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KSED procedure

Calculates the standard error for K function differences under random labelling (M.A. Mugglestone, S.A. Harding, B.Y.Y. Lee, P.J. Diggle & B.S. Rowlingson).

Option

`PRINT` = string token Controls printed output (`summary`); default `summ`

Parameters

`Y1` = variates Vertical coordinates of the first spatial point patterns; no default – this parameter must be set Horizontal coordinates of the first spatial point patterns; no default – this parameter must be set Vertical coordinates of the second spatial point patterns; no default – this parameter must be set Horizontal coordinates of the second spatial point patterns; no default – this parameter must be set Vertical coordinates of the polygons; no default – this parameter must be set Horizontal coordinates of the polygons; no default – this parameter must be set Vectors of distances to use; no default – this parameter must be set Variates to receive the values of the standard error of the difference between the K functions for the two patterns under random labelling Saves the variance-covariance matrix Saves the variance of Khat for first spatial point pattern Saves the variance of Khat for second spatial point pattern Saves the covariance of Khat for the two samples

Description

The K function, or reduced second-order moment function, relates to the distribution of the inter-event distances between all ordered pairs of events in a spatial point pattern (see Diggle 1983). The procedure `KHAT` can be used to obtain an approximately unbiased estimator of K(s) for an observed pattern, and this may be used to investigate the degree of clustering/regularity in the pattern. Patterns consisting of two different types of events may be separated into two patterns, one for each type of event. The difference between the K functions for the two univariate patterns may then be used to investigate whether the two types of events display similar degrees of clustering/regularity. (If the difference between the K functions is positive (negative) then the first pattern is more (less) strongly clustered than the second.)

The term random labelling is used to represent the hypothesis that the spatial distributions of different types of events within an overall pattern are completely random. The expected value of the difference between two K functions under random labelling is zero. The standard error of the estimated difference can be obtained using the method of Diggle & Chetwynd (1991).

The procedure `KSED` calculates the standard error for the difference between two K functions under random labelling. The data required by the procedure are the coordinates of two spatial point patterns (specified by parameters `X1`, `Y1`, `X2` and `Y2`), the coordinates of a polygon containing the points (specified by the parameters `XPOLYGON` and `YPOLYGON`) and a vector of distances at which to estimate the K functions (specified by the parameter `S`). The standard error for the difference between the two K functions for each distance in `S` can be saved using the parameter `KSED`. The `VCOVARIANCE` parameter can be used to save the variance-covariance matrix for the difference between the K functions for the two patterns. The variances for the two K functions can be saved using the `VK11` and `VK12` parameters. The covariance for the K function of the two point patterns can be saved using the `VK12` parameter.

Printed output is controlled using the `PRINT` option. The default setting of `summary` prints the distances at which the standard error is calculated and the values of the standard error under the headings `S` and `KSED`. The variance and covariance for the two K functions are also displayed under the headings `VK11`, `VK22` and `VK12`.

Option: `PRINT`.

Parameters: `Y1`, `X1`, `Y2`, `X2`, `YPOLYGON`, `XPOLYGON`, `S`, `KSED`, `VCOVARIANCE`, `VK1`, `VK2`, `VK12`.

Method

A procedure `PTCHECKXY` is called to check that `X1` and `Y1` have identical restrictions. Similar checks are made on `X2` and `Y2`, and on `XPOLYGON` and `YPOLYGON`. The procedure then calls `PTCLOSEPOLYGON` to close the polygon specified by `XPOLYGON` and `YPOLYGON`. The `SORT `function is then used to create a variate containing the distances in `S` arranged in ascending order. (The original variate is left unchanged.) The procedure then calls `APPEND` to combine the horizontal coordinates for both patterns, and again to combine the vertical coordinates. The coordinates of the closed polygon, the sorted values of `S` and the combined coordinates for the two patterns are then passed to the Fortran program using a procedure `PTPASS`. This program calculates the variance-covariance matrix for the difference between the K functions for the two patterns and the variance of the K function for each sample. Finally, the standard error for the difference between the two K functions is obtained using the `CALCULATE` directive by taking the square root of the values on the diagonal of the variance-covariance matrix.

Action with `RESTRICT`

The variates `X1`, `Y1`, `X2`, `Y2`, `XPOLYGON`, `YPOLYGON` and `S` may be restricted as long as `X1` has the same restriction as `Y1`, `X2` has the same restriction as `Y2`, and `XPOLYGON` has the same restriction as `YPOLYGON`. Only the subset of values specified by each restriction will be included in the calculations.

References

Diggle, P.J. (1983). Statistical Analysis of Spatial Point Patterns. Academic Press, London.

Diggle, P.J. & Chetwynd, A.G. (1991). Second-order analysis of spatial clustering. Biometrics, 47, 1155-63.

Procedures: `FHAT`, `GHAT`, `KHAT`, `KSTHAT`, `K12HAT`.

Commands for: Spatial statistics.

Example

```CAPTION   'KSED example'; STYLE=meta
VARIATE   pykx,pyky
0.077 0.045 0.324 0.008 0.332 0.018 0.354 0.008 0.366 0.008
0.534 0.057 0.561 0.045 0.648 0.059 0.739 0.020 0.943 0.047
0.012 0.170 0.326 0.134 0.344 0.154 0.715 0.178 0.852 0.138
0.887 0.162 0.905 0.107 0.136 0.237 0.156 0.235 0.273 0.277
0.312 0.253 0.504 0.257 0.682 0.275 0.779 0.241 0.923 0.257
0.040 0.322 0.073 0.372 0.308 0.391 0.338 0.362 0.514 0.316
0.652 0.310 0.664 0.310 0.763 0.322 0.779 0.340 0.834 0.352
0.259 0.407 0.514 0.431 0.670 0.443 0.709 0.470 0.802 0.472
0.899 0.455 0.138 0.522 0.202 0.575 0.265 0.516 0.308 0.545
0.352 0.601 0.451 0.545 0.518 0.549 0.585 0.561 0.680 0.557
0.729 0.573 0.729 0.545 0.109 0.656 0.130 0.640 0.630 0.686
0.638 0.646 0.638 0.636 0.725 0.664 0.889 0.609 0.073 0.791
0.245 0.743 0.458 0.787 0.484 0.751 0.696 0.747 0.747 0.713
0.909 0.745 0.945 0.771 0.132 0.838 0.277 0.818 0.350 0.814
0.385 0.881 0.739 0.808 0.765 0.893 0.909 0.804 0.970 0.877
0.364 0.980 0.980 0.913 :
VARIATE   metx,mety
0.024 0.071 0.043 0.030 0.103 0.097 0.107 0.057 0.130 0.099
0.132 0.059 0.166 0.026 0.166 0.012 0.285 0.051 0.204 0.063
0.036 0.119 0.047 0.146 0.081 0.168 0.190 0.128 0.202 0.138
0.213 0.115 0.227 0.174 0.269 0.182 0.008 0.241 0.036 0.292
0.043 0.283 0.059 0.213 0.245 0.294 0.257 0.209 0.261 0.263
0.294 0.213 0.385 0.206 0.411 0.245 0.431 0.213 0.455 0.237
0.490 0.285 0.431 0.020 0.455 0.457 0.468 0.099 0.480 0.012
0.486 0.083 0.502 0.047 0.431 0.154 0.478 0.132 0.504 0.126
0.504 0.113 0.536 0.107 0.565 0.146 0.609 0.119 0.617 0.154
0.636 0.158 0.672 0.164 0.727 0.119 0.763 0.172 0.771 0.182
0.775 0.170 0.818 0.125 0.840 0.119 0.840 0.099 0.921 0.136
0.982 0.154 0.543 0.026 0.601 0.073 0.648 0.028 0.666 0.040
0.779 0.047 0.842 0.008 0.852 0.047 0.949 0.032 0.581 0.267
0.672 0.217 0.706 0.219 0.763 0.300 0.767 0.209 0.779 0.225
0.818 0.209 0.838 0.296 0.885 0.219 0.949 0.241 0.014 0.328
0.051 0.368 0.057 0.356 0.154 0.308 0.178 0.322 0.285 0.374
0.356 0.375 0.482 0.314 0.530 0.350 0.585 0.316 0.648 0.352
0.660 0.342 0.704 0.381 0.739 0.330 0.846 0.383 0.850 0.308
0.917 0.308 0.921 0.346 0.974 0.348 0.974 0.312 0.990 0.332
0.036 0.470 0.036 0.421 0.103 0.466 0.144 0.494 0.170 0.439
0.575 0.455 0.595 0.415 0.652 0.474 0.751 0.502 0.773 0.486
0.838 0.427 0.858 0.482 0.889 0.494 0.931 0.427 0.931 0.403
0.937 0.417 0.972 0.419 0.051 0.534 0.065 0.589 0.083 0.573
0.117 0.553 0.237 0.557 0.243 0.542 0.259 0.593 0.302 0.579
0.302 0.551 0.304 0.601 0.320 0.589 0.326 0.532 0.368 0.518
0.431 0.551 0.462 0.514 0.500 0.542 0.551 0.530 0.652 0.585
0.662 0.577 0.767 0.545 0.806 0.597 0.826 0.538 0.874 0.549
0.006 0.668 0.028 0.648 0.233 0.636 0.304 0.656 0.405 0.652
0.425 0.680 0.435 0.648 0.510 0.644 0.534 0.670 0.634 0.609
0.656 0.640 0.672 0.652 0.727 0.613 0.931 0.656 0.968 0.623
0.022 0.708 0.063 0.787 0.103 0.711 0.273 0.715 0.324 0.757
0.346 0.709 0.375 0.800 0.381 0.773 0.435 0.777 0.437 0.745
0.557 0.771 0.595 0.708 0.652 0.747 0.822 0.759 0.850 0.794
0.885 0.785 0.953 0.763 0.004 0.879 0.020 0.854 0.071 0.874
0.109 0.800 0.113 0.822 0.134 0.822 0.152 0.834 0.162 0.877
0.202 0.810 0.225 0.852 0.257 0.806 0.300 0.850 0.328 0.806
0.356 0.903 0.409 0.875 0.437 0.905 0.447 0.852 0.455 0.842
0.488 0.850 0.510 0.905 0.526 0.854 0.557 0.087 0.617 0.846
0.642 0.858 0.652 0.814 0.751 0.893 0.820 0.822 0.862 0.820
0.903 0.899 0.907 0.854 0.923 0.834 0.043 0.913 0.051 0.990
0.089 0.970 0.089 0.931 0.164 0.933 0.184 0.937 0.204 0.929
0.249 0.964 0.281 0.935 0.281 0.921 0.287 0.988 0.411 0.945
0.449 0.929 0.494 0.931 0.508 0.998 0.518 0.929 0.542 0.992
0.648 0.980 0.711 0.951 0.755 0.925 0.830 0.986 0.858 0.949
0.273 0.427 0.356 0.427 0.494 0.409 0.561 0.500 0.939 0.992
0.460 0.036 :
VARIATE   xpoly; VALUES=!(0,1,1,0)
&         ypoly; VALUES=!(0,0,1,1)
&         s; VALUES=!(0.01,0.02...0.1)
KHAT      [PRINT=*] Y=pyky; X=pykx; YPOLYGON=ypoly; XPOLYGON=xpoly;\
S=s; KVALUES=kpyknotic
KHAT      [PRINT=*] Y=mety; X=metx; YPOLYGON=ypoly; XPOLYGON=xpoly;\
S=s; KVALUES=kmetaphase
CALCULATE kdiff = kpyknotic - kmetaphase
KSED      [PRINT=*] Y1=pyky; X1=pykx; Y2=mety; X2=metx;\
YPOLYGON=ypoly; XPOLYGON=xpoly; S=s; KSED=sekdiff
PRINT     s,kdiff,sekdiff
```
Updated on March 7, 2019