Point Pattern Analysis: Randomness

12. Point Pattern Analysis: Randomness#

Topics

Where is the center of a geographic distribution
- Mean center
How features are distributed around its center
- Standard distance
- Standard deviation ellipse

Characterizing the randomness of a set of points in space
- Random
- Uniform (Dispersed)
- Clustered

Analysis on forms helps understand the processes that produced the patterns.
Clustered patterns:
- Contagion process where a particular location attracts a number of points.
Dispersed/uniform patterns:
- Some form of competition in space where points repel one another.
Random patterns:
- Results of random processes.
- Any point is equally likely to occur at any location.
- The position of any point is not affected by the position of any other point.

Initially developed by ecologists studying the spatial distribution of plants.
A regular grid is overlain over the region of interest.
The number of points in each quadrat is recorded.
The observed number of points per quadrat is compared with the theoretical distribution generated by a complete spatial random (CSR) process.
If the observed and theoretical distributions are similar, then the observed distribution could be generated by a random process.

For a random process, the probability of finding k points in a quadrat follows the Poisson distribution.
\(P(k) = \frac{\lambda^k e^{-\lambda}}{k!}\)
- \(\lambda\) = intensity (total points / total quadrats).

A common way to describe the pattern in quadrat analysis.
For a Poisson (random) distribution, the variance equals the mean (\(VMR = 1\)).
Clustered pattern:
- Many quadrats with 0 points and few with many points.
- Variance is greater than the mean (\(VMR > 1\)).
Uniform pattern:
- Most quadrats have a similar number of points.
- Variance is less than the mean (\(VMR < 1\)).

Result depends on quadrat size and orientation.
A measure of dispersion (VMR) is not a measure of the actual point locations within quadrats.
Small sample size issues.

Developed by Clark and Evans (1954).
Uses the distance between a point and its nearest neighbor to characterize the pattern.
Provides a more refined measure than quadrat analysis because it uses the actual coordinates of every point.

\(NNI = \frac{d(NN)}{d(random)}\)
\(d(NN)\): The average observed distance between each point and its nearest neighbor.
\(d(random)\): The expected average distance if the pattern were random.
- \(d(random) = \frac{1}{2\sqrt{\rho}}\) (where \(\rho\) is the density of points).

\(NNI = 1\): The pattern is random.
\(NNI < 1\): The pattern is clustered (observed distances are shorter than random).
\(NNI > 1\): The pattern is dispersed/uniform (observed distances are longer than random).
- Maximum \(NNI = 2.149\) (perfectly hexagonal/uniform distribution).

Clark and Evans proposed a Z-test for significance.
\(Z = \frac{d(NN) - d(random)}{SE}\)
- \(SE\) is the standard error of the random distance.
Null Hypothesis: The distribution is from a random distribution.
If the Z-value is outside the critical range (-1.96 to 1.96 for a 0.05 confidence level), the null hypothesis is rejected.

Nearest Neighbor:
- No quadrat size problem to be concerned with.
- Uses exact locations.
Both methods:
- Still have the issue of the study area boundary (the “edge effect”).
- Points near the edge might have a nearest neighbor outside the study area that is not counted.