Fundamentals: Maps as Outcomes of Processes

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Chapter 3

• Fundamentals Maps as Outcomes of Processes

By Mindy Syfert
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Processes and Patterns

• Maps are considered as outcomes of processes
• A map is one of the possible patterns that might
  have been generated by a hypothesized process.
• Spatial patterns are potential realizations

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Processes and the Patterns They Make

• Deterministic Processes
• Often mathematical
• The spatial pattern produces the same outcome at
  each location
• EX. z 2x 3y
• Stochastic Processes
• Random element included to make the process
  unpredictable
• Thus, many different patterns can result

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Stochastic Process

• Independent Random Process (IRP) / Complete
  Spatial Randomness (CSR)
• When points are randomly placed so that each
  location has equal probability of receiving a
  point and the positioning of any point is
  independent of the positioning of any other
  points
• Ex Using the dice to place points in a grid.
• This spatial process described mathematically
• P(k,n,x) (n k) (1/x)k (x-1/x)n-k
• This is a binomial expression and is not very
  practiced, but Poisson distribution can be a good
  approximate

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Two Ways Real Processes Differ From IRP/CSR

• First-Order effect variations in the density of
  a process across space
• Ex some oak species prefer soil derived from
  limestone and are clustered in this area more
  than in a neighboring soil derived from mudstone
• Second-Order effect- interaction between
  locations
• Ex Woolly Adelgid infesting and killing Eastern
  Hemlocks- nearby trees are infested before ones
  further away

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Distinct Aspects of Spatial Patterns

• First-order and second-order effects shift a
  process from being stationary to changing over
  space

• Weakness close to impossible to distinguish from
  variation in the environment or interaction
  between point events by the analysis of spatial
  data

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More processes

• Anisotropic directional effects in spatial
  variation of data
• Ex Again, Woolly Adelgid infestation on Eastern
  Hemlocks- the infestation rate is directional
  from southern US to northern US
• Isotropic- NO directional effects in spatial
  variation of data
• Ex If the Woolly Adelgid infestation had no
  direction, the infestation rate would simply
  spread outward

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Stochastic Processes in lines

• More difficult to figure out the frequencies of
  path lengths for IRP than for point patterns
• Reasons for this
• Points patterns are discrete with equal
  probability and path lengths have a continuous
  probability density function
• Path lengths depend on the shape of where they
  are crossing
• Statistician pay little attention to
  path-generating processes
• However, IRP for path lengths are useful for line
  direction
• Ex. geologists looking at orientations of the
  particles to indicate processes- for instance,
  sand dunes

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Key ideas

• Any map can be regarded as the outcome of a
  spatial process
• Although spatial processes can be deterministic
  (one outcome) we often think in stochastic
  processes, which include random elements (many
  different patterns).

• IRP idea can be applied to all entity types
  (point, line, area and field)
• IRP/CSR allows mathematics to be used for
  long-run average outcomes of spatial processes

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Questions

• What makes the stochastic process different from
  the deterministic process?
• Define IRP/CSR when dealing with a point pattern.
• Explain the difference between first-order and
  second-order effects.