Geography 625

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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 Fa
ll 2006
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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

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1. Revisit IRP/CSR
Independent random process (IRP) Complete spatial
randomness (CSR)

1. 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.
2. Independence the positioning of any point is
   independent of the positioning of any other point.

and
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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.

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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/fa
020203.htm)
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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.

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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.
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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

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3. Introduction to point pattern analysis
Requirements for a set of events to constitute a
point pattern

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

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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.
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4. Describing a Point Pattern
First-order or second order?
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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
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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
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5. Density-based point pattern measures
Crude density/Overall intensity
The crude density changes depending on the study
area
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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?

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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.

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5. Density-based point pattern measures
-Quadrat Count Methods
Other shapes of quadrats
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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
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5. Density-based point pattern measures
-Density Estimation
Bandwidth r
If r is too large, then ? If r is too small,
then?
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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)
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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

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6. Distance-based point pattern measures

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

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6. Distance-based point pattern measures
-Nearest-Neighbor Distance
Euclidean distance
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6. Distance-based point pattern measures
-Nearest-Neighbor Distance
If clustered, has a higher or lower
value?
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6. Distance-based point pattern measures
-Distance Functions G function
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6. Distance-based point pattern measures
-Distance Functions G function
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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.

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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)

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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.

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6. Distance-based point pattern measures
-Comparisons between G and F functions
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6. Distance-based point pattern measures
-Comparisons between G and F functions
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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.
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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

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6. Distance-based point pattern measures
-Distance Functions K Function
is the study area event density
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6. Distance-based point pattern measures
-Distance Functions K Function
Clustered?
Evenly distributed?
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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.
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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?
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7. Assessing Point Patterns Statistically
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7. Assessing Point Patterns Statistically
-Quadrat Counts
Independent random process (IRP) Complete spatial
randomness (CSR)
A
and
B
Mean
Variance
The variance/mean (VMR) is expected to be 1.0 if
the distribution is Poisson.
How about mean gt variance? mean lt
variance?
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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
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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
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7. Assessing Point Patterns Statistically
-Quadrat Counts
A
VMR Variance/Mean 0.9375/1.25
0.75
B
Clustered? Random? Dispersed?
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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 gt 1 then dispersed, else if R lt 1 then
clustered?
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7. Assessing Point Patterns Statistically
-G and F Functions
Clustered
Evenly Spaced
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7. Assessing Point Patterns Statistically
K Functions
IRP/CSR