Point Pattern Analysis

Question 1

Methods of PPA

Answer 1

• Quadrat Analysis and & Poisson Distribution
• Nearest Neighbour
• Ripley’s K-function
• Spatial Autocorrelations (LISA)

Question 2

PPA

Answer 2

• Only concerned with location
• Indication of underlying spatial process
• Exploratory
• May lead to spatial regression and geostatistical analysis
• Ex. Crime data (burglaries), Biology (bird nests), Epidemiology (disease incidence)

Question 3

PPA is only concerned with what?

Answer 3

• Location
• establish pattern of occurrence and evaluate possible causes
• Don’t need what the value is until the stats stage

Question 4

John Snow

Answer 4

• Not GoT
• Modern epidemiology and spatial pattern analysis
• Cholera in London, deaths at address and pattern
• Linked deaths to contaminated wells, not air transmission

Question 5

What is point pattern analysis needed for?

Answer 5

Objective, quantitative measures of spatial pattern because visual interpretation is not enough
- Ex. Crime analysts cannot necessarily pick out true clusters of crime just by looking at a map

Question 6

What is the Null hypothesis?

Answer 6

• No pattern present

- Random

Question 7

What is the Alternative hypothesis?

Answer 7

• Pattern present

- Some stats can tell if clustered or dispersed

Question 8

What are the must haves for PPA?

Answer 8

• Proper coordinates (Location, not centroid or polygon, or areal units, or ‘representative over area)
• Proper projection (preferably preserve distances)
• Study are ‘objectively determined’ (recall edge effects, MAUP)

Question 9

What can projections distort/preserve?

Answer 9

Angles, Area, Shape, or Distance

Question 10

Shape, examples for Physical Geography and Human Geography

Answer 10

• Physical: Woodlands rectangles

- Human: Administrative boundary polygons

Question 11

Study Area: Why is it better to have geometrically regular shapes?

Answer 11

• Easier calibration and model fit

Question 12

What are some possible subsequent analysis for PPA?

Answer 12

• Trend Surface

- Issue -> Edge effects

Question 13

Why must distance be preserved in projections?

Answer 13

• Result can be skewed if distance not correct

Question 14

Same area, different phenomena

Answer 14

Attribute

Question 15

Different area, same phenomenon

Answer 15

Location

Question 16

PPA Exploration prereqs?

Answer 16

• Scatter plot
• Visual (location, binary data)
• Outliers (measurement error)

Question 17

Poisson Distribution

Answer 17

• Compare observations to poisson (observed vs. expected if random)
• Probability of an event happening rarely, if at all, and if it does occur, time and place of occurrence are independent and random
• No spatial or temporal autocorrelation

Question 18

Poisson eqn

Answer 18

Mu = sigma^2

Question 19

CSR

Answer 19

Complete Spatial Randomness

- Poisson density function

Question 20

Poisson Density Function

Answer 20

p (x) = e^-lambda x lambda^x/x!

• Lambda = density = mean occurrence for time unit
• ! (factorial) = number of permutations of X

Question 21

Spatial Poisson Distribution ‘Poisson Process’

Answer 21

• No interactions btwn subareas, whether inhibitory or attraction
• No possibility of multiple groupings of individuals w/in each subarea (no point clusters)
• No tendency for neighbouring areas to display similar traits

Question 22

Chi-squared test, X^2

Answer 22

• Accepts or rejects null hypothesis
• = VMR (m - 1) or Sum of (observed - expected)^2/expected
• where m is number of quadrats
• Use table to get critical chi value
• is test <> critical value for significance level used

Question 23

What is a possible problem with too small quadrats?

Answer 23

• Leave black areas or create clustering

- Size Matters!

Question 24

PPA analysis /correlation of x, y data

Answer 24

• Locational, z is not important at this point
• More about spatial relationships on data distribution
• Look at z for more robust analysis as to why that spatial pattern may exist

Question 25

Steps of Quadrat Analysis

Answer 25

- The scoring of points that fall into a quadrat - Divide study area into quadrats - Count points in each quadrat - Sum the results - Compare with CSR (poisson) - Test against chi-square

Question 26

Optimal quadrat size

Answer 26

= (2 x Area)/n

Question 27

Quadrat analysis: Property

Answer 27

Mean = Variance

Question 28

VMR

Answer 28

Variance Mean Ratio = Variance/ mean cell frequency = n, number of points/obs divided by 'm' (number of cells/quadrats)

Question 29

VAR

Answer 29

- Number of points per cell | - Variance of the frequency

Question 30

QA: Index of dispersion

Answer 30

VAR = [((Sum of fi xi^2) - (Sum of fi xi)^2)/m]/ (m-1) - fi is frequency of cells with i number of points - xi is the number of points per cell - m is number of quadrats

Question 31

QA, Null hypothesis

Answer 31

VMR = 1 | - Point pattern is random

Question 32

QA, VMR does not = 1

Answer 32

Point pattern is not random

Question 33

QA, Alternative hypothesis

Answer 33

- VMR > 1, point pattern is more clustered than random | - VMR < 1, Point pattern is more dispersed than random

Question 34

NNA

Answer 34

- Nearest Neighbour Analysis - Measurement of distance btwn one point and its neighbours - Next step from PPA - Mean of observed distances is compared to an expected avg distance based on random poisson distribution - Null is random, alt is more or less dispersed than random - Test statistic to see if result is significant

Question 35

NNA eqn

Answer 35

Sum of Nearest Neighbour Distances (NND)/ number of obs/points - NND is distance to the next nearest point

Question 36

NND = 0

Answer 36

Perfect clustering | - The closer to 0 NND is, the more clustered it becomes

Question 37

NNDr = 1/2 x sq root of density

Answer 37

NNDr, Random clustering

Question 38

NND = 1/sq. root of density

Answer 38

Regular square lattice

Question 39

NNDd= 1.07453/sq. root of density

Answer 39

Regular hexagonal lattice

Question 40

NNA, adjustment for edge effects

Answer 40

- NND = 1/2 x sq. root (A/n) plus (0.514 plus 0.412/sq. root of n) x p/n - and Sigma^2 = 0.070 x (A/n^2) plus 0.035p((sq. root of A)/n^5/2)

Question 41

NNA, null hypothesis

Answer 41

Ho: NND = NNDr | - Point pattern is random

Question 42

NNA, Alternative hypotheses

Answer 42

- NND does not = NNDr, point pattern is not random - NND > NNDr, point pattern is more dispersed - NND < NNDr, point pattern is more clustered

Question 43

NNA Test statistic

Answer 43

Zn = (NND - NNDr)/ sigma of NND | - Sigma of NND = 0.26136/sq. root of (n x density)

Question 44

NNA, standard nearest neighbour index

Answer 44

``` R = NND/NNDr R = 2.149, perfect dispersion R = 1.5, more dispersed than random R = 1, random R = 0.5, more clustered than random R = 0, perfectly clustered ```

Question 45

Ripley's K-Function

Answer 45

- One step further than NNA, finds where distances are located - For Distance, d - Avg number of events found in circle of radius d around event, divided by mean intensity of the process - Mean intensity is number of events divided by study area

Question 46

Ripley's K-Function Formula

Answer 46

- K (d) = ...complex (Area/number of points^2 x sum of C(si, d) - C (si, d) is a circle of radius d centered at si - Take lambda =n/a out of equation by inverting to a/n - Uses search radius bands for number of points that fall in each ring - Averages number of points w/in distance

Question 47

What happens if K-Function distance is too small?

Answer 47

Clustering

Question 48

K-Function results

Answer 48

- ArcGIS outputs a graph, distance vs. K(d) - Expected looks like straight line - Clustered when observed greater than expected - Dispersed when observed less than expected - Cluster size and separation distance (flat line on graph?)

Question 49

What do you do after analyzing point patterns?

Answer 49

- Test for spatial autocorrelation | - (After Quadrat, NNA, and K-function)