Chapter 3 Fundamentals: Maps as Outcomes of Processes By: Mindy Syfert

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

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

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

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

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

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

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

Key ideas 1. Any map can be regarded as the outcome of a spatial process 2. Although spatial processes can be deterministic (one outcome) we often think in stochastic processes, which include random elements (many different patterns). 3. IRP idea can be applied to all entity types (point, line, area and field) 4. IRP/CSR allows mathematics to be used for long-run average outcomes of spatial processes

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.