Processes & Patterns Spatial Data Models 5/10/2019 © J.M. Piwowar

Maps: From Processes to Patterns Processes  Patterns Maps have an inherent ability to suggest patterns in the phenomena they represent. An observed map pattern is only one of the possible patterns that might have been generated by a process. Patterns provide clues to the process(es) that created them. A spatial process is a description of how a spatial pattern might be generated. © J.M. Piwowar Processes & Patterns

Deterministic Processes A process that always produces the same pattern. e.g. z = 2x + 3y The value of z at location (3,4) will always be 18, no matter how many times the process is realized. © J.M. Piwowar Processes & Patterns

Deterministic Processes Spatial Data Models 5/10/2019 Deterministic Processes Universal Soil Loss Equation (USLE) A=R*K*L*S*C*P A = Computed soil loss per unit area R = Rainfall factor K = Soil erodibility factor L = Slope length factor S = Slope percent factor C = Cover and management factor P = Support practice factor NDVI increases with increases in biomass Supervised classification: given the same set of training data, the classification will always come out the same Geometric correction: given the same ground control points, the correction will always come out the same © J.M. Piwowar Processes & Patterns © J.M. Piwowar

Spatial Data Models 5/10/2019 Stochastic Processes A process whose outcome is subject to some degree of random variation. e.g. z = 2x + 3y + random The value of z at location (3,4) will be unpredictable each time the process is realized. Notice that this map (pattern) isn’t random: There is still a trend from southwest to northeast, but it has a local chance component added © J.M. Piwowar Processes & Patterns © J.M. Piwowar

Spatial Data Models 5/10/2019 Stochastic Processes With 100 cells, there are 2100 different realizations (patterns) from this process: >1,000,000,000,000,000,000,000,000,000,000 There are 2^100 = 1.27 x 10^30 possible realizations of this pattern (with 100 cells). © J.M. Piwowar Processes & Patterns © J.M. Piwowar

Classic Stochastic Processes Spatial Data Models 5/10/2019 Classic Stochastic Processes Independent Random Process (IRP) Complete Spatial Randomness (CSR) Each map is a realization of a process that selects random points from a fixed, uniform probability distribution. If the world were completely random, geography would be irrelevant. © J.M. Piwowar Processes & Patterns © J.M. Piwowar

Independent Random Process (IRP) Spatial Data Models 5/10/2019 Independent Random Process (IRP) Equal Probability Any point has an equal probability of being in any location. Any area of the map has an equal probability of receiving a point. Independence The location of any point is independent of the location of any other point. If the world were completely random, geography would be irrelevant. © J.M. Piwowar Processes & Patterns © J.M. Piwowar

Processes & Patterns It is the process that is random, not the pattern. Maps produced by stochastic processes often display spatial patterns. Spatial patterns in reality rarely occur due to chance. © J.M. Piwowar Processes & Patterns

Processes & Patterns In spatial analysis & modelling we: Spatial Data Models 5/10/2019 Processes & Patterns In spatial analysis & modelling we: Observe patterns and try to determine the process that could have created them (an inductive approach); or Understand a process and try to determine the range of patterns that can be realized from it (a deductive approach). We will examine the 2nd option first. © J.M. Piwowar Processes & Patterns © J.M. Piwowar

Predicting the Pattern Generated by a Process Example: A study area divided into quadrats (equal-sized, non-overlapping regions) . What are the observed frequencies of events (number of events/quadrat)? © J.M. Piwowar Processes & Patterns

Predicting the Pattern Generated by a Process Spatial Data Models 5/10/2019 Predicting the Pattern Generated by a Process 0.125=1/8 0.25=2/8 0.375=3/8 © J.M. Piwowar Processes & Patterns © J.M. Piwowar

Predicting the Pattern Generated by a Process Can we determine (statistically) what the expected frequency distribution should be? © J.M. Piwowar Processes & Patterns

Predicting the Pattern Generated by a Process e.g. there are 10 events distributed over 8 quadrats. What are the probabilities of having 0, 1, 2 … 10 events occurring in a particular quadrat? © J.M. Piwowar Processes & Patterns

Predicting the Pattern Generated by a Process What is the probability of having a specific event occurring in a particular quadrat? P(event A in shaded quadrat) = 1/8. © J.M. Piwowar Processes & Patterns

Predicting the Pattern Generated by a Process What is the probability of not having a specific event occurring in a particular quadrat? P(event A not in shaded quadrat) = 7/8. © J.M. Piwowar Processes & Patterns

Predicting the Pattern Generated by a Process What is the probability of having a specific event to be the only event occurring in a particular quadrat? © J.M. Piwowar Processes & Patterns

Predicting the Pattern Generated by a Process What is the probability of having a specific event to be the only event occurring in a particular quadrat? © J.M. Piwowar Processes & Patterns

Predicting the Pattern Generated by a Process What is the probability of having any 1 of the 10 events to be the only event occurring in a particular quadrat? © J.M. Piwowar Processes & Patterns

Predicting the Pattern Generated by a Process What is the probability of having any 2 of the 10 events to be the only events occurring in a particular quadrat? How many possible combinations of choosing 2 events from the 10 are there? © J.M. Piwowar Processes & Patterns

Predicting the Pattern Generated by a Process What is the probability of having any 2 of the 10 events to be the only events occurring in a particular quadrat? How many possible combinations of choosing 2 events from the 10 are there? How many possible combinations are there of choosing k events from a set of n events? © J.M. Piwowar Processes & Patterns

Predicting the Pattern Generated by a Process Spatial Data Models 5/10/2019 Predicting the Pattern Generated by a Process The expected quadrat count distribution for IRP/CSR conforms to the binomial distribution. This can also be approximated with the Poisson distribution (it is easier to calculate). © J.M. Piwowar Processes & Patterns © J.M. Piwowar

Predicting the Pattern Generated by a Process © J.M. Piwowar Processes & Patterns

Predicting the Pattern Generated by a Process © J.M. Piwowar Processes & Patterns

Predicting the Pattern Generated by a Process © J.M. Piwowar Processes & Patterns

Conclusions There is good agreement between the observed and expected frequency distributions. The expected frequency distribution is an independent random process (IRP). Therefore, our observed point distribution is a realization of a random process. © J.M. Piwowar Processes & Patterns

Summary It is possible to describe a spatial process mathematically. We can predict the spatial pattern generated by the independent random process (IRP) and use this to determine if an observed pattern is a likely realization of that process. © J.M. Piwowar Processes & Patterns