Geog. 579: GIS and Spatial Analysis Lecture 25 Overheads Page: 5/5
Lecture 25: Spatial Indices and Landscape Measures (I)
TOPICS:
1. Spatial Centroids
2. Point Pattern Analysis
REFERENCE:
Chapter 6 of Quantitative Geography by Fortheringham, Brunsdon and Charlton (2000)
1. Spatial Centroids:
(1) Area weighted average
(2) Geometric centroid (mean centre)
(3) Spatial Centroid (spatial mean) (the balance point)
where fi is the gravity density function over area i.
2. Point Pattern Analysis:
2.1 What is point pattern analysis?
1) What is a point pattern?
A set of two-dimensional points (events) with definite spatial locations (x, y), (Examples: distributions of burglaries, occurrences of certain diseases)
2) What is point pattern analysis?
Determine whether a set of points exhibit some patterns (clustering or correlation) which may be the results of certain processes
It is another form of analyzing spatial autocorrelation associated with points (particularly categorical data)
2.2 Methods:
2.2.1 Visual methods:
1) Scatter Plots (maps): (Figure 6.2)
2) Bubble Plots (proportional symbol maps) (Figure 6.3)
3) Animated maps: (Wisc-Alcohol Events)
2.2.2 Summary Statistics:
1) Mean centre:
Geometric centroid for a set of points:
2) Standard distance
3) Mean nearest neighbour distance
Average, over all points, of the distance from one point to the closest point to it in the data set.
(Figure 6.5)
2.2.3: Modelling approaches:
2.2.3.1 First order intensity analysis
First-order properties are related to the expected values associated with individual points (or areas) in a study area
1) Notation:
R: study area
E: event set
n: the number of events in E
ei: Single event i whose coordinates are (xi, yi)
A: sub set of study area and |A| is the area size of A
N(A, E): number of events in A
T(A,E): expected number of events in A
C(x, r): circle of radius r around point x
2) Mean intensity over an area and intensity at a point
Mean Intensity:
Intensity at point (x):
Changes as the total number in the whole area changes
3) Probability density:
The probability of an event will occur at a given point:
Difference between probability density function and the Probability function:
If relative spatial distribution is under concern, then compare the probability density function.
If prevalence of a crime over another crime, then compare the intensity functions.
4) Index of Dispersion:
(Given a grid, the counts of events in each grid)
where gj is the event counts in grid j, and is the mean count (total / number of squares).
If Complete Spatial Randomness (CSR):
I) constant intensity function over the area
II) events are independent of each other.
Then the number of events in a grid follows Poisson distribution (which means that the mean and sample variance are the same), that is, I = 1.
If I much less than 1, then dispersion or regularity (less variability than expected under CSR).
If I much greater than 1, then clustering.
If n > 6 and greater than 1, then (n-1) I follows χ2 n-1 distribution and statistical testing is possible.
Example: (Redwood Figure and Count Table):
N = 25,
=2.48,
I = 2.22,
(n-1) I = 24*2.22=53.28
at 99% χ2 24 = 42.98;
reject Null Hypothesis (what is the null hypothesis?)
Meaning clustering.
2.2.3.2 Second-order Intensity Analysis:
Second-order properties examine the correlations or covariances between events occurring at two distinct points or regions in the study area
Examples of second order processes are seed dispersion, organized crimes.
Independence measure:
The Burglary data set: (Data Table)
The product of the expected first order intensities of the two areas:
The second order intensity measure:
When is much greater than positive spatial correlation between the two areas is implied
When they are the same, no correlation is implied
When When is much less than negative correlation implies.
The Burglary Example:
(Table 6.3)