Processes & Patterns

Processes & Patterns Processes & Patterns

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. 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. Processes & Patterns

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 Deterministic Processes Processes & Patterns

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. Processes & Patterns

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 Processes & Patterns

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. Processes & Patterns

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. Processes & Patterns

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. Processes & Patterns

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). Processes & Patterns

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)? Processes & Patterns

Predicting the Pattern Generated by a Process Processes & Patterns

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

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? Predicting the Pattern Generated by a Process Processes & Patterns

Predicting the Pattern Generated by a Process 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. 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. Processes & Patterns

Processes & Patterns