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

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

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

14. 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

15. 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

16. 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

17. 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?
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Processes & Patterns

18. 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

19. 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

20. 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

21. 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

22. 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

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

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

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

26. 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

27. 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