Dataplot Vol 1 Vol 2

COMPLETE SPATIAL RANDOMNESS

Name:
COMPLETE SPATIAL RANDOMNESS
Type:
Analysis Command
Purpose:
Perform several tests for complete spatial randomness in the two-dimensional case.
Description:
In spatial analysis, a common first step is to test the data for complete spatial randomness. If the data exhibits complete spatial randomness, this implies that there is no underlying structure in the data and therefore little to be gained from further analysis.

In the univariate case, spatial randomness implies that the data points can be modeled with a uniform distribution. Likewise, in the two-dimensional case, spatial randomness implies that the data can be modeled with a bivariate uniform distribution with zero correlation between the two dimensions. So a quick graphical assessment of spatial randomness can be obtained by simply plotting the points. If there is complete spatial randomness, this plot should show no obvious structure.

This command implements the following formal tests for complete spatial randomness:

1. BIVARIATE CRAMER VON-MISES TEST

The Cramer Von-Mises test is a common test for assessing distributional goodness of fit for a univariate dataset. Specifically, it can be used to test for univariate uniformity.

Zimmerman (1993, see References below) describes an extension of the Cramer Von-Mises test for the case of bivariate uniformity. He modified the bivariate Cramer Von-Mises test so that it is invariant to which vertex of the enclosing rectangle is used as the origin.

For this test, we have a set of n points defined in (X,Y). We also need the coordinates of the enclosing rectangle (if these are not given, they will be based on the data minimum and maximums). The test statistic is

$$\begin{array}{lcl} \bar{w} & = & \frac{1}{4n}\sum_{i=1}^{n}{\sum_{i=1}^{n} {(1 - |u_{i} - u_{j}|) (1 - |v_{i} - v_{j}|)}} \\ & & - 0.5 \sum_{i=1}^{n}{(u_{i}^{2} - u_{i} - 0.5) (v_{i}^{2} - v_{i} - 0.5)} + \frac{n}{9} \end{array}$$

where

$$u_{i} = X_{i}/\mbox{XMAX}$$
$$v_{i} = Y_{i}/\mbox{YMAX}$$

The critical values for the test are taken from tables in the Zimmerman article and are provided for alpha levels 0.50, 0.75, 0.85, 0.90, 0.95, 0.98, and 0.99.

2. MEAN NEAREST NEIGHBORS TEST

The mean nearest neighbor distance test was first described by Clark and Evans (1954, see Refereces below). Dataplot implements a modified version of the Clark-Evans test due to Donnelly (1978) that is described in Zimmerman (1993).

Given a set of N points (X,Y), the test statistic is

$$T = \frac{\bar{Z} - \mu_{\bar{Z}}}{\sigma_{\bar{Z}}}$$

where

$$\bar{Z} = \frac{1}{n}\sum_{i=1}^{n}{Z_{i}}$$

Zi is the distance from the i-th event to it's nearest neighbor

$$\mu_{\bar{Z}} = 0.5 n^{-1/2} + 0.206 n^{-1} + 0.164 n^{-3/2}$$

$$\sigma_{\bar{Z}} = 0.070 n^{-2} + 0.148 n^{-5/2}$$

When there is complete spatial randomness, this statistic follows an approximately standard normal distribution.

3. POLLARD TEST

The Pollard test is also based on nearest neighbors. However, instead of the first nearest neighbor, we check the first through fifth nearest neighbors. The test implemented in Dataplot is a modified version of Pollard's test that is described in Fortin and Dale (2005).

The Pollard test statistic is

$$P_{j} = C_1 [n \ln(C_2) - C_3]/C_4$$

where

$$C_1 = 12 j^{2} n$$

$$C_2 = \sum_{i=1}^{n}{\frac{X_{ij}{2}}{n}}$$

$$C_3 = \sum_{i=1}^{n}{\ln(X_{ij}^{2})}$$

$$C_4 = (6 j n + n + 1) (n-1)$$

and where

j denotes the j-th nearest neighbor

Xij is the distance from the i-th point to it's j-th nearest neighbor

j is 1, 2, 3, 4, or 5

Values of the test statistic near 1 indicate complete spatial randomness. Values less than 1 indicate overdispersion and values greater than 1 indicate underdispersion.

(n-1) Pj has an approximately chi-square distribution with (n-1) degrees of freedom.

Note that there are many ways in which the data can be non-random. In particular, a broad distinction is typically made between 1) random (i.e., complete spatial randomness); 2) underdispersed (clumped or aggregated); and 3) overdispersed (spaced or regular). In addition, non-randomness can be scale dependent. That is, non-randomness may appear either "locally" or "globably". For example, a set of points may appear random if examined in smaller subsets (i.e., local) but not if examined as a whole.

A large number of tests have been developed to test for spatial randomness. These tests vary in what types of non-randomness they are sensitive to. According to Zimmerman, tests based on nearest neighbors tend to be sensitive to "local" non-randomness while being relatively insensitive to "global" characteristics. So these tests are quite good at detecting aggregation and regularity but not good at detecting heterogeneity. On the other hand, the bivariate Cramer Von Mises test is more senstive to global characteristics and less sensitive to local characteristics. So it tends to be good at detecting heterogeneity but not as good at detecting aggregation and regularity. The Pollard test expands the "local" neighborhood by looking at the first through fifth nearest neighbors rather than just the single nearest neighbor. So taken together, this combination of tests should be able to detect many different types of non-randomness.

Syntax:
COMPLETE SPATIAL RANDOMNESS <x> <y>
<SUBSET/EXCEPT/FOR qualification>
where <x> is a response variable;
<y> is a factor identifier variable;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.
Examples:
COMPLETE SPATIAL RANDOMNESS X Y
Note:
For the bivariate Cramer Von-Mises test, the boundaries of the enclosing rectangle can be specified by entering the commands

LET XMIN = <value>
LET XMAX = <value>
LET YMIN = <value>
LET YMAX = <value>
Note:
The following statistics are also supported:

LET A = BIVARIATE CRAMER VON MISES TEST X Y
LET A = BIVARIATE CRAMER VON MISES TEST CV95 X Y
LET A = BIVARIATE CRAMER VON MISES TEST CV05 X Y

LET A = MEAN NEAREST NEIGHBOR DISTANCE TEST X Y
LET A = MEAN NEAREST NEIGHBOR DISTANCE CDF X Y
LET A = MEAN NEAREST NEIGHBOR DISTANCE PVALUE X Y

LET A = POLLARD <ONE/TWO/THREE/FOUR/FIVE> TEST X Y
LET A = POLLARD <ONE/TWO/THREE/FOUR/FIVE> CDF X Y
LET A = POLLARD <ONE/TWO/THREE/FOUR/FIVE> PVALUE X Y

Dataplot statistics can be used in a number of commands. For details, enter

Default:
None
Synonyms:
None
Related Commands:
 GOODNESS OF FIT = Perform a univariate goodness of fit test. INDEPENDENT UNIFORM RANDOM NUMBERS = Generate multivariate uniform random numbers when there is zero correlation. MULTIVARIATE UNIFORM RANDOM NUMBERS = Generate multivariate uniform random numbers when there is non-zero correlation.
Reference:
Dale Zimmerman (1993), "A Bivariate Cramer-Von Mises Type of Test For Spatial Randomness", Journal of the Royal Statistical Society, Series C, Applied Statistics, Vol. 42, No. 1, pp. 43-54.

Clark and Evans (1954), "Distance to Nearest Neighbor as a Measure of Spatial Relationships in Populations", Ecology, 35, pp. 23-30.

Donnelly (1978), "Simulations to Determine the Variance and Edge-Effect of Total Nearest-Neighbor Distance", in Simulation Studies in Archaeology (ed. Hodder), pp. 91-95, London: Cambridge University Press.

Fortin and Dale (2005), "Spatial Analysis: A Guide for Ecologists", Cambridge University Press, pp. 34-35.

Pollard (1971), "On Distance Estimators of Density in Randomly Distributed Forests", Biometrics, 27, pp. 991-1002.

Liu (2001), "A Comparison of Five Distance-Based Methods for Pattern Analysis", Journal of Vegetation Science, 12, pp. 411-416.

Applications:
Spatial Analysis
Implementation Date:
2014/1
Program:

. Step 1:   Generate some random points.
.
let lowlim = data 0 0
let upplim = data 1 1
let n = 40
let m = independent uniform random numbers lowlim upplim n
let y = m1
let x = m2
.
. Step 2:   Plot the uniform random numbers
.
char circle
char fill on
char hw 0.5 0.375
line blank
plot y x
.
. Step 3:   Complete Spatial Randomness Test
.
set write decimals 3
complete spatial randomness test y x

The following output is generated
             Bivariate Cramer Von-Mises Test
for Complete Spatial Randomness

First Response Variable: Y
Second Response Variable: X

H0: Complete Spatial Randomness
Ha: Not Complete Spatial Randomness

Number of Observations:                              40
Data Minimum for X:                               0.001
Data Maximum for X:                               0.973
Data Minimum for Y:                               0.035
Data Maximum for Y:                               0.970

Test Statistic Value:                             0.177

Percent Points of the Reference Distribution
-----------------------------------
Percent Point               Value
-----------------------------------
0.01    =          0.043
0.02    =          0.049
0.05    =          0.057
0.10    =          0.066
0.15    =          0.075
0.25    =          0.088
0.50    =          0.122
0.75    =          0.171
0.85    =          0.206
0.90    =          0.234
0.95    =          0.281
0.98    =          0.342
0.99    =          0.389

Conclusions (Two-Tailed Test)
--------------------------------------------------------
Lower            Upper
Alpha   Critical Value   Critical Value     Conclusion
--------------------------------------------------------
20%            0.066            0.234      Accept H0
10%            0.057            0.281      Accept H0
4%            0.049            0.342      Accept H0
2%            0.043            0.389      Accept H0

Mean Nearest Neighbors Test
for Complete Spatial Randomness

First Response Variable: Y
Second Response Variable: X

H0: Complete Spatial Randomness
Ha: Not Complete Spatial Randomness

Number of Observations:                              40

Test Statistic Value:                            -0.028
Test Statistic CDF:                               0.489
Test Statistic P-Value:                           0.978

Two-Tailed Test for Complete Spatial Randomness

H0: Complete Spatial Randomness
------------------------------------------------------------
Null
Significance           Test       Critical     Hypothesis
Level      Statistic    Value (+/-)     Conclusion
------------------------------------------------------------
50%         -0.028          0.674         ACCEPT
75%         -0.028          1.149         ACCEPT
80%         -0.028          1.282         ACCEPT
90%         -0.028          1.645         ACCEPT
95%         -0.028          1.960         ACCEPT
99%         -0.028          2.576         ACCEPT
99.9%         -0.028          3.290         ACCEPT

Pollard Statistic Test (index = 1)
for Complete Spatial Randomness

First Response Variable: Y
Second Response Variable: X

H0: Complete Spatial Randomness
Ha: Not Complete Spatial Randomness

Number of Observations:                              40
Nearest Neighbor Index:                               1

Test Statistic Value:                             1.014
Test Statistic CDF:                               0.489
Test Statistic P-Value:                           0.978

Two-Tailed Test for Complete Spatial Randomness

H0: Complete Spatial Randomness
---------------------------------------------------------------------------
Lower          Upper           Null
Significance           Test       Critical       Critical     Hypothesis
Level      Statistic    Value (+/-)    Value (+/-)     Conclusion
---------------------------------------------------------------------------
50%         39.545         32.737         44.539         ACCEPT
75%         39.545         29.138         49.292         ACCEPT
80%         39.545         28.196         50.659         ACCEPT
90%         39.545         25.695         54.572         ACCEPT
95%         39.545         23.653         58.119         ACCEPT
99%         39.545         19.995         65.475         ACCEPT
99.9%         39.545         16.272         74.724         ACCEPT

Pollard Statistic Test (index = 2)
for Complete Spatial Randomness

First Response Variable: Y
Second Response Variable: X

H0: Complete Spatial Randomness
Ha: Not Complete Spatial Randomness

Number of Observations:                              40
Nearest Neighbor Index:                               2

Test Statistic Value:                             0.760
Test Statistic CDF:                               0.489
Test Statistic P-Value:                           0.978

Two-Tailed Test for Complete Spatial Randomness

H0: Complete Spatial Randomness
---------------------------------------------------------------------------
Lower          Upper           Null
Significance           Test       Critical       Critical     Hypothesis
Level      Statistic    Value (+/-)    Value (+/-)     Conclusion
---------------------------------------------------------------------------
50%         29.628         32.737         44.539         REJECT
75%         29.628         29.138         49.292         ACCEPT
80%         29.628         28.196         50.659         ACCEPT
90%         29.628         25.695         54.572         ACCEPT
95%         29.628         23.653         58.119         ACCEPT
99%         29.628         19.995         65.475         ACCEPT
99.9%         29.628         16.272         74.724         ACCEPT

Pollard Statistic Test (index = 3)
for Complete Spatial Randomness

First Response Variable: Y
Second Response Variable: X

H0: Complete Spatial Randomness
Ha: Not Complete Spatial Randomness

Number of Observations:                              40
Nearest Neighbor Index:                               3

Test Statistic Value:                             0.998
Test Statistic CDF:                               0.489
Test Statistic P-Value:                           0.978

Two-Tailed Test for Complete Spatial Randomness

H0: Complete Spatial Randomness
---------------------------------------------------------------------------
Lower          Upper           Null
Significance           Test       Critical       Critical     Hypothesis
Level      Statistic    Value (+/-)    Value (+/-)     Conclusion
---------------------------------------------------------------------------
50%         38.923         32.737         44.539         ACCEPT
75%         38.923         29.138         49.292         ACCEPT
80%         38.923         28.196         50.659         ACCEPT
90%         38.923         25.695         54.572         ACCEPT
95%         38.923         23.653         58.119         ACCEPT
99%         38.923         19.995         65.475         ACCEPT
99.9%         38.923         16.272         74.724         ACCEPT

Pollard Statistic Test (index = 4)
for Complete Spatial Randomness

First Response Variable: Y
Second Response Variable: X

H0: Complete Spatial Randomness
Ha: Not Complete Spatial Randomness

Number of Observations:                              40
Nearest Neighbor Index:                               4

Test Statistic Value:                             0.814
Test Statistic CDF:                               0.489
Test Statistic P-Value:                           0.978

Two-Tailed Test for Complete Spatial Randomness

H0: Complete Spatial Randomness
---------------------------------------------------------------------------
Lower          Upper           Null
Significance           Test       Critical       Critical     Hypothesis
Level      Statistic    Value (+/-)    Value (+/-)     Conclusion
---------------------------------------------------------------------------
50%         31.750         32.737         44.539         REJECT
75%         31.750         29.138         49.292         ACCEPT
80%         31.750         28.196         50.659         ACCEPT
90%         31.750         25.695         54.572         ACCEPT
95%         31.750         23.653         58.119         ACCEPT
99%         31.750         19.995         65.475         ACCEPT
99.9%         31.750         16.272         74.724         ACCEPT

Pollard Statistic Test (index = 5)
for Complete Spatial Randomness

First Response Variable: Y
Second Response Variable: X

H0: Complete Spatial Randomness
Ha: Not Complete Spatial Randomness

Number of Observations:                              40
Nearest Neighbor Index:                               5

Test Statistic Value:                             0.867
Test Statistic CDF:                               0.489
Test Statistic P-Value:                           0.978

Two-Tailed Test for Complete Spatial Randomness

H0: Complete Spatial Randomness
---------------------------------------------------------------------------
Lower          Upper           Null
Significance           Test       Critical       Critical     Hypothesis
Level      Statistic    Value (+/-)    Value (+/-)     Conclusion
---------------------------------------------------------------------------
50%         33.831         32.737         44.539         ACCEPT
75%         33.831         29.138         49.292         ACCEPT
80%         33.831         28.196         50.659         ACCEPT
90%         33.831         25.695         54.572         ACCEPT
95%         33.831         23.653         58.119         ACCEPT
99%         33.831         19.995         65.475         ACCEPT
99.9%         33.831         16.272         74.724         ACCEPT

.
. Step 4:   Demonstrate how to extract individual statistics
.
let tval  = mean nearest neighor distance test   x y
let tcdf  = mean nearest neighor distance cdf    x y
let tpval = mean nearest neighor distance pvalue x y
print tval tcdf tpval

PARAMETERS AND CONSTANTS--

TVAL    --          0.489
TCDF    --         -0.028
TPVAL   --          0.978

let bval  = bivariate cramer von mises test x y
let cv95  = bivariate cramer von mises 95 critical value x y
let cv05  = bivariate cramer von mises 05 critical value x y
print bval cv95 cv05

PARAMETERS AND CONSTANTS--

BVAL    --          0.177
CV95    --          0.281
CV05    --          0.057

let xmin = 0
let xmax = 1
let ymin = 0
let ymax = 1
let bval2  = bivariate cramer von mises test x y

PARAMETERS AND CONSTANTS--

BVAL2   --          0.145
CV95    --          0.281
CV05    --          0.057
.
let pol1     = pollard one test   x y
let pol1cdf  = pollard one cdf    x y
let pol1pval = pollard one pvalue x y
print pol1 pol1cdf pol1pval

PARAMETERS AND CONSTANTS--

POL1    --          1.014
POL1CDF --          0.554
POL1PVAL--          0.891

.
let pol2     = pollard two test   x y
let pol2cdf  = pollard two cdf    x y
let pol2pval = pollard two pvalue x y
print pol2 pol2cdf pol2pval

PARAMETERS AND CONSTANTS--

POL2    --          0.760
POL2CDF --          0.139
POL2PVAL--          0.279

.
let pol3     = pollard three test   x y
let pol3cdf  = pollard three cdf    x y
let pol3pval = pollard three pvalue x y
print pol3 pol3cdf pol3pval

PARAMETERS AND CONSTANTS--

POL3    --          0.998
POL3CDF --          0.527
POL3PVAL--          0.947

.
let pol4     = pollard four test   x y
let pol4cdf  = pollard four cdf    x y
let pol4pval = pollard four pvalue x y
print pol4 pol4cdf pol4pval

PARAMETERS AND CONSTANTS--

POL4    --          0.814
POL4CDF --          0.211
POL4PVAL--          0.423

.
let pol5     = pollard five test   x y
let pol5cdf  = pollard five cdf    x y
let pol5pval = pollard five pvalue x y
print pol5 pol5cdf pol5pval

 PARAMETERS AND CONSTANTS--

POL5    --          0.867
POL5CDF --          0.296
POL5PVAL--          0.591



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Date created: 09/11/2014
Last updated: 09/11/2014