Download POINT PATTERN ANALYSIS (PPA) * by DongMei Chen and Arthur

*
*
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POINT PATTERN ANALYSIS (PPA)
*
by
DongMei Chen and Arthur Getis
Department of Geography
San Diego State University
San Diego, CA 92118-4493
[email protected]
[email protected]
May 12, 1998
*
TABLE OF CONTENTS
*
**
*Introduction*
*Routines *
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
Basic descriptive statistics
Nearest neighbor analysis
Refined nearest neighbor analysis
K-function (or second order analysis)
Weighted K-function
Cluster: Knox space-time
Join-count statistics
Global Moran’s I
Global Geary’s c
General Getis-Ord’s G(d)
Local Moran’s
Local
Local
Local K-Function
*
POINT PATTERN ANALYSIS (PPA)
Introduction
*
This Point Pattern Analysis (PPA) software package is written and
compiled in C and is used to describe and help analyze point patterns.
It consists of 14 different analysis routines. These represent a variety
of basic descriptive statistics and include: nearest neighbor analysis,
refined nearest neighbor analysis, K-function, weighted K-function,
space-time Knox, Join-Count statistics, Global Moran’s I and Geary’s c,
general Getis-Ord’s G, local Moran’s , local and , and local K-function.
This manual contains a brief description of each analysis as well as
input and output information.
This PPA package can be run in both DOS (or a DOS program in WINDOWS)
and UNIX. The memory requirement for running PPA depends on the size of
the data set.
In order to run this package, you need to copy the executable files
ppa.exe (for DOS), or ppa (for UNIX) to a new directory that you want to
work on and then type
**
* ppa *
followed by a Return. This will clear the screen and a welcome message
will appear. Press /Enter /key, then choose the desired routine.
All routines (except Join-Count) are designed for data in three columns
X, Y and Z, where X, Y are coordinates, and Z represents the value at
site X, Y or time. If all Z values are weighted similarly, that is, they
are to be evaluated as single, unweighted points, a column of 1s should
make up the Z column. If a weight matrix file is used, it should be
organized as an N by N matrix in the order of your input data.
All the results will appear on the screen (UNIX) and be saved to a file
that you name. PPA will empty this file before it saves any new output.
**
*Instruction *
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Call for directory /ppa/
Type /ppa/
Press Enter key to begin
Choose routine member
Enter input data
Enter output data file name (any name will do)
To continue, enter 1 (computing will now take place)
Enter 0 to quit
Call for output file
*1. Basic Descriptive Statistics (BDS)
*
In this option, the minimum, maximum, mean, standard deviation,
skewness, and kurtosis are calculated. Standard deviation measures
dispersion from the mean, skewness measures the extent to which the bulk
of the values in a distribution are concentrated to one side or the
other of the mean, kurtosis measures the extent to which values are
concentrated in one part of a frequency distribution. The formulas for
these parameters are the following:
Mean =
Standard Deviation Std(Z) =
Skewness =
Kurtosis =
where N is the total number of points.
**
* *
*/
Input
/*
The data file contains N rows of X, Y coordinates and Z values.
*/
Output
/*
The output lists the total number of points in the files, the minimum,
maximum, mean, standard deviation, skewness and kurtosis values for X, Y
and Z.
*
2. Nearest Neighbor Analysis
*
Nearest neighbor analysis examines the distances between each point and
the closest point to it, and then compares these to expected values for
a random sample of points from a CSR (complete spatial randomness)
pattern.
*/
Formula:
/
a) The mean nearest neighbor distance
*
[1]
where N is the number of points. / /is the nearest neighbor distance for
point i.
*
b) The expected value of the nearest neighbor distance in a random
pattern
*
[2]
where A is the area and B is the length of the perimeter of the study
area.
**
*c) The variance *
[3]
Equations [2] and [3] contain a correction factor to account for the
boundary effect based on Donnelly (1978).
*/
Input
/*
You’ll be asked to enter the input data file, which should contain N
rows of X, Y coordinates, and Z values. Make Z values all equal to 1
representing points.
*/
Output
/*
The output file lists a) the input data file, b) the total number of
points, c) the minimum and maximum of the X, and Y coordinates, d) the
size of study area, e) the observed mean nearest neighbor distance, f)
the expected average nearest neighbor distance, g) the variance, and h)
Z statistic (standard normal variate). A negative Z score indicates
clustering; a positive score means dispersion or evenness.
*/
Limitation
/*
Equations [2] and [3] cannot be used for irregularly shaped study areas.
In this program, the study area is a regular rectangle or a square. A is
calculated by (Xmax - Xmin) * (Ymax - Ymin).
**
*3.* *Refined Nearest Neighbor Analysis*
Refined nearest neighbor analysis involves comparing the complete
distribution function of the observed nearest neighbor distances,, with
the distribution function of the expected nearest neighbor distances for
CSR, . The program finds the largest absolute difference , and tests for
significance based on a Monte Carlo test.
*//*
*/Formula/*
a) is obtained by taking the nearest neighbor distances, , and the
nearest distances to study boundary, , for each point i. The program
ranks from the smallest to the largest. For every distance of interest,
, the program counts the number of points for which , and the number of
points for which . The observed proportion of the nearest neighbor
distances less than or equal to some chosen distance is decided by
equation [1].
[1]
Where N is the total number of points.
b) The proportion of expected nearest neighbor distances less than or
equal to r for an unbounded CSR pattern is:
[2]
Where:
e is the mathematical constant 2.718283
is the mathematical constant 3.141593
r is the specified distance
is the estimated point density (N/A)
c)
Where Max | | means the largest absolute value obtained for
corresponding values of r.
*/
Input
/*
You’ll be asked to enter the input data file, which should contain rows
of X, Y coordinates representing points, and Z values made up of 1s.
*/
Output
/*
The output file includes three parts: the first part lists a) the input
data file, b) the total number of points, c) the minimum and maximum of
X, and Y coordinates, and d) the size of study area.
The second part is a table of the following form:
r
(distance)
Observed number of points () for which
Observed number of points () for which
Observed proportion
Expected proportion
:
:
:
*
*
If for each r, >, a clustered pattern is indicated, whereas < indicates
a regular pattern of points.
The last part shows (the largest absolute value obtained for ), r (the
distance for ), and its significance. If F is greater than P, then
clustering is implied.
*/
Limitation
/*
In this program, the study area is a regular rectangle or a square. The
area is calculated by (Xmax - Xmin) * (Ymax - Ymin).
*
4. K-Function
*
K-Function is also called second-order analysis to indicate that the
focus is on the variance, or second moment, of interevent distances. It
considers all combinations of pairs of points. It compares the number of
observed pairs with the expectation at all distances, taking into
consideration the density of points, the borders, and the size of the
sample.
*//*
*/Formula/*
[1]
Where:
A is the size of study area,
N is the number of points,
d is the distance,
is the number of j points within distance d of all i points. /k(i, j)/
is the weight, which is estimated by
a) If no edge corrections,
in case
otherwise
b) If a point i is closer to one boundary than it is to a point j, the
border correction is employed.
where e is the distance to the nearest edge.
c) If a point i is closer to two right angle boundaries than it is to a
point j, the weighting formula is
are the distances to the nearest vertical and horizontal borders
respectively.
The expectation for a CSR pattern of L(d) is d.
*//*
*/Input/*
1. The input data file, which should include the X, Y coordinates of
points, and Z values ( a column of 1s).
2. The maximum distance (dmax) that you want. Usually a statistically
unbiased maximum distance is less than the circumradius of the study
area, or one-half the lesser of the length or the width of a rectangular
study area.
3. The number of increments.
4. The number of permutations for creating the confidence envelope.
5. Output file.
*/
Output
/*
This program calculates the distances /d(i,j)/ between all combinations
of two points, and calculate the /k(i, j)/ for all pairs, and then
calculate the /L(d)/ for all d. The program will randomly generate the N
points in the whole study area M times, and get the minimum and maximum
of /L(d)/ for the envelope. The output lists the input data file, the
total number of points, the minimum and maximum of x and y coordinates,
the size of study area, and the following table
distance d
Observed /L(d)/
//
/L(d)/ - d
Minimum /L(d)/
Maximum /L(d)/
:
:
:
*/
Limitation
/*
The boundary correction formulas used here are inappropriate for
irregular borders. In this program we assume the study area is a
rectangle or a square.
*
5. Weighted K-Function
*
The weighted K-function was developed by Getis (1982) based on the
K-Function. It considers both location and the value of a point. The
statistical test is based on the assumption of CSR, and performed on
independent simulations of all values in fixed locations of the study
area.
*/
Formulas
/*
[1]
[2]
Equation [1] includes the i points’ interaction with all points,
including itself while in [2] i does not equal j.
Where A is the size of the study area,
/z(i)/ is the weight of the point i,
/k(i, j)/ is the border correction value, the same as that defined for
the K-Function.
*//*
*/Input/*
1. The input data file, contains rows of the (x, y) coordinates and the
z values of points.
2. The maximum distance (dmax); usually statistical unbiased maximum
distance is less than the circumradius of the study area, or one-half
the lesser of the length or the width of a rectangular study area.
3. The number of increments.
4. The number of permutations for creating the confidence envelope.
5. Output file.
*/
Output
/*
This program calculates the distances /d(i,j)/ between all combinations
of two points, and calculates the /k(i, j)/ for all pairs, and then
calculates the /L(d)/ and /L*(d)/ for all d. For the confidence
envelope, the program will randomly assign the Z values of each point to
the N points M times, and find the minimum and maximum of /L(d)/ and
/L*(d)/. The output lists the input data file, the total number of
points, the minimum and maximum of X, and Y coordinates, the size of
study area, and the following two tables showing /L(d)/ and /L*(d)/
respectively:
distance (d)
Observed /L(d)/
//
/Minimum L(d)/
//
/Maximum L(d)/
:
:
:
distance (d)
Observed L*(d)
Minimum L*(d)
Maximum L*(d)
:
:
:
*/
Limitation
/*
The boundary correction formulas used here are inappropriate for
irregular borders. In this program we assume the study area is a regular
rectangle or a square.
*
6. Knox Statistic for Space-Time Clustering
*
The Knox approach is used to test whether there is a significant cluster
during a defined distance and time period. First it counts the number of
point pairs as either close or distant in space and /or time, then
calculates the P-value.
*/
Formula
/*
For a certain distance /d /and time period /t/, the Knox statistic
calculates the following number:
/d(i, j)/ is the distance of point i and j,
/t(i, j)/ is the time interval of point i and j,
: the number of point pairs /(i, j)/ with /d(i, j) d/, and /t(i, j) t/,
: the number of point pairs /(i, j)/ with /d(i, j) d/, and /t(i, j) > t/,
: the number of point pairs /(i, j)/ with /d(i, j) > d/, and /t(i, j) t/,
: the number of point pairs /(i, j)/ with /d(i, j) > d/, and /t(i, j) >
t/,
/N /is the total number of point pairs ()
The P-value is:
where ,
*/
Input
/*
1. Input data file, which should record X, Y coordinates of points and T
the times attached to each points (time elapsed in days, or years or
minutes, etc).
2. The time interval.
3. The distance interval.
4. Output file.
*/
Output
/*
1. The input data file name,
2. The total number of points,
3. The minimum and maximum of X, and Y coordinates, and time.
4. The time and distance intervals.
5. The number of point pairs tabulated as
//
/t(i, j) <= t/
//
/t(i, j) > t/
Space only
//
/d(i, j) <= d/
//
/d(i, j) > d/
Time only
//
/N/
6. EN11 is the expected value of .
6. The P-value. Low P-values (e.g., 0.01) represent significant
time-space clustering.
*
7. Join-Count Statistics for Spatial Autocorrelation (Free sampling
model)
*
Join-Count statistics are the simplest measure of spatial
autocorrelation. They are used for a binary variable ( 1 or 0 ). The two
values of the variable are referred to as "black" (/B/) and "white"
(/W/). A join links two neighboring areas. So the possible types of
joins are black-black (/BB/), black-white (/BW/), and white-white
(/WW/). Join counts are counts of the numbers of /BB/, /BW/, and /WW/
joins in the study area, and these numbers are compared to the expected
numbers of /BB, BW/ and /WW/ joins under the null hypothesis of no
spatial autocorrelation.
*/
Formulas
/*
The observed number of /BB, BW/ and /WW/ joins are given by
[1]
[2]
[3]
Where is the binary value, 1 for black, 0 for white,
w(i, j) is the binary weight, 1 if two areas are contiguous, 0 otherwise.
For different assumptions about the data, the theoretical expressions
for /E(BB), E(BW)/ and /E/(/WW/) will vary. Under the free sampling
model, the expected /BB, BW/ and /WW/ are:
and is the number of areas with B values.
The variances are
Where
*/
Input
/*
1. Input data file, which records the binary value for each area.
2. Input weight matrix file, which is an N by N weight matrix with 1 for
contiguous areas, 0 otherwise.
3. Output file.
*/
Output
/*
1. The total number of areas,
2. The total number of black areas,
3. The total number of white areas,
4. The observed number of BB, BW and WW joins,
5. The expected number of BB, BW, and WW joins,
6. The variance of BB, BW joins,
7. The z-statistics of BB, BW joins.
**
*8. Global Moran’s /I /*
*9. Global Geary’s /c /*
Moran’s /I /and Geary’s /c/ are well known for testing for spatial
autocorrelation. They represent two special cases of the general
cross-product statistic that measures spatial autocorrelation. Moran’s
/I/ is produced by standardizing the spatial autocovariance by the
variance of the data using a measure of the connectivity of the data.
Geary’s /c/ uses the sum of squared differences between pairs of data
values as its measure of covariation.
*/
Formula
/*
[1]
[2]
Where is the mean of , ,
The expected value of Moran’s /I /is /-1/(N-1)./ Values of /I/ that
exceed /-1/(N-1)/ indicate positive spatial autocorrelation, in which
similar values, either high values or low values, are spatially
clustered. Values of /I/ below /-1/(N-1)/ indicate negative spatial
autocorrelation, in which neighboring values are dissimilar.
The theoretical expected value for Geary’s c is 1. A value of Geary’s c
of less than 1 indicates positive spatial autocorrelation, while a value
larger than 1 points to negative spatial autocorrelation.
The variances of /I/ and /c/ will vary for different assumptions about
the data. Under the randomization assumption, the variance of /I /and /c
/are
where
The values of Moran’s /I/ and Geary’s /c/ depend on the w(i,j), which
are specified by the spatial weighting scheme chosen. In this program,
two weighting schemes can be selected:
a. The /w(i, j)/ are equal to the values in the input N by N matrix
taken from the spatial weight matrix file that the user has prepared.
b. The , where A is a constant (usually set at 1), /d(i,j)/ is the
distance between the /i/th and /j/th points; /m/ is a parameter
representing the friction of distance selected a /priori/.
In order to evaluate spatial trends in the pattern, sometimes it is
necessary to identify spatial autocorrelation at several levels of
spatial separation ( in the form of a spatial correlogram), such as for
different orders or distances of neighboring points. In this program,
two different correlograms for /I/ and /c/ are available, one is the
change by bands (Figure 1a), and the other is by distance increments
(Figure 1b).
. .
(a): Bands (b): Increments
Figure 1: in (a), points found in the band represented by the shadowed
concentric circle are related to the /i/th point shown at the center.
The correlogram shows the relationship of points in this band and
further bands to each of the /i/ points taken together. In (b), points
found in the shadowed region are related to the /i/th point at the
center. The correlogram shows the /cumulative /relationship of points at
a series of distances from the /i /points.
*/
Input for I and/or c
/*
1. Input data file contains the X, Y coordinates and the value at each
point.
2. Input whether you have a spatial weight matrix file.
3. Select the weighting scheme. If you select /b/, you’ll be asked to
enter the /A /and /m/ parameter.
4. Select whether you want to calculate a single /I/ (or /c/), or
correlogram. If correlogram is selected, you will be asked to enter the
maximum distance (dmax), steps you want (nstep), and whether by bands or
increments.
*/
Output for Moran’s I
/*
The output depends on the your input. For each distance range, this
program will output
a. the total number of points,
b. observed /I/,
c. expected /I/,
d. the variance,
e. /z/ value
*/
Output for Geary’s c
/*
The output depends on the your input. For each distance range, this
program will output
a. the total number of points,
b. observed /c/,
c. the variance,
d. /z/ value
**
*10*. *General G(d) statistic*
The /G /statistics were developed by Getis and Ord (1992). It is a
multiplicative measure of overall spatial association of values which
fall within a critical distance of each other. It can only be computed
for positive variables.
*/
Formula
/*
For a chosen critical distance d, /G(d)/ is
where is the value of ith point,
is the weight for point i and j for distance d.
The expected mean value of G(d)
The variance of G(d)
where
A positive z-value for /G(d)/ indicates spatial clustering of high
values, and a negative z-value indicates spatial clustering of low
values.
*/
Input
/*
1. Input data file, which records the X, Y coordinate and the value of
points.
2. The maximum distance.
3. The number of increments.
4. The output file.
/
*
Output
*/
1. The number of points.
2. The distance and its corresponding G(d), expected G(d), Variance and
Z value (standard variates).
**
*11.* *Local Spatial Autocorrelation Statistics*
Local spatial autocorrelation statistics are observation-specific
measures of spatial association. They focus on the location of
individual points, and allow for the decomposition of global or general
statistics into the contribution by each individual observation. It can
be used to detect the local spatial clustering around an individual
location, spatial nonstationarity, the outline of spatial regimes,
especially in cases where global statistics may fail to detect these
patterns, or where a single measure of global association may contribute
little meaning
**
*Local Moran’s *
According to Anselin (1995), a local Moran statistic for a point i is
defined as
where
For a randomization hypothesis, the expected value is
The variance is as
where
/
Note: this statistics is calculated for bands only in this package.
*
Input
*/
1. Input data file, which record the X, Y coordinates and the values of
points.
2. The distance used.
3. The parameter /m/ used in weighting scheme ().
4. Output file.
*
Output
*
1. The number of points,
2. The distance used,
3. The local Moran’s /Ii/, the expected value, variance, /z/-value for
each point.
point
E()
Var()
//
/z/
:
:
**
*12. Local *
*13. Local * * *
**and* * were developed by Ord and Getis (1995). They indicate the
extent to which a location is surrounded by a cluster of high or low
values. The **statistic excludes the value at i from the summation while
the **includes the value at i. Positive **or ** indicate spatial
clustering of high values, whereas negative **or **indicate spatial
clustering of low values.
*/
Formulas
/*
where
Where
Both **and* * are asymptotically normally distributed as d increases.
Under the null hypothesis that there is no association, the expectation
value is 0, the variance is 1. If the underlying data are normally
distributed, we can consider their values as standard variates.
*/
Input
/*
1. Input data file,
2. The maximum distance,
3. The number of increments,
4. Output file.
*/
Output
/*
1 The number of points,
2 The distance used,
3 The **or* *for each point.
point
**
**or* *
:
**
* *
*14. Local K-Function*
The local K-Function was developed by Getis (1984). It is similar to the
global K-function analysis, but differs in that the local K-function
only considers those pairs of points having as one of its members a
given point i.
*//*
*/Formula/*
Where:
A is the size of study area,
N is the number of points,
d is the distance,
is the summation over all points that are within distance d of point i,
and it includes a boundary correction that is the same as that in the
K-function program.
*//*
*/Input/*
1. The input data file includes the X, Y coordinates of points, and Z
values ( assign 1s).
2. The maximum distance (dmax) that you want, usually statistically
unbiased maximum distance is less than the circumradius of the study
area, or one-half the lesser of the length or the width of a rectangle
study area.
3. The number of increments.
4. The number of permutations for creating the confidence envelope.
5. Output file.
*/
Output
/*
This program calculates the number of pairs of points between /i/ and
all points within /d/, and calculate the /k(i, j)/, and then calculate
the /Li(d)/ for each /i /and /d/. The program will randomly generate the
N points in the whole study area M times, and found the minimum and
maximum of /L(d)/ for the envelope. The output lists the input data
file, the total number of points, the minimum and maximum of X and Y
coordinates, and the size of the study area. For each distance, the
output lists the distance, the minimum and maximum /L(d)/ , and the
following table.
Points
Observed /Li(d)/
//
/Li(d) - d/
1
2
:
*
References
*
Anselin, L. (1995) /SpaceStat Tutorial: A Workbook for Using SpaceStat
in the Analysis of Spatial Data/. NCGIS, Santa Barbara.
Anselin, L. (1995) The Local Indicators of Spatial Association – LISA,
/Geographical Analysis/, 27: 93-115.
Boots, B.N. and Getis, A (1988) /Point Pattern Analysis/, Sage: Newbury
Park, CA.
Cliff, A.D. and Ord, J.K (1973) /Spatial Autocorrelation/, Pion: London.
Cliff, A.D. and Ord, J.K (1981) /Spatial Processes: Models and
Applications/, Pion:London.
Cressie, N.A. (1991) /Statistics for Spatial Data,/ John Wiley:
Chichester.
Diggle, P. and Chetwynd, A.G (1991) Second-order analysis of spatial
clustering, /Biometrics /47:1155-1163.
Gatrell, A.C, Bailey, T.C., Diggle, P and Rowlingson, B.S.(1996) Spatial
Point Pattern Analysis and its Application in Geographical Epidemiology,
/Trans. Inst Br Geogr/ NS 2: 256-274.
Getis, A (1984) Interaction Modeling Using Second-order Analysis.
/Environment and Planning/ A 16: 173-183.
Getis, A and J. Franklin (1987) Second-order Neighborhood Analysis of
Mapped Point Patterns. /Ecology/, 68(3): 473-477.
Getis, A and Ord, J.K (1992) The Analysis of Spatial Association by Use
of Distance Statistics, /Geographical Analysis/, 24: 189-206.
Getis, A and Ord, J.K (1996) Local Spatial Statistics: An Overview. In
/Spatial Analysis: Modeling in a GIS Environment/, edited by P. Longley
and M. Batty. John Wiley & Sons: New York.
Ord, J.K and Getis, A., (1995) Local Spatial Autocorrelation Statistics:
Distribution Issues and an Application, /Geographical Analysis/_,_
27(4): 286-306.