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Geog418SptStts > Point Pattern Analysis > Flashcards

Point Pattern Analysis Flashcards

1
Q

Methods of PPA

A
  • Quadrat Analysis and & Poisson Distribution
  • Nearest Neighbour
  • Ripley’s K-function
  • Spatial Autocorrelations (LISA)
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2
Q

PPA

A
  • 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)
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3
Q

PPA is only concerned with what?

A
  • Location
  • establish pattern of occurrence and evaluate possible causes
  • Don’t need what the value is until the stats stage
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4
Q

John Snow

A
  • Not GoT
  • Modern epidemiology and spatial pattern analysis
  • Cholera in London, deaths at address and pattern
  • Linked deaths to contaminated wells, not air transmission
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5
Q

What is point pattern analysis needed for?

A

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

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6
Q

What is the Null hypothesis?

A
  • No pattern present

- Random

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7
Q

What is the Alternative hypothesis?

A
  • Pattern present

- Some stats can tell if clustered or dispersed

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8
Q

What are the must haves for PPA?

A
  • 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)
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9
Q

What can projections distort/preserve?

A

Angles, Area, Shape, or Distance

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10
Q

Shape, examples for Physical Geography and Human Geography

A
  • Physical: Woodlands rectangles

- Human: Administrative boundary polygons

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11
Q

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

A
  • Easier calibration and model fit
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12
Q

What are some possible subsequent analysis for PPA?

A
  • Trend Surface

- Issue -> Edge effects

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13
Q

Why must distance be preserved in projections?

A
  • Result can be skewed if distance not correct
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14
Q

Same area, different phenomena

A

Attribute

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15
Q

Different area, same phenomenon

A

Location

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16
Q

PPA Exploration prereqs?

A
  • Scatter plot
  • Visual (location, binary data)
  • Outliers (measurement error)
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17
Q

Poisson Distribution

A
  • 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
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18
Q

Poisson eqn

A

Mu = sigma^2

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19
Q

CSR

A

Complete Spatial Randomness

- Poisson density function

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20
Q

Poisson Density Function

A

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

  • Lambda = density = mean occurrence for time unit
  • ! (factorial) = number of permutations of X
21
Q

Spatial Poisson Distribution ‘Poisson Process’

A
  • 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
22
Q

Chi-squared test, X^2

A
  • 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
23
Q

What is a possible problem with too small quadrats?

A
  • Leave black areas or create clustering

- Size Matters!

24
Q

PPA analysis /correlation of x, y data

A
  • 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
25
Q

Steps of Quadrat Analysis

A
  • 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
26
Q

Optimal quadrat size

A

= (2 x Area)/n

27
Q

Quadrat analysis: Property

A

Mean = Variance

28
Q

VMR

A

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

29
Q

VAR

A
  • Number of points per cell

- Variance of the frequency

30
Q

QA: Index of dispersion

A

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
31
Q

QA, Null hypothesis

A

VMR = 1

- Point pattern is random

32
Q

QA, VMR does not = 1

A

Point pattern is not random

33
Q

QA, Alternative hypothesis

A
  • VMR > 1, point pattern is more clustered than random

- VMR < 1, Point pattern is more dispersed than random

34
Q

NNA

A
  • 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
35
Q

NNA eqn

A

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

36
Q

NND = 0

A

Perfect clustering

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

37
Q

NNDr = 1/2 x sq root of density

A

NNDr, Random clustering

38
Q

NND = 1/sq. root of density

A

Regular square lattice

39
Q

NNDd= 1.07453/sq. root of density

A

Regular hexagonal lattice

40
Q

NNA, adjustment for edge effects

A
  • 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)
41
Q

NNA, null hypothesis

A

Ho: NND = NNDr

- Point pattern is random

42
Q

NNA, Alternative hypotheses

A
  • NND does not = NNDr, point pattern is not random
  • NND > NNDr, point pattern is more dispersed
  • NND < NNDr, point pattern is more clustered
43
Q

NNA Test statistic

A

Zn = (NND - NNDr)/ sigma of NND

- Sigma of NND = 0.26136/sq. root of (n x density)

44
Q

NNA, standard nearest neighbour index

A 
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
45
Q

Ripley’s K-Function

A
  • 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
46
Q

Ripley’s K-Function Formula

A
  • 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
47
Q

What happens if K-Function distance is too small?

A

Clustering

48
Q

K-Function results

A
  • 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?)
49
Q

What do you do after analyzing point patterns?

A
  • Test for spatial autocorrelation

- (After Quadrat, NNA, and K-function)

Geog418SptStts (6 decks)

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