto and the parallel through Montreal, near Burke's Falls, Ontario
  • the MAT point (assuming travel on the surface by great circle paths) was in a school yard in Richmond Hill, Ontario
  • What use are centers?
    • for tracking change in geographic distributions, e.g. the march of the US Center of Population westward is still worth national news coverage
    • for identifying most efficient locations for activities
    • location at the MAT minimizes travel
    • a central facility should be located to minimize travel to the geographic distribution that it serves
    • should we use continuous or discrete space?
    • this technique was considered so important to central planning in the Soviet Union in the early 20th century that an entire laboratory was founded
    • the Mendeleev Centrographic Laboratory flourished in Leningrad around 1925
    • centers are often used as simplifications of complex objects
    • at the lower levels of census geography in many countries
    • e.g. ED in US, EA in Canada, ED in UK
    • to avoid the expense of digitizing boundaries
    • e.g. land parcel databases
    • or where boundaries are unknown or undefined
    • e.g. ZIPs
    • in the census examples, common practice is to eyeball a centroid
    • some very effective algorithms have been developed for redistributing population from centroids
    Measures of dispersion:
    • what you would want to know if you could have two measures of a geographical distribution
    • the spread of the distribution around its center
    • average distance from the center
    • measures of dispersion are used to indicate positional accuracy
    • the error ellipse
    • the Tissot indicatrix
    • the CMAS
    Potential measures:
    • a measure which increases with the weight of geographic objects and with proximity to them
    • calculated as:
        V = summation of (w(i)/d(i))
      where i is an object, d is the distance to the object and w is its weight

      the summation can be carried out at any location

      V can be mapped - a "potential" map

    Potential is a useful measure of:
    • the market share obtainable by locating at a point
    • the best location is the place of maximum potential
    • population pressure on a recreation facility
    • accessibility to a geographic distribution
    • e.g. a network of facilities
    • potential measures omit the "alternatives" factor
    • imply that market share can potentially increase without limit
    • potential measures have been used as predictors of growth
      • economic growth most likely in areas of highest potential
    • potential calculation exists as a function in SPANS GIS
    • the objects used to calculate potential must be discrete in an empty space
      • adding new objects will increase potential without limit
      • it makes no sense to calculate potential for a set of points sampled from a field
      • potential makes sense only in the object view
    Potential measures and density estimation
    think of a scatter of points representing people
    how to map the density of people?

    replace each dot by a pile of sand, superimposing the piles

    the amount of sand at any point represents the number and proximity of people

    the shape of the pile of sand is called the kernel function

    example - density estimation and Chicago crime


    Measures of shape:

    • shape has many dimensions, no single measure can capture them all
    • many measures of shape try to capture the difference between compact and distended
    • many of these are based on a comparison of the shape's perimeter with that of a circle of the same area
    • e.g. shape = perimeter / (3.54 * sqrt(area))
    • this measure is 1.0 for a circle, larger for a distended shape
    • all of these measures based on perimeter suffer from the same problem
    • within a GIS, lines and boundaries are represented as straight line segments between points
    • this will almost always result in a length that is shorter than the real length of the curve, unless the real shape is polygonal
    • consequently the measure of shape will be too low, by an undetermined amount
    • shape (compactness) is a useful measure to detect gerrymanders in political districting

    Spatial Interpolation

    Spatial interpolation is defined as a process of determining the characteristics of objects from those of nearby objects

    • of guessing the value of a field at locations where value has not been measured
    The objects are most often points (sample observations) but may be lines or areas

    The attributes are most often interval-scaled (elevations) but may be of any type

    From a GIS perspective, spatial interpolation is a process of creating one class of objects from another class

    Spatial interpolation is often embedded in other processes, and is often used as part of a display process

  • e.g. to contour a surface from a set of sample points, it is necessary to use a method of spatial interpolation to determine where to place the contours among the points
    Many methods of spatial interpolation exist:

    Distance-weighted interpolation

    Known values exist at n locations i=1,...,n

    The value at a location xi is denoted by z(xi)

    We need to guess the value at location x, denoted by z(x)

    The guessed value is an average over the known values at the sample points

    • the average is weighted by distance so that nearby points have more influence.
    Let d(xi,x) denote the distance from location x, where we want to make a guess, to the ith sample point.

    Let w[d] denote the weight given to a point at distance d in calculating the average.

    The estimate at x is calculated as:

    z(x) = summation over every point i (w[d(xi,x)] z(xi)) / summation over every point i (w[d(xi,x)])

    in other words, the average weighted by distance.

    The simplest kind of weight is a switch - a weight of 1 is given to any points within a certain distance of x, and a weight of 0 to all others

  • this means in effect that z(x) is calculated as the average over points within a window of a certain radius.
    Better methods include weights which are continuous, decreasing functions of distance such as an inverse square:

    w[d] = d-2

    All of the distance weighted methods (e.g IDW) share the same positive features and drawbacks. They are:

    • easy to implement and conceptually simple
    • adaptable - the weighting function can be changed to suit the circumstances. It is even possible to optimize the weighting function in this sense:
      Suppose the weighting function has a parameter, such as the size of the window

      Set the window size to some test value

      Then select one of the sample points, and use the method to interpolate at that point by averaging over the remaining n-1 sample values

      Compare the interpolated value to the known value at that point. Repeat for all n points and average the errors

      Then the best window size (parameter value) is the one which minimizes total error

      In most cases this will be a non-zero and non-infinite value.

    • all interpolated values must lie between the minimum and maximum observed values, unless negative weights are used
    • This means that it is impossible to extrapolate trends
    • If there is no data point exactly at the top of a hill or the bottom of a pit, the surface will smooth out the feature
    • the interpolated surface cannot extrapolate a trend outside the area covered by the data points - the value at infinity must be the arithmetic mean of the data points
     Although distance-weighted methods underlie many of the techniques in use, they are far from ideal

    Polynomial surfaces

    A polynomial function is fitted to the known values - interpolated values are obtained by evaluating the function

      e.g. planar surface - z(x,y) = a + bx + cy
      e.g. cubic surface - z(x,y) = a + bx + cy + dx2 + exy + fy2 + gx3 + hx2y + ixy2 + jy3
    • easy to use
    • useful only when there is reason to expect that the surface can be described by a simple polynomial in x and y
    • very sensitive to boundary effects

    Most real surfaces are observed to be spatially autocorrelated - that is, nearby points have values which are more similar than distant points.

    The amount and form of spatial autocorrelation can be described by a variogram, which shows how differences in values increase with geographical separation

    Observed variograms tend to have certain common features - differences increase with distance up to a certain value known as the sill, which is reached at a distance known as the range.

    To make estimates by Kriging, a variogram is obtained from the observed values or past experience

    • Interpolated best-estimate values are then calculated based on the characteristics of the variogram.
    • perhaps the most satisfactory method of interpolation from a statistical viewpoint
    • difficult to execute with large data sets
    • decisions must be made by the user, requiring either experience or a "cookbook" approach
    • a major advantage of Kriging is its ability to output a measure of uncertainty of each estimate
      • This can be used to guide sampling programs by identifying the location where an additional sample would maximally decrease uncertainty, or its converse, the sample which is most readily dropped.
    Locally-defined functions

    Some of the most satisfactory methods use a mosaic approach in which the surface is locally defined by a polynomial function, and the functions are arranged to fit together in some way

    With a TIN data structure it is possible to describe the surface within each triangle by a plane

    • Planes automatically fit along the edges of each triangle, but slopes are not continuous across edges
    • This can be arranged arbitrarily by smoothing the appearance of contours drawn to represent the surface
    • Alternatively a higher-order polynomial can be used which is continuous in slopes.
    Another popular method fits a plane at each data point, then achieves a smooth surface by averaging planes at each interpolation point

    Hypothesis tests:
    • compare patterns against the outcomes expected from well-defined processes
    • if the fit is good, one may conclude that the process that formed the observed pattern was like the one tested
    • unfortunately, there will likely be other processes that might have formed the same observed pattern
    • in such cases, it is reasonable to ignore them as long as a) they are no simpler than the hypothesized process, and b) the hypothesized process makes conceptual sense
    • the best known examples concern the processes that can give rise to certain patterns of points
    • attempts to extend these to other types of objects have not been as successful
    Point pattern analysis
    • a commonly used standard is the random or Poisson process
    • in this process, points are equally likely to occur anywhere, and are located independently, i.e. one point's location does not affect another's
    • CSR = complete spatial randomness
    • a real pattern of points can be compared to this process
    • most often, the comparison is made using the average distance between a point and its nearest neighbor
    • in a random pattern (a pattern of points generated by the Poisson process) this distance is expected to be 1/(2 * sqrt(density)) where density is the number of points per unit area, and area is measured in units consistent with the measurement of distance
    • when the number of points is limited, we would expect to come close to this estimate in a random pattern
    • theory gives the limits within which average distance is expected to lie in 95% of cases
    • if the actual average distance falls outside these limits, we conclude that the pattern was not generated randomly
    There are two major options for non-random patterns:
    • the pattern is clustered
    • points are closer together than they should be
    • the presence of one point has made other points more likely in the immediate vicinity
    • some sort of attractive or contagious process is inferred
    • the pattern is uniformly spaced
    • points are further apart than they should be
    • the presence of one point has made other points less likely in the vicinity
    • some sort of repulsive process is inferred, or some sort of competition for space
    Unfortunately it is easy for this process of inference to come unstuck
    • the process that generated the pattern may be non-random, but not sufficiently so to be detectable by this test
    • this false conclusion is more likely reached when there is little data - the more data we have, the more likely we are to detect differences from a simple random process
    • in statistics, this is known as a Type II error - accepting the null hypothesis when in fact it is false
    • the process may be non-random, but not in either of the senses identified above - contagious or repulsive
    • points may be located independently, but with non-uniform density, so that points are not equally likely everywhere
    • it is possible to hypothesize more complex processes, but the test becomes progressively weaker at confirming them






    Section 3 - Spatial interaction models - What they are and where they're used - Calibration and "what-if" - Trade area analysis and market penetration:

    • The Huff model and variations.
    • Site modeling for retail applications - regression, analog, spatial interaction.
    • Modeling the impact of changes in a retail system.
    • Calibrating spatial interaction models in a GIS environment.
    What is a spatial interaction model?
    • a model used to explain, understand, predict the level of interaction between different geographic locations
    • examples of interactions:
    • migration (number of migrants between pairs of states)
    • phone traffic (number of calls between pairs of cities)
    • commuting (number of vehicles from home to workplace)
    • shopping (number of trips from home to store)
    • recreation (number of campers from home to campsite)
    • trade (amount of goods between pairs of countries)
    • interaction is always expressed as a number or quantity per unit of time
    • interaction occurs between defined origin and destination
    • these may be the same or different classes of objects
    • e.g. the same class in the case of migration between states
    • e.g. different classes in the case of journeys to shop or work
    • the matrix of interactions can be square or rectangular

    Interaction is believed to be dependent on:
    • some measure of the origin (its propensity to generate interaction)
    • some measure of the destination (its propensity to attract interaction)
    • some measure of the trip (its propensity to deter interaction)
    • these measures are assumed to multiply
     Let: i denote an origin object (often an area)

    j denote a destination object (a point or area)

    I*ij denote the observed interaction between i and j, measured in appropriate units (e.g. numbers of trips, flow of goods, per defined interval of time)

    Iij denote the interaction predicted by the spatial interaction model

    • if the model is good (fits well), the predicted interactions per interval of time will be close in value to the observed interactions
    • each Iij will be close to its corresponding I*ij
    Ei denote the emissivity of the origin area i

    Aj denote the attraction of the destination area j

    Cij denote the deterrence of the trip between i and j (probably some measure of the trip length or cost)

    a a constant to be determined

    Then the most general form of spatial interaction model is:

    Iij = a Ei Aj Cij

    • that is, interaction can be predicted from the product of a constant, emissivity, attraction and deterrence
    The model began life in the mid 19th century as an attempt to apply laws of gravitation to human communities - the gravity model
    • such ideas of social physics have long since gone out of fashion, but the name is still sometimes used
    • even in the form above, the model bears some relationship to Newton's Law of Gravitation
     In any application of the model, some aspects are assumed to be unknown, and determined by calibration
    • e.g. the value of a might be unknown in a given application
    • its value would be calibrated by finding the value that gives the best fit between the observed interactions and the interactions predicted by the model
    • the conventional measure of fit is the total squared difference between observation and prediction, that is, the summation over i and j of (Iij - I*ij)2
    • this is known as least squares calibration
    • other unknowns might be the method of calculating deterrence (Cij) from distance, or the attraction value to give to certain retail stores
    Measurement of the variables:


    • deterrence is often strongly related to distance
    • the further the distance, the less interaction and thus the lower Cij
    • a common choice is a decreasing function of distance:
      • Cij = dij-b            (Cij = 1 / dijb)
        or Cij = exp (-bdij)            (exp denotes 2.71828 to the power)
    • generally the fit of the model is not sufficiently good to distinguish between these two, that is, to identify which gives the better fit
    • the negative exponential has a minor technical advantage in not creating problems when dij = 0 (origin and destination are the same place)
    • the b parameter is unknown and must be calibrated
    • its value depends on the type of interaction, and also probably on the region
    • b has units in the negative exponential case (1/distance) but none in the negative power case
    • other measures of deterrence include:
    • some function of transport cost
    • some function of actual travel time
    • in either case the function used is likely to be the negative power or negative exponential above
    • there are examples where distance has a positive effect on interaction
    • how to measure the propensity of each origin to emit interaction?
    • gross population Pi
    • some more appropriate measure weighting each cohort, e.g. age and sex cohorts
    • some cohorts are more likely to interact than others
    • gross income
    • Ei could be treated as unknown and calibrated
    • the propensity of each destination to attract interaction
    • could be unknown and calibrated
    • for shopping models, gross floor area of retail space is often used
    • some forms of interaction are symmetrical
    • flow from origin to destination equals reverse flow
    • e.g. phone calls
    • requires Ei and Aj to be the same, e.g. population
    The Huff model
  • what happens when a new destination is added?
  • interactions with existing destinations are unaffected
  • assumes outflow from origins can increase without limit
  • in practice, in many applications flow from origin to existing destinations will be diverted
  • we need some form of "production constraint"
  • Huff proposed this change:
    • summing interaction to all destinations from a given origin:
      • Iij = Ei
    • that is, total interaction from an origin will always equal Ei regardless of the number and locations of destinations
    • flow will now be partially diverted from existing destinations to new ones
    • Ei is now the total outflow, can be set equal to the total of observed outflows from origin i
    • the Huff model is consistent with the axiom of Independence of Irrelevant Alternatives (IIA)
    • the ratio of flows to two destinations from a given origin is independent of the existence and locations of other destinations
    Because of its production constraint, the Huff model is very popular in retail analysis
  • it is often desirable to predict how much business a new store will draw from existing ones
  • e.g. how much will a new mall draw business away from downtown?
  • Other "what if" questions:
    • population of a neighborhood increases by x%
    • ethnic mix of a neighborhood changes
    • a new bridge is constructed
    • an earthquake takes a freeway out of operation
    • a mall adds new space
    • an anchor store moves out
    • a store changes its signage
    Site modeling for retail applications
  • three major areas:
  • use of the spatial interaction model
  • analog techniques
  • regression models
    • the business done by a new store or an old store operating under changed circumstances is best estimated by finding the closest analog in the chain
    • requires a search
    • criteria include:
    • physical characteristics of each store
    • intangibles such as management, signage
    • local market area
    • a GIS can help compare market areas (local densities, street layouts, traffic patterns)
    • a multi-media GIS can help with the intangibles
    • bring up images of site, layout, signage...
    • identify all of the factors affecting sales, and construct a model to predict based on these factors
    • an enormous range of factors can affect sales
    • some factors are exogenous
    • determined by external, physical, measurable variables
    • some of these travel with the store if it moves (site factors), others are attributes of place (situation factors)
    • other factors are endogenous
    • determined by crowding, types of customers, trends, advertizing
    • unpredictable, determined by the state of the system
    Exogenous factors:
    • site layout - on a corner? parking spaces, etc.
    • inside layout
    • trade area - number of households in primary, etc
    • characteristics of neighborhood
    Example model:

    Sales per 2-week period for convenience store:

          + 4542 if gas pumps on site
          + 3172 if major shopping center in vicinity

          + 3990 if side street traffic is transient

          + 3188 per curb cut on side street

          + 2974 if store stands alone

          - 1722 per lane on main street

    • use of surrogate variables
    • problems in use of model for prediction in planning
    Calibration of the spatial interaction model
    • many different circumstances
    • major issues involved in calibration
    • specific tools are available
    • possible to use standard tools in e.g. SAS, GLIM
    • calibration possible using aggregate flows or individual choices
  • transformations to make the right hand side of the equation a linear combination of unknowns, the left hand side known
  • Linearization of the unconstrained model:
    • suppose the Ei are known, the Aj unknown
    • the constant a can be absorbed into the Aj (i.e. find aAj)
    • suppose we use the negative power deterrence function
    •  Iij = Ei Aj / dijb
    • move the Ei to the left:
      • Iij/Ei = Aj / dijb
  • take the logs of both sides:
  •  log (Iij/Ei) = log Aj - b log dij
    •   now a trick - introduce a set of dummy variables uijk, set to 1 if j=k, otherwise zero:
     log (Iij/Ei) = uij1 log A1 + uij2 log A2 + ... - b log dij
    • now the left hand side is all knowns, the right hand side is a linear combination of unknowns (the logs of the As and b)
    • the model can now be calibrated (the unknowns can be determined) using ordinary multiple regression in a package like SAS
    • it may be easier to avoid linearizing altogether by using the nonlinear regression facilities in many packages
    The objective function:
    • normally, we would try to maximize the fit of the observed and predicted interactions
    • linearization changes this
    • e.g. we minimize the squared differences between observed and predicted values of log (Iij/Ei) if ordinary regression is used on the linearized form above
    • this is easy in practice, but makes no sense
    • intuitively, an error of 30 in a prediction of 1000 trips is much more acceptable than an error of 30 in a prediction of 10 trips
    • these ideas are formalized in the technique of Poisson regression, which assumes that Iij is a count of events, and sets up the objective function accordingly
    • the function minimized to get a good fit is roughly the difference between observed and predicted, squared, divided by the predicted flow






    Section 4 - Spatial dependence - Looking at causes and effects in a geographical context:

    • Spatial autocorrelation - what is it, how to measure it with a GIS.
    • The independence assumption and what it means for modeling spatial data.
    • Applying models that incorporate spatial dependence - tools and applications.
    Two concepts:

    Spatial dependence

    • what happens at one place depends on events in nearby places
    • all things are related but nearby things are more related than distant things (Tobler's first law of geography)
    • positive spatial dependence:
    • nearby things are more alike than things are in general
    • negative spatial dependence:
    • nearby things are less alike than things are in general
    • conceptual problems with negative spatial dependence
    • e.g. the chessboard
    • spatial autocorrelation measures spatial dependence
    • an index, rather than a parameter of a process
    • dependence between discrete objects, or dependence in a continuous field?
    • a world without positive spatial dependence would be an impossible world
      • impossible to map
      • impossible to describe, live in
      • hell is a place with no spatial dependence
    Geary index:
  • compares the squared differences in value between neighboring objects with overall variance in values
  • Moran index:
    • calculates the product of values in neighboring objects
    • related to Geary but not in a simple algebraic sense


    Calculation of the Geary index of spatial autocorrelation


      a is the mean of x values

      wij = 1 if i,j adjacent, else 0

      c is 1 if neighbors vary as much as the sample as a whole

      c < 1 if neighbors are more similar than the sample as a whole (positive dependence)

      c > 1 if neighbors are less similar (negative dependence)

    c = 3 x 16 / (2 x 10 x 2) = 48 / 40 = 1.2
    • i.e. neighboring values are slightly more similar than one would expect if the values were randomly allocated to the four areas
    Continuous space
  • see the discussion of variograms and Kriging
  • the term geostatistics is normally associated with continuous space, spatial statistics more with discrete space
    Measures of spatial dependence can be calculated in GIS:
    • Idrisi calculates autocorrelation over a raster
    • code has been written to calculate autocorrelation in ARC/INFO (see NCGIA Technical Paper 91-5)
    More extensive codes have been written using the statistical packages, e.g. MINITAB, SAS
    • contact Dan Griffith, Syracuse University; Luc Anselin, University of Illinois
    • some of these fail to take advantage of GIS capabilities, for generating input data and displaying output
    Spatial heterogeneity:
    • suppose there is a relationship between number of AIDS cases and number of people living in an area
    • the form of this relationship will vary spatially
    • in some areas the number of cases per capita will be higher than in others
    • we could map the constant of proportionality
    • spatial heterogeneity describes this geographic variation in the constants or parameters of relationships
    • when it is present, the outcome of an analysis depends on the area over which the analysis is made
    • often this area is arbitrarily determined by a map boundary or political jurisdiction

    Geographically weighted regression (GWR)

    fits a model such as y = a + bx
    but assumes that the values of a and b will vary geographically

    determines a and b at any point by weighting observations inversely by distance from that point



    Geographical brushing:

    • a technique of ESA
    • a user-defined window is moved over the map
    • analysis occurs only within the window
    Conventional analysis (analysis done aspatially, e.g. using a statistical package) assumes independence (no spatial dependence) and homogeneity (no spatial heterogeneity)
    • e.g. regression analysis assumes that the observations (cases) are statistically independent
    • this violates the first law of geography
    • in general, analysis in space is very different from conventional statistical analysis (although this is very often carried out on spatial data)
    An example:
    • the relationship between land devoted to growing corn and rainfall in a Midwestern state like Kansas
    • rainfall available at 50 weather stations
    • percent of land growing corn available for 100 counties
    • use a method of spatial interpolation to estimate rainfall in each county from the weather station data
    • plot one variable against the other, and perhaps fit a regression equation
    • how many data points are there?
    • the more data points, the more significant the results
    • 100 (the number of counties)?
    • 50 (the real number of weather observations)?
    • something in between?
    • more data points can be invented by intensifying the sample network using spatial interpolation, but no more real data has been created by doing so
    • both variables are strongly spatially autocorrelated, violating an assumption of regression
    • the significance of the analysis is now uncertain
    • methods of spatial regression try to overcome this problem in a systematic way
    • see Spacestat

    An example

    Crime and income in Los Angeles
    rate of car thefts (per sq km per year)

    median annual income in thousands

    per census tract

    5,000 observations
    b = increase in car thefts per sq km per thousand dollars median income
      = -0.22

    R2 = percentage of variation in car thefts explained by income
       = 0.26

    is this significant?

    is it significant at the 95% level of confidence?

    in a population of millions of census tracts, exhibiting the same range of rates of car thefts and median incomes, but no relationship between them (b = 0, R2 = 0), could a sample of 5,000 census tracts have exhibited the same
    degree of apparent relationship, or more, purely by chance?

    but, but, but...we don't have a random sample of a larger population
    there are only 5,000 tracts in LA and we have all there is

    A related issue - the MAUP

    • many statistics are reported by averaging or summing over polygons - e.g. populations of counties, average elevation
    • it is commonly necessary to interpolate such values to new polygons which do not coincide
    • e.g. from census tracts with known populations to school districts
    • source zones have known populations
    • populations of target zones are unknown
    • the best method of solving this problem is to create a continuous surface from the source data, then to integrate this surface to the new target areas
    Various assumptions can be made about the underlying surface:
    • density is constant within source zones
    • density is constant within target zones
    • density is constant within some third set of control zones
    • density varies smoothly (Tobler's Pycnophylactic interpolation)
    Analysis carried out on modifiable units can produce frightening results
    • e.g. Openshaw and Taylor
    • 99 counties of Iowa
    • two variables - % over 65, and % Republican
    • correlation for the counties was .3466
    Results of analysis using some alternative reporting zones:
      6 Republican-proposed congressional districts .4823

      6 Democrat-proposed congressional districts .6274

      6 existing congressional districts .2651

      6 urban/rural regional types .8624

      6 functional regions .7128

    By regrouping the counties into larger regions, Openshaw and Taylor were able to generate a vast range of outcomes of the analysis:
    • e.g. 48 regions - correlations between -.548 and +.886
    • e.g. 12 regions - correlations between -.936 and +.996
    What to do?
    • evaluate the range?
    • are we asking the right question?
    • is scale part of the question rather than a mere matter of implementation?






    Section 5 - Site selection - Locational analysis and location/allocation - Other forms of operations research in spatial analysis - Spatial decision support systems - Linking spatial analysis with GIS to support spatial decision-making:

    • Shortest path, traveling salesman, traffic assignment.
    • What is location/allocation, and where can it be applied?
    • Modeling the process of retail site selection. Criteria.
    • Electoral districting and sales territories.
    • What is an SDSS? What are its component parts? How does it compare to a GIS or a DSS? Why would you want one? Building SDSS.
    • Examples of SDSS use - site selection, districting.
    Methods of analysis on networks

    A spatial database can be used to support the solution of a variety of network problems, including optimal location, routing and vehicle scheduling

  • these include:
  • Routing:
          Shortest path problem
          Traveling salesman problem and variants
          Trans-shipment problem
          Hitchcock transportation problem
          Traffic assignment problem
          P-median problem
          Coverage problems
          Minimax location problems
          Plant location problem
    • many of these are implemented in current GIS, e.g. network extensions in ArcGIS, TransCAD and GIS*PLUS from Caliper Corp.
      • additional code interfaced with GIS has been developed
    • solution of many of these problems raises a number of issues of data modeling
      • some of these have been raised earlier in the example of modeling a street network for shortest path analysis
    Example: Brine disposal in the Petrolia, Ontario oil field
    • oil extraction from the field generates large quantities of waste fluid
    • there are 14 active producers in the field, each operating a single extraction facility
    • the only effective method of disposal is by pumping to a formation below the oil producing layer
    • options include:
    • a single, central disposal facility
    • requiring each producer to install a facility
    • some intermediate configuration of shared facilities
    One disposal well per producer:
    • maximum capital cost
    • zero transport cost
    One central facility:
    • minimum capital cost
    • maximum transport cost
    The location-allocation problem:
  • find locations for one or more central facilities and allocate producers to them in order to minimize the total of capital and transport costs
    Two alternatives for transport of waste brine to central facilities: pipe and truck.

    Pipe cost:

      C1 = A D0 / B + C
        A installed cost per metre

        D0 distance in metres

        B pipe life in years

        C pump cost per year

    Truck cost:
      C2 = E D V0 / (365 F) + Q V0 (H + D0/(1000P)) / G
        E holding period, days

        D holding capacity, m3

        V0 volume of brine, m3 per year

        F life of holding capacity, years

        Q truck cost, $/hour

        H time to load and unload truck

        P speed of truck in km/hour

        G truck load, m3

    Disposal well cost:
      New well - $50-$75,000

      Success rate 60-80%

      Buildup of residual hydrocarbons

      Corrosion of pipes

    Slide: Petrolia area

    Slide: Transport cost functions

    GIS implementation:

    Network of streets and rights of way - potential routes for trucks/pipes

    Links with attributes of length

    Nodes with attributes of volume produced - producer sites plus other potential well locations

    GIS database with nodes and links and associated attributes:

    • data input functions (editing)
    • data display - graphics, plots
    • storage of geographic data
    • provides data to the analysis module
    Analysis module interacting with GIS database
    • obtains nodes and links from the GIS
    • performs analysis, reports results directly to the user
    • includes several heuristic methods for solving the optimization problem
    • allows the user access to the display/analysis functions of the GIS
    An analysis module supported by a GIS database provides a spatial decision support system (SDSS) tailored to specific, advanced forms of spatial analysis

    Location-allocation analysis module:

    1. Finds shortest paths between points on network (could be a GIS function)
    2. Define and modify model parameters
    3. Use paths and parameters to calculate transport costs
    4. Search for optimum solution using add, drop and swap heuristics
    5. Evaluate solutions and print results

    Facility cost
    Transport cost
    $/m3 brine
    $/m3 oil
    All producers
    Central by truck
    Central any nodes
    Central any producers
    Existing disposal wells

    % pipe
    % truck
    Optimum sites
    Cost $000s
    Pipe cost A
    Pipe life B
    Pump cost C
    Well cost R
    Life of well S
    Brine ratio U
    Other examples of complex GIS-based analysis:
      Vehicle routing and scheduling

      Traffic modeling

      Corridor location for pipelines/powerlines/highways

      Runoff modeling based on DEM

      Load balancing in electrical networks

    Spatial search

    Boolean search

    Search through an attribute table to find objects satisfying a set of criteria Example:

    Forest stands - area object type, non-overlapping

    Attributes:   area (reserved)   species


    For each stand, compare species and age to desired criteria. Dissolve and merge boundaries between neighboring stands if both fit the criteria

    Use tables to obtain estimated yield for given species/age and area

    Generate a map showing merged groups of cuttable stands, with new IDs, plus a table showing yield for each group.

    Topological overlay

    Two or more coverages can be overlayed to obtain new object types with concatenated attributes. This allows Boolean search and related operations to be conducted on multiple object types, i.e. with more information available.


    Add soil moisture information, from a separate coverage, to the criteria used to identify cuttable stands.   Buffer zone generation

    A buffer zone allows Boolean searches to include criteria based on distance


    A stand is cuttable only if it is not less than 200m from the nearest stream/lake In many cases it is not possible to reduce all criteria to simple yes/no requirements. e.g. from those stands satisfying criteria 1 and 2, select that stand which minimizes total cost (sum of criteria 3, 4 and 5)   When all non-conditional criteria are commensurate (dollars) they can be summed.

    In many cases criteria are not commensurate and cannot be summed.


    1. Timber extraction/hauling costs - direct $ costs

    2. Environmental cost of extraction - intangible

    3. Road construction cost - $, but long-term benefits

    Decision Theory provides methods for determining:

    Single Utility Functions (SUFs) for each criterion

    Multiple Utility Functions (MUFs) to combine criteria.

    Both SUFs and MUFs can be determined by experimental designs involving groups of decision-makers

    Decision theoretic methods can be incorporated into GIS technology. The GIS is used to evaluate the criteria for each alternative, then to weigh them using SUFs and MUFs to arrive at a decision.

    A model for spatial analysis with a GIS

    Example of multi-stage GIS analysis

    Generation of a Recreation Opportunity Spectrum (ROS) map for a National Forest 1:24,000 quad (7.5 minute)

    Problem: generate zones and associated ROS classes for Forest Service land based on distance from transportation features, with urban exclusions.

    Data needed:

    D1: Roads and railways (1:24,000) - line objects

    D2: Forest Service ownership map (1:24,000) - area objects

    D3: City and town boundaries map (1:24,000) - area objects

      GIS functions:
    Reclassify attributes (B2)

    Dissolve and merge (B3)

    Generate corridors (B24)

    Topological overlay (B50)

    Measure size of areas (B35)

    Centroid calculation and sequential numbering (B8)

    Plot (A12)

    Create list and report (B1)

    Steps to make product: 1. Using the forest service ownership data, reclassify area objects as forest land / not forest land. (B2)

    2. Dissolve boundaries between polygons with the same value of the forest land / not forest land attribute, and merge polygons (B3)

    3. Using the transportation map, generate corridors 0.5 miles wide around all roads and railways. (B24)

    4. Using the transportation map, generate corridors 1.0 miles wide around all roads and railways. (B24)

    5. Topologically overlay the results of 2, 3 and 4 and concatenate the attributes, to obtain polygons with the following attributes:

    forest land / not forest land

    within/outside 0.5 mile corridor

    within/outside 1.0 mile corridor (B50)

    6. Topologically overlay the urban boundary map, and concatenate attributes, adding urban/non-urban to the list in 5. (B50)

    7. Reclassify the area objects resulting from 6 according to the following rules:

     Class        Criteria

    Null                 not forest land

    RMU               forest land and urban

    SPM                forest land, non-urban and within 0.5 miles of road/rail

    SPN                 forest land, non-urban, outside 0.5 mile and inside 1.0 mile corridors

    P                      forest land, non-urban, outside both 0.5 mile and 1.0 mile corridors (B2)

      8. Dissolve and merge adjacent polygons with the same class (B3)   9. Measure areas of polygons resulting from 8 (B35)

    10. Reclassify polygons of class SPM according to the following rules:

      Class        Criteria

    SPM             Areas of less than 2500 acres

    RN                Areas of more than 2500 acres (B2)

    11. Calculate centroids and sequentially number polygons (B8)

    12. Plot classified polygons with classes and numbers assigned in 11, plus roads and railways and urban areas (A12)

    13. Create a list of all polygons, with IDs, areas and classes. (B1)

    Summary sequence of operations:

    Initial data sets: D1, D2, D3

    1. B2 on D2 -> E1

    2. B3 on E1 -> E2

    3. B24 on D1 -> E3

    4. B24 on D1 -> E4

    5. B50 on E2, E3, E4 -> E5

    6. B50 on E5, D3 -> E6

    7. B2 on E6 -> E7

    8. B3 on E7 -> E8

    9. B35 on E8 -> E9

    10. B2 on E9 -> E10

    11. B8 on E10 -> E11

    12. A12 on E11, D1, D3

    13. B1 on E11

    Many GIS applications require complex decision rules in reclassification operations.

    e.g. finding the most cuttable stand of timber: Criterion 1.         Area of stand > 100 acres (B35)

    2.         More than 100m from stream/lake (B24)

    3.         Subrules based on slope, aspect and soil mechanics determine method of timber extraction.

    4.         Analysis of existing roads and terrain leads to estimates of costs of constructing new
                roads and hauling timber to mill

    5.         Subrules based on costs of replanting, silviculture


    • GIS technology useful in designing sales areas, analyzing trade areas of stores
    • similar applications occur in politics
    • design of voting districts (apportionment, gerrymandering) has enormous impact on outcome of elections
    • major interest in reapportionment after 1990 census
    • GIS applications in these areas are still at early stage
    Characteristics of application area
    • scale:
    • street centerline, census reporting zones - i.e. 1:24,000 and smaller
    • data at block group/enumeration district scale (250 households) is needed for locating smaller commercial operations like gas stations and convenience stores
    • data at census tract scale (2,000 households) is good for the location of larger facilities like supermarkets and fast food outlets
    • data sources:
    • much reliance on existing sources of digital data
    • especially TIGER and DIME
    • similar data available in other countries
    • additional data added to standard datasets by vendors
    • e.g. updating TIGER files by digitizing new roads, correcting errors
    • e.g. adding ZIP code boundaries, locations of existing retailers
    • functionality:
    • dissolve and merge operations, e.g. to build voting districts out of small building blocks
    • modeling, e.g. to predict consumer choices, future population growth
    • overlay operations, e.g. to estimate populations of user-defined districts, correlate ZIP codes with census zones
    • point in polygon operations, e.g. to identify census zone containing customer's residence
    • mapping, particularly choropleth and point maps of consumers
    • geocoding, address matching
    • data quality:
    • more concern with accuracy of statistics, e.g. population counts, than accuracy of locations
    Types of applications
    • districting
    • designing districts for sales territories, voting
    • objective is to group areas so that they have a given set of characteristics
    • "geographical spreadsheets" allow interactive grouping and analysis of characteristics
    • e.g. Geospreadsheet program from GDT
    • site selection
    • evaluating potential locations summarizing demographic characteristics in the vicinity
    • e.g. tabulating populations within 1 km rings
    • searching for locations that meet a threshold set of criteria
    • e.g. a minimum number of people in the appropriate age group are within trading distance
    • market penetration analysis
    • analyzing customer profiles by identifying characteristics of neighborhoods within which customers live
    • targeting
    • identifying areas with appropriate demographic characteristics for marketing, political campaigns
    • many data vendors and consulting companies active in the field, many large retailers
    • no organization unique to the field
    • American Demographics is influential magazine
    Districting example
    • GIS has applications in design of electoral districts, sales territories, school districts
    • each area of application has its own objectives, goals
    • this example looks at designing school districts
    • the Catholic school system of London, Ontario, Canada provides elementary schools for Kindergarten through Grade 8 to a city of approx. 250,000
    • about 25% of school children attend the Catholic system
    • 27 elementary schools were open prior to the study
    • population data is available for polling subdivisions from taxation records
    • approx. 700 polling subdivisions have average population of 350 each
    • forecasts of school age populations are available for 5, 10, 15 years from the base year at the polling subdivision level
    • children are bussed to school if their home location is more than 2 miles away, or if the walking route to school involves significant traffic hazard
    • minimal changes to the existing system of school districts
    • minimal distances between home and school, and minimal need for bussing
    • long-term stability in school district boundaries
    • preservation of the concepts of community and parish - if possible a school should serve an identifiable community, or be associated with a parish church
    • maintenance of a viable minimal enrollment level in each school, defined as 75% of school capacity and > 200 enrollment
    Technical requirements
    • digitized boundaries of the polling subdivision "building blocks"
    • an attribute file of building blocks giving current and forecast enrollment data
      • for forecasting, we must include developable tracts of land outside the current city limits, plus potential "infill" sites within the limits
      • 748 polygons
      • development tracts are isolated areas outside the contiguous polling subdivisions
      • infill sites are shown as points
    • the ability to merge building blocks and dissolve boundaries to create school districts
      • school districts are not required to be conterminous
      • if necessary a school can serve several unconnected subdistricts
    • a table indicating whether walking or bussing is required for each building-block/school combination
    Slide: City and development areas

    Current districts

  • "starbursts" show allocations of building blocks to 29 current schools (includes two special education centers)
  • note bussed areas in NW and SW - separate enclaves of recent high-density housing allocated to distant schools
  • this strategy allows an expanding city to deal with
  • dropping school populations in the core leading to an excess of capacity
  • rising school populations in the periphery but lack of funds for new school construction
  • without constantly adjusting boundaries
    Slide: Current districts

    Projections of enrollment based on current school districts

    • rapid increase in developing areas, e.g. St Joseph's (#3), St Thomas More (#4) NW
    • decrease in maturing areas of periphery, e.g. St Jude's (#8) - SW area
    • rejuvenation in some inner-city schools due to infilling, e.g. St Martin's (#15) - lower center
    • stagnation in other inner-city schools, e.g. St Mary's (#17), decline e.g. St John's (#14) - center
    • general strategy - begin with current allocations, shift building blocks between districts in order to satisfy objectives
    • requires interaction between graphic display and tabular output
    • quick response to "what if this block is reassigned to the school over here?"
    • implementation allowed School Board members to make changes during meetings, observe results immediately
    • using map on digitizer tablet, tables on adjacent screen
    • one of the alternative plans developed
    • note:
      • assumes closure of 6 schools
      • rise in enrollment as percent of capacity
      • stability of projections through time
      • reduction in number of "non-viable" schools (<200 enrollment)
      • increase in percent not assigned to nearest school
      • increase in average distance traveled

    Slide: Projected enrollments

    Slide: Planned enrollments