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Intermediate

Geographic Information Science

Week4: Point Pattern Analysis

Instructor: Changshan Wu

Department of Geography

The University of Wisconsin-Milwaukee

Fall 2006

- 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

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

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.

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)

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.

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.

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

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)

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.

First-order or second order?

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

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

5. Density-based point pattern measures

Crude density/Overall intensity

The crude density changes depending on the study area

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?

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.

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

5. Density-based point pattern measures

-Density Estimation

Bandwidth r

If r is too large, then ?

If r is too small, then?

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)

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

6. Distance-based point pattern measures

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

6. Distance-based point pattern measures

-Nearest-Neighbor Distance

If clustered, has a higher or lower value?

6. Distance-based point pattern measures

-Distance Functions: G function

6. Distance-based point pattern measures

-Distance Functions: G function

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.

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)

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.

6. Distance-based point pattern measures

-Comparisons between G and F functions

6. Distance-based point pattern measures

-Comparisons between G and F functions

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.

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

6. Distance-based point pattern measures

-Distance Functions: K Function

is the study area event density

6. Distance-based point pattern measures

-Distance Functions: K Function

Clustered?

Evenly distributed?

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.

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?

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?

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

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

7. Assessing Point Patterns Statistically

-Quadrat Counts

A

VMR = Variance/Mean

= 0.9375/1.25

= 0.75

B

Clustered?

Random?

Dispersed?

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?

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