Pattern Statistics

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Pattern Statistics

• Michael F. Goodchild
• University of California
• Santa Barbara

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Outline

• Some examples of analysis
• Objectives of analysis
• Cross-sectional analysis
• Point patterns

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What are we trying to do?

• Infer process
• processes leave distinct fingerprints on the
  landscape
• several processes can leave the same fingerprints
• enlist time to resolve ambiguity
• invoke Occam's Razor
• confirm a previously identified hypothesis

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Alternatives

• Expose aspects of pattern that are otherwise
  invisible
• Openshaw
• Cova
• Expose anomalies, patterns
• Convince others of the existence of patterns,
  problems, anomalies

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Cross-sectional analysis

• Social data collected in cross-section
• longitudinal data are difficult to construct
• difficult for bureaucracies to sustain
• compare temporal resolution of process to
  temporal resolution of bureaucracy
• Cross-sectional perspectives are rich in context
• can never confirm process
• though they can perhaps falsify
• useful source of hypotheses, insights

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What kinds of patterns are of interest?

• Unlabeled objects
• how does density vary?
• do locations influence each other?
• are there clusters?
• Labeled objects
• is the arrangement of labels random?
• or do similar labels cluster?
• or do dissimilar labels cluster?

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First-order effects

• Random process (CSR)
• all locations are equally likely
• an event does not make other events more likely
  in the immediate vicinity
• First-order effect
• events are more likely in some locations than
  others
• events may still be independent
• varying density

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Second-order effects

• Event makes others more or less likely in the
  immediate vicinity
• clustering
• but is a cluster the result of first- or
  second-order effects?
• is there a prior reason to expect variation in
  density?

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Testing methods

• Counts by quadrat
• Poisson distribution

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Deaths by horse-kick in the Prussian army

• Mean m 0.61, n 200

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Towns in Iowa

• 1173 towns, 154 quadrats 20mi by 10mi

Chisquare with 8 df 12.7 Accept H0
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Distance to nearest neighbor

• Observed mean distance ro
• Expected mean distance re 1/2?d
• where d is density per unit area
• Test statistic

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Towns in Iowa

• 622 points tested
• 643 per unit area
• Observed mean distance 3.52
• Expected mean distance 3.46
• Test statistic 0.82
• Accept H0

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But what about scale?

• A pattern can be clustered at one scale and
  random or dispersed at another
• Poisson test
• scale reflected in quadrat size
• Nearest-neighbor test
• scale reflected in choosing nearest neighbor
• higher-order neighbors could be analyzed

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Weaknesses of these simple methods

• Difficulty of dealing with scale
• Second-order effects only
• density assumed uniform
• Better methods are needed

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K-function analysis

• K(h) expected number of events within h of an
  arbitrarily chosen event, divided by d
• How to estimate K?
• take an event i
• for every event j lying within h of i
• score 1

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Allowing for edge effects
score lt 1
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The K function

• In CSR K(h) ?h2
• So instead plot

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What about labeled points?

• How are the points located?
• random, clustered, dispersed
• How are the values assigned among the points?
• among possible arrangments
• random
• clustered
• dispersed

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Moran and Geary indices
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