1 Analysis of Point (Event) Data II Describing the Spatial ... - Capita

Analysis of Point (Event) Data
II
Frequency
1
0.9
0.8
0.7
0.6
0.5
CE/ENVE 424
G
E
F
0.4
0.3
0.2
0.1
0
0 2 4 6 8 10
Distance
Describing the Spatial Pattern of Events
We can describe the spatial pattern of a
events (point) dataset using:
• Summary statistics
• Density based analysis
• Simple (quadrant count)
• Kernels and kernel functions
• Distance based analysis
• Nearest neighbor
• Distance functions
• G (event-to-event)
• F (point-to-event)
• K (multiple radius distances)

Relating Intensity Patterns
The intensity patterns derived using the methods previously discussed provide
a meaningful analysis endpoint, particularly when comparing patterns among
data sets
However, we’re frequently interested in more explicit methods for describing the
a point spatial pattern
The most common approach is to test against complete spatial
randomness (CSR):
• Is the event pattern significantly more clustered than what
would be expected from a random distribution?
• Is the event pattern more uniformly spaced than would be
expected from a random distribution?
Complete Spatial Randomness
A process is considered random if its intensity (average
number of events per unit area) is constant over the region
of interest.
1) the chance of a given x,y point existing is equal to the
chance any other point existing (uniform probability
distribution)
2) the existence of a x,y point is independent of the
existence of any other point
These two conditions constitute an independent random
process (IRP) or complete spatial randomness (CSR)

Complete Spatial Randomness
CSR is a baseline hypothesis (null hypothesis) against which is assessed
whether an observed pattern is evenly spaced, clustered, or random.
In testing for CSR, we define a model for CSR. We could simulate a the
pattern for number of events over a region of interest using the model. We
can then compare the spatial distribution of the modeled patterns with our
observed pattern.
The standard model to use in testing spatial point patterns follow a Poisson
distribution.
The probability of observing k events in one unit area in our region is
approximated by:
6
k
λ
λ
Mean = Variance
5
−
P(
k)
=
Quadrant Count
e
k!
n
λ = e ≈ 2.
718
a
So the ratio of
mean/variance can
be used to
determine if the
pattern is random
Frequency
4
3
2
1
0
1 2 3 4 5 6 7
Bin
Recall from the last lecture, a quadrant count is conducted by superimposing a
regular grid over data, counting the number of events in each grid cell and divide
the count by its cell area to get intensity.
47 grid cells
Mean cell count
47
µ = = 1.
175
40
variance
mean
=
Variance:
1 ∑=
n
2
s = µ
n i 1
85.
775
= =
40
2.
1444
1.
175
=
1.
825
( ) 2
k −
2.
1444
A s 2 to µ ratio greater than 1 indicates
clustering

Nearest Neighbor
7
The expected value of mean nearest neighbor is: E(
d )
1
9
8
20
4
3
5
2
6
11
10
12
event
nearest
neighbor dmin
1 3 10
2 5 2
3 5 1
4 3 1.5
5 3 1
6 5 1.5
7 8 1
8 7 1
9 7 2
10 11 1
11 10 1
12 10 1.5
G Function
20
The G Function provides a
cumulative frequency
distribution of the nearest
neighbor event-event distances
7
1
9
8
. [ d ( s ) d]
no min i <
G( d ) =
n
4
3
5
2
6
11
10
12
Frequency
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
d
n
∑
i=
= 1
min
d
24 . 5
= =
12
min
n
( s )
2.
04
i
1
=
2 λ
Calculating the ratio of our observed mean
nearest neighbor distance and the expected
distance provides a measure of clustering
E
0
0 2 4 6 8 10
Distance
dmin
R =
1/
2
R =
[ G(
d ) ]
0.
5
λ
2.
04
= 0.
71
12 /( 20X
20)
A ratio less than 1
indicates clustering
CSR Expected G(d):
= 1−
e
−λπd
Distance G(d) Count G(d) Freq. E(G) Freq E(G) Count
0 0 0.000 0.000 0.000
1 6 0.500 0.090 1.079
2 5 0.917 0.314 2.690
3 0 0.917 0.572 3.093
4 0 0.917 0.779 2.482
5 0 0.917 0.905 1.519
6 0 0.917 0.966 0.734
7 0 0.917 0.990 0.285
8 0 0.917 0.998 0.090
9 0 0.917 1.000 0.023
10 1 1.000 1.000 0.005
2

F Function
The F Function provides a
cumulative frequency
distribution of the nearest
neighbor point-event distances
7
1
9
8
. [ dmin
( p , S ) d ]
no
i <
F( d ) =
m
K Function
n
λ =
a
4
3
5
2
6
11
10
12
Frequency
CSR Expected F(d):
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
E
[ F(
d ) ]
G E
= 1−
e
F
0
0 2 4 6 8 10
Distance
−λπd
The K function uses all distances between events and provides a measure
of spatial dependence over a wider range of scales.
K(
d)
n
∑
i=
= 1
no
. [ S ∈C(
s , d ) ]
nλ
i
[ K(
d ) ]
E =
λπ 2
d
λ
2
= πd
( d )
d
K
L( d ) = −
π
2

L Function
7
1
9
8
4
3
5
2
6
11
10
12
Distance K(d) L(d) E(K(d)) E(L(d))
0 0.000 0.000 0.000 0
1 16.667 1.303 3.142 0
2 30.556 1.119 12.566 0
3 30.556 0.119 28.274 0
4 30.556 -0.881 50.265 0
5 30.556 -1.881 78.540 0
6 30.556 -2.881 113.097 0
7 30.556 -3.881 153.938 0
8 30.556 -4.881 201.062 0
9 30.556 -5.881 254.469 0
10 33.333 -6.743 314.159 0
Multi Variant Analysis
An L(d) of 0 is expected
An L(d) above the “zero line” indicates there are
more events at that separation distance than
expected
An L(d) below the “zero line” indicates there are
fewer events at that separation distance than
expected
Sometimes you have multiple data sets and wish to know if the
spatial patterns among them are similar.
Cross functions are variants on the previously discussed
distance-based analysis methods.
2
1
0
0 2 4 6 8 10
-1
For G(d), the distance of interest is between events in one dataset
and events in another dataset
For K(d), counts the number of events one dataset based on
distances from events in another dataset
L(d)
-2
-3
-4
-5
-6
-7
Distance

Incorporating Temporal Dimension
Intensity is defined as the number of events per unit area in unit time
Are events clustered in space and time?
Distance is a statistical distance in units of physical distance X time
distance.
One method for determining whether there is space-time dependence
is to calculate
1) a K(d,t) for space*time distance
2) a K(d) and K(t) separately
3) K(d) * K(t)
4) K(d,t) – K(d)*K(t)
Examples
Phytopathology Vol. 92, No. 4, 2002 361-377
Background
Florida has been substantially impacted by Asiatic citrus canker, a disease
that can cause defoliation, dieback and fruit drop
Objectives
Does removing healthy citrus trees within a 38 m radius of infected trees
curtail further spread of the disease
Methods Used
K function for testing complete spatial randomness (CSR)

American Journal of Tropical Medicine and Hygiene Vol. 58, No 3, 1998 287-298
Background
Dengue fever, a viral disease transmitted by mosquitoes, can spread
rapidly and is without a vaccine. Cases tend to be clustered because it is assumed
that female mosquitoes rarely travel further than 50-100 m in their lifetime.
Objectives
What is the spatial-temporal pattern of the diesease
Methods Used
K function for testing complete spatial randomness (CSR)

Projects and Articles
Begin searching for articles to get ideas
Also try to understand the availability of data
This weekend, the class website will be updated with data sources, links
to online journals, references to other journals
If you come across a good resource, please let me know and I’ll add it to
the website