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Geography 625. Intermediate Geographic Information Science. Week4: Point Pattern Analysis. Instructor : Changshan Wu Department of Geography The University of Wisconsin-Milwaukee Fall 2006. Outline. Revisit IRP/CSR First- and second order effects Introduction to point pattern analysis

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slide1

Geography 625

Intermediate

Geographic Information Science

Week4: Point Pattern Analysis

Instructor: Changshan Wu

Department of Geography

The University of Wisconsin-Milwaukee

Fall 2006

slide2

Outline

  • Revisit IRP/CSR
  • First- and second order effects
  • Introduction to point pattern analysis
  • Describing a point pattern
  • Density-based point pattern measures
  • Distance-based point pattern measures
  • Assessing point patterns statistically
slide3

1. Revisit IRP/CSR

Independent random process (IRP)

Complete spatial randomness (CSR)

  • Equal probability: any point has equal probability of being in any position or, equivalently, each small sub-area of the map has an equal chance of receiving a point.
  • Independence: the positioning of any point is independent of the positioning of any other point.

and

slide4

2. First- and second order effects

IRP/CSR is not realistic

  • The independent random process is mathematically elegant and forms a useful starting point for spatial analysis, but its use is often exceedingly naive and unrealistic.
  • If real-world spatial patterns were indeed generated by unconstrained randomness, geography would have little meaning or interest, and most GIS operations would be pointless.
slide5

2. First- and second order effects

1. First-order effect

  • The assumption of Equal probability cannot be satisfied
  • The locations of disease cases tends to cluster in more densely populated areas
  • Plants are always clustered in the areas with favored soils.

From (http://www.crimereduction.gov.uk/toolkits/fa020203.htm)

slide6

2. First- and second order effects

2. Second-order effect

  • The assumption of Independence cannot be satisfied
  • New developed residential areas tend to near to existing residential areas
  • Stores of McDonald tend to be far away from each other.
slide7

2. First- and second order effects

In a point process the basic properties of the process are set by a single parameter, the probability that any small area will receive a point – the intensity of the process.

First-order stationary: no variation in its intensity over space.

Second-order stationary: no interaction between events.

slide8

3. Introduction to point pattern analysis

Point patterns, where the only data are the locations of a set of point objects, represent the simplest possible spatial data.

  • Examples
  • Hot-spot analysis for crime locations
  • Disease analysis (patterns and environmental relations)
  • Freeway accident pattern analysis
slide9

3. Introduction to point pattern analysis

Requirements for a set of events to constitute a point pattern

  • The pattern should be mapped on the plane (prefer to preserve distance between points)
  • The study area should be determined objectively.
  • The pattern should be an enumeration or census of the entities of interest, not a sample
  • There should be a one-to-one correspondence between objects in the study area and events in the pattern
  • Event locations must be proper (should not be the centroids of polygons)
slide10

4. Describing a Point Pattern

Point density (first-order or second-order?)

Point separation (first-order or second-order?)

When first-order effects are marked, absolute location is an important determinant of observations, and in a point pattern clear variations across space in the number of events per unit area are observed.

When second-order effects are strong, there is interaction between locations, depending on the distance between them, and relative location is important.

slide11

4. Describing a Point Pattern

First-order or second order?

slide12

4. Describing a Point Pattern

A set of locations S with n events

s1 (x1, y1)

The study region A has an area a.

Mean Center

Standard Distance: a measure of how dispersed the events are around their mean center

slide13

4. Describing a Point Pattern

A summary circle can be plotted for the point pattern, centered at

with radius d

If the standard distance is computed separately for each axis, a summary ellipse can be obtained.

Summary circle

Summary ellipse

slide14

5. Density-based point pattern measures

Crude density/Overall intensity

The crude density changes depending on the study area

slide15

5. Density-based point pattern measures

-Quadrat Count Methods

  • Exhaustive census of quadrats that completely fill the study region with no overlaps
  • The choice of origin, quadrat orientation, and quadrat size affects the observed frequency distribution
  • If quadrat size is too large, then ?
  • If quadrat size is too small, then?
slide16

5. Density-based point pattern measures

-Quadrat Count Methods

  • 2. Random sampling approach is more frequently applied in fieldwork.
  • It is possible to increase the sample size simply by adding more quadrats (for sparse patterns)
  • May describe a point pattern without having complete data on the entire pattern.
slide17

5. Density-based point pattern measures

-Quadrat Count Methods

Other shapes of quadrats

slide18

5. Density-based point pattern measures

-Density Estimation

The pattern has a density at any location in the study region, not just locations where there is an event

This density is estimated by counting the number of events in a region, or kernel, centered at the location where the estimate is to be made.

Simple density estimation

C(p,r) is a circle of radius r centered at the location of interest p

slide19

5. Density-based point pattern measures

-Density Estimation

Bandwidth r

If r is too large, then ?

If r is too small, then?

slide20

5. Density-based point pattern measures

-Density Estimation

Density transformation

1) visualize a point pattern to detect hot spots

2) check whether or not that process is first-order stationary from the local intensity variations

3) Link point objects to other geographic data (e.g. disease and pollution)

slide21

5. Density-based point pattern measures

-Density Estimation

Kernel-density estimation (KDE)

Kernel functions: weight nearby events more heavily than distant ones in estimating the local density

  • IDW
  • Spline
  • Kriging
slide22

6. Distance-based point pattern measures

  • Look at the distances between events in a point pattern
  • More direct description of the second-order properties
slide23

6. Distance-based point pattern measures

-Nearest-Neighbor Distance

Euclidean distance

slide24

6. Distance-based point pattern measures

-Nearest-Neighbor Distance

If clustered, has a higher or lower value?

slide25

6. Distance-based point pattern measures

-Distance Functions: G function

slide26

6. Distance-based point pattern measures

-Distance Functions: G function

slide27

6. Distance-based point pattern measures

-Distance Functions: G function

The shape of G-function can tell us the way the events are spaced in a point pattern.

  • If events are closely clustered together, G increases rapidly at short distance
  • If events tend to evenly spaced, then G increases slowly up to the distance at which most events are spaced, and only then increases rapidly.
slide28

6. Distance-based point pattern measures

-Distance Functions: F function

  • Three steps
  • Randomly select m locations {p1, p2, …, pm}
  • Calculate dmin(pi, s) as the minimum distance from location pi to any event in the point pattern s
  • 3) Calculate F(d)
slide29

6. Distance-based point pattern measures

-Distance Functions: F function

  • For clustered events, F function rises slowly at first, but more rapidly at longer distances, because a good proportion of the study area is fairly empty.
  • For evenly distributed events, F functions rises rapidly at first, then slowly at longer distances.
slide30

6. Distance-based point pattern measures

-Comparisons between G and F functions

slide31

6. Distance-based point pattern measures

-Comparisons between G and F functions

slide32

6. Distance-based point pattern measures

-Distance Functions: K Function

The nearest-neighbor distance, and the G and F functions only make use of the nearest neighbor for each event or point in a pattern

This can be a major drawback, especially with clustered patterns where nearest-neighbor distances are very short relative to other distances in the pattern.

K functions (Ripley 1976) are based on all the distances between events in S.

slide33

6. Distance-based point pattern measures

-Distance Functions: K Function

  • Four steps
  • For a particular event, draw a circle centered at the event (si) and with a radius of d
  • Count the number of other events within the circle
  • Calculate the mean count of all events
  • This mean count is divided by the overall study area event density
slide34

6. Distance-based point pattern measures

-Distance Functions: K Function

is the study area event density

slide35

6. Distance-based point pattern measures

-Distance Functions: K Function

Clustered?

Evenly distributed?

slide36

6. Distance-based point pattern measures

-Edge effects

Edge effects arise from the fact that events near the edge of the study area tend to have higher nearest-neighbor distances, even though they might have neighbors outside of the study area that are closer than any inside it.

slide37

7. Assessing Point Patterns Statistically

A clustered pattern is likely to have a peaky density pattern, which will be evident in either the quadrat counts or in strong peaks on a kernel-density estimated surface.

An evenly distributed pattern exhibits the opposite, an even distribution of quadrat counts or a flat kernel-density estimated surface and relatively long nearest-neighbor distances.

But, how cluster? How dispersed?

slide39

7. Assessing Point Patterns Statistically

-Quadrat Counts

Independent random process (IRP)

Complete spatial randomness (CSR)

A

B

and

Mean

Variance

The variance/mean (VMR) is expected to be 1.0 if the distribution is Poisson.

How about mean > variance?

mean < variance?

slide40

7. Assessing Point Patterns Statistically

-Quadrat Counts

For a particular observation

Mean = number of events / study area

n is the number of events

x is the number of quadrats

A

B

slide41

7. Assessing Point Patterns Statistically

-Quadrat Counts

Variance

2 * (0 – 1.25)2 = 3.125

k = 0:

k = 1:

3 * (1 – 1.25)2 = 0.1875

k = 2:

2 * (2 – 1.25)2 = 1.125

A

B

k = 3:

1 * (3 – 1.25)2 = 3.0625

slide42

7. Assessing Point Patterns Statistically

-Quadrat Counts

A

VMR = Variance/Mean

= 0.9375/1.25

= 0.75

B

Clustered?

Random?

Dispersed?

slide43

7. Assessing Point Patterns Statistically

-Nearest Neighbor Distances

The expected value for mean nearest-neighbor distance for a IRP/CSR is

The ratio R between observed nearest-neighbor distance to this value is used to assess the pattern

If R > 1 then dispersed, else if R < 1 then clustered?

slide44

7. Assessing Point Patterns Statistically

-G and F Functions

Clustered

Evenly Spaced