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# K12HAT procedure

Calculates an estimate of the bivariate K function (M.A. Mugglestone, S.A. Harding, B.Y.Y. Lee, P.J. Diggle & B.S. Rowlingson).

### Option

`PRINT` = string token What to print (`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 each polygon; no default – this parameter must be set Horizontal coordinates of each polygon; no default – this parameter must be set Vectors of distances to use; no default – this parameter must be set Variates to receive the estimated values of the bivariate K functions

### Description

The bivariate K function, or reduced second-order moment function, relates to the distribution of inter-event distances in a spatial point pattern consisting of two different types of events (see Diggle 1983). Suppose that the two types of events are classified as type j (j = 1, 2), then there are four bivariate K functions, Kij(s) (i, j = 1, 2), each defined as the expected number of type j events within distance s of an arbitrary type i event, divided by the overall density of type j events per unit area. Each Kjj(s) corresponds exactly to the function K(s) for a univariate process, and can be used to investigate whether an observed pattern is random, clustered or regular (see the procedure `KHAT`). The two remaining functions, K12(s) and K21(s), can be used to investigate whether the spatial patterns of the two types of events are independent or positively / negatively correlated.

Assuming that the bivariate process is stationary and isotropic (meaning that its properties are invariant under translations and rotations of the coordinate space), then K12(s) = K21(s). An approximately unbiased estimator for K12(s) which incorporates corrections for edge effects can, therefore, be obtained by taking a weighted sum of separate estimators for K12(s) and K21(s) (these being analogous to the edge-corrected estimator for the K function of a univariate process – see the procedure `KHAT`). The final estimator for K12(s), due to Lotwick & Silverman (1982), is given by

K2(s) = (n2 × K2(s) + nl × K1(s)) / (n1 + n2),

where K2(s) and K1(s) are the separate estimators for K12(s) and K21(s), and nj (j = 1, 2) is the number of type j events.

For independent processes, the expected number of type j events which lie within a distance s of an arbitrary type i event is simply the area of a circle of radius s, multiplied by the overall density of events. Thus, the bivariate K function for independent processes is given by

K12(s) = π × (s2).

The bivariate K function for positively (negatively) correlated processes will tend to be larger (smaller) than the values given by the above expression, at least for small distances.

The procedure `K12HAT` calculates Lotwick & Silverman’s (1982) estimator for K12(s) given the coordinates of two spatial point patterns (specified by the 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 (specified by the parameter `S`). The output of the procedure is a vector of estimates of K12(s) corresponding to the distances in `S`. The estimated bivariate K function can be saved using the parameter `KVALUES`.

Printed output is controlled using the `PRINT` option. The default setting of `summary` prints the distances at which the bivariate K function is estimated and the estimates themselves under the headings `S` and `KVALUES`.

Option: `PRINT`.

Parameters: `Y1`, `X1`, `Y2`, `X2`, `YPOLYGON`, `XPOLYGON`, `S`, `KVALUES`.

### Method

A procedure `PTCHECKXY` is called to check that `X1` and `Y1` have identical restrictions. Similar checks are made on `X2` and `Y2`, and `XPOLYGON` and `YPOLYGON`. Procedure `PTCLOSEPOLYGON` is called 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 a procedure `PTPASS` to call a Fortran program to calculate the bivariate K function.

### 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.

Lotwick, H.W. & Silverman, B.W. (1982). Methods for analysing spatial processes of several types of points. Journal of the Royal Statistical Society, Series B, 44, 406-413.

### See also

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

Commands for: Spatial statistics.

### Example

```CAPTION   'K12HAT example'; STYLE=meta
VARIATE   hickx,hicky
READ      [SETNVALUES=yes] hickx,hicky
0.069 0.014 0.049 0.057 0.094 0.015 0.106 0.036 0.130 0.126
0.081 0.079 0.027 0.068 0.027 0.069 0.034 0.081 0.066 0.132
0.040 0.115 0.031 0.108 0.040 0.074 0.041 0.074 0.039 0.156
0.023 0.207 0.098 0.142 0.084 0.176 0.053 0.221 0.035 0.300
0.036 0.310 0.050 0.284 0.027 0.327 0.108 0.307 0.134 0.328
0.143 0.353 0.128 0.379 0.105 0.411 0.129 0.385 0.036 0.358
0.030 0.360 0.028 0.354 0.031 0.376 0.030 0.453 0.068 0.453
0.075 0.490 0.047 0.474 0.023 0.486 0.069 0.481 0.076 0.422
0.082 0.458 0.089 0.444 0.107 0.437 0.146 0.444 0.089 0.473
0.078 0.470 0.084 0.513 0.110 0.516 0.132 0.518 0.122 0.539
0.124 0.555 0.124 0.563 0.103 0.537 0.091 0.555 0.118 0.554
0.027 0.539 0.079 0.562 0.025 0.605 0.026 0.587 0.030 0.571
0.039 0.594 0.047 0.582 0.054 0.580 0.077 0.594 0.070 0.635
0.054 0.617 0.042 0.589 0.069 0.567 0.081 0.620 0.092 0.575
0.102 0.571 0.147 0.569 0.131 0.618 0.098 0.620 0.120 0.608
0.081 0.671 0.102 0.676 0.116 0.680 0.118 0.683 0.094 0.701
0.079 0.648 0.065 0.671 0.047 0.701 0.053 0.709 0.035 0.701
0.030 0.692 0.026 0.684 0.047 0.666 0.064 0.694 0.032 0.711
0.024 0.727 0.027 0.741 0.035 0.725 0.040 0.732 0.045 0.721
0.070 0.712 0.045 0.760 0.032 0.772 0.080 0.721 0.078 0.725
0.092 0.732 0.097 0.720 0.136 0.763 0.127 0.773 0.106 0.747
0.131 0.752 0.109 0.772 0.145 0.787 0.095 0.804 0.129 0.807
0.142 0.844 0.135 0.843 0.100 0.794 0.010 0.795 0.027 0.787
0.028 0.799 0.029 0.814 0.064 0.838 0.009 0.819 0.027 0.877
0.038 0.866 0.037 0.882 0.054 0.851 0.066 0.867 0.070 0.903
0.068 0.912 0.063 0.910 0.035 0.900 0.034 0.899 0.024 0.907
0.001 0.864 0.054 0.865 0.061 0.863 0.077 0.867 0.085 0.855
0.116 0.882 0.120 0.886 0.136 0.863 0.142 0.881 0.127 0.899
0.115 0.905 0.101 0.921 0.137 0.911 0.123 0.893 0.101 0.923
0.123 0.925 0.136 0.978 0.127 0.939 0.115 0.942 0.114 0.950
0.108 0.965 0.105 0.959 0.104 0.975 0.081 0.974 0.078 0.952
0.026 0.939 0.035 0.942 0.071 0.931 0.076 0.953 0.065 0.981
0.064 0.981 0.047 0.749 0.041 0.960 0.031 0.965 0.031 0.963
0.052 0.949 0.207 0.918 0.183 0.908 0.172 0.908 0.175 0.880
0.216 0.892 0.219 0.853 0.160 0.864 0.198 0.852 0.278 0.912
0.245 0.912 0.229 0.896 0.240 0.889 0.246 0.896 0.274 0.890
0.219 0.945 0.244 0.947 0.279 0.979 0.288 0.924 0.234 0.943
0.209 0.922 0.168 0.922 0.163 0.932 0.170 0.763 0.163 0.767
0.154 0.767 0.159 0.756 0.157 0.723 0.179 0.748 0.166 0.724
0.281 0.780 0.258 0.768 0.229 0.815 0.238 0.816 0.249 0.829
0.248 0.835 0.279 0.804 0.265 0.802 0.225 0.792 0.228 0.789
0.153 0.831 0.162 0.846 0.167 0.812 0.176 0.815 0.174 0.831
0.203 0.839 0.211 0.827 0.160 0.784 0.186 0.845 0.155 0.607
0.193 0.568 0.174 0.598 0.166 0.610 0.268 0.592 0.258 0.590
0.260 0.582 0.253 0.703 0.276 0.677 0.289 0.672 0.207 0.700
0.156 0.670 0.162 0.680 0.148 0.661 0.161 0.489 0.209 0.495
0.224 0.477 0.228 0.487 0.222 0.457 0.220 0.445 0.238 0.550
0.247 0.519 0.229 0.522 0.221 0.521 0.254 0.498 0.262 0.524
0.159 0.555 0.212 0.527 0.158 0.527 0.201 0.521 0.193 0.325
0.216 0.325 0.286 0.408 0.289 0.409 0.223 0.402 0.229 0.400
0.211 0.395 0.212 0.255 0.184 0.288 0.198 0.062 0.183 0.0
0.148 0.015 0.185 0.013 0.251 0.066 0.256 0.013 0.271 0.016
0.258 0.019 0.234 0.012 0.242 0.126 0.242 0.136 0.268 0.108
0.265 0.088 0.253 0.104 0.256 0.071 0.398 0.154 0.416 0.167
0.424 0.156 0.413 0.182 0.370 0.251 0.419 0.273 0.385 0.273
0.290 0.293 0.394 0.307 0.404 0.312 0.415 0.306 0.417 0.326
0.372 0.368 0.302 0.008 0.320 0.061 0.294 0.066 0.406 0.008
0.289 0.095 0.339 0.131 0.409 0.408 0.404 0.418 0.335 0.407
0.354 0.422 0.310 0.438 0.332 0.426 0.317 0.465 0.338 0.477
0.394 0.436 0.417 0.450 0.427 0.436 0.430 0.447 0.474 0.477
0.329 0.532 0.330 0.531 0.335 0.508 0.353 0.561 0.323 0.613
0.348 0.620 0.347 0.667 0.409 0.649 0.356 0.709 0.304 0.701
0.331 0.718 0.307 0.716 0.323 0.737 0.331 0.727 0.333 0.746
0.342 0.725 0.347 0.727 0.358 0.725 0.360 0.769 0.342 0.771
0.344 0.761 0.346 0.755 0.302 0.752 0.304 0.769 0.318 0.781
0.333 0.763 0.387 0.722 0.425 0.719 0.370 0.775 0.405 0.847
0.419 0.819 0.408 0.833 0.398 0.832 0.390 0.839 0.367 0.830
0.304 0.785 0.308 0.792 0.314 0.802 0.328 0.787 0.329 0.847
0.312 0.654 0.321 0.865 0.325 0.882 0.323 0.902 0.294 0.909
0.380 0.856 0.389 0.927 0.425 0.973 0.384 0.972 0.366 0.973
0.307 0.985 0.291 0.986 0.412 0.946 0.465 0.897 0.463 0.915
0.440 0.987 0.461 0.987 0.497 0.958 0.438 0.756 0.553 0.760
0.576 0.833 0.475 0.834 0.436 0.841 0.513 0.703 0.538 0.522
0.459 0.486 0.490 0.489 0.498 0.471 0.501 0.435 0.501 0.446
0.444 0.435 0.578 0.501 0.543 0.470 0.556 0.465 0.540 0.452
0.448 0.542 0.459 0.537 0.491 0.522 0.475 0.522 0.459 0.527
0.492 0.561 0.456 0.530 0.438 0.350 0.473 0.332 0.473 0.308
0.556 0.354 0.437 0.421 0.446 0.412 0.508 0.410 0.482 0.403
0.487 0.182 0.439 0.157 0.534 0.254 0.569 0.127 0.495 0.119
0.439 0.104 0.459 0.094 0.693 0.006 0.686 0.008 0.708 0.133
0.657 0.128 0.577 0.105 0.608 0.098 0.619 0.132 0.609 0.126
0.631 0.168 0.621 0.201 0.620 0.203 0.617 0.190 0.708 0.184
0.671 0.218 0.619 0.347 0.659 0.354 0.694 0.376 0.682 0.377
0.601 0.385 0.585 0.404 0.616 0.491 0.615 0.492 0.636 0.496
0.639 0.492 0.682 0.442 0.712 0.458 0.706 0.476 0.696 0.466
0.672 0.476 0.658 0.494 0.714 0.501 0.661 0.525 0.673 0.534
0.660 0.562 0.629 0.516 0.617 0.537 0.604 0.543 0.588 0.547
0.584 0.595 0.609 0.597 0.712 0.584 0.703 0.618 0.667 0.702
0.666 0.715 0.579 0.718 0.641 0.739 0.602 0.753 0.695 0.737
0.686 0.751 0.672 0.762 0.656 0.721 0.659 0.795 0.598 0.839
0.619 0.841 0.584 0.855 0.629 0.910 0.585 0.854 0.654 0.851
0.654 0.897 0.661 0.937 0.683 0.948 0.680 0.987 0.677 0.923
0.663 0.958 0.627 0.952 0.639 0.935 0.640 0.983 0.640 0.983
0.623 0.986 0.608 0.979 0.596 0.989 0.639 0.983 0.767 0.922
0.733 0.902 0.747 0.883 0.779 0.873 0.839 0.913 0.813 0.960
0.848 0.981 0.852 0.979 0.854 0.984 0.858 0.944 0.721 0.978
0.782 0.974 0.776 0.947 0.728 0.924 0.737 0.718 0.838 0.782
0.811 0.769 0.821 0.752 0.831 0.744 0.855 0.844 0.823 0.812
0.820 0.805 0.828 0.813 0.827 0.811 0.819 0.813 0.815 0.819
0.758 0.819 0.768 0.831 0.742 0.648 0.789 0.620 0.793 0.585
0.771 0.597 0.763 0.583 0.748 0.604 0.831 0.650 0.803 0.637
0.842 0.635 0.816 0.591 0.804 0.588 0.798 0.706 0.803 0.700
0.806 0.701 0.811 0.665 0.801 0.680 0.832 0.674 0.744 0.703
0.786 0.711 0.750 0.671 0.769 0.495 0.738 0.471 0.756 0.472
0.764 0.470 0.777 0.443 0.753 0.442 0.748 0.453 0.738 0.446
0.731 0.439 0.802 0.465 0.848 0.479 0.823 0.438 0.804 0.545
0.827 0.544 0.830 0.538 0.842 0.555 0.831 0.509 0.741 0.537
0.782 0.550 0.774 0.510 0.727 0.526 0.841 0.338 0.803 0.413
0.816 0.395 0.746 0.407 0.790 0.393 0.725 0.422 0.784 0.389
0.727 0.201 0.773 0.185 0.832 0.149 0.802 0.155 0.854 0.268
0.832 0.222 0.829 0.223 0.825 0.233 0.846 0.256 0.778 0.279
0.789 0.253 0.729 0.231 0.782 0.014 0.794 0.032 0.823 0.015
0.785 0.078 0.748 0.123 0.950 0.021 0.926 0.052 0.943 0.093
0.951 0.080 0.907 0.115 0.895 0.149 0.908 0.148 0.976 0.181
1.000 0.199 0.991 0.199 0.970 0.211 0.944 0.205 0.972 0.264
0.923 0.278 0.916 0.277 0.997 0.234 0.880 0.227 0.898 0.254
0.866 0.293 0.881 0.289 0.906 0.305 0.896 0.340 0.892 0.343
0.883 0.325 0.874 0.330 0.893 0.315 0.913 0.310 0.929 0.315
0.968 0.285 0.932 0.324 0.924 0.333 0.948 0.341 0.942 0.364
0.949 0.359 0.962 0.370 0.998 0.421 0.987 0.292 0.963 0.323
0.951 0.325 0.946 0.325 0.964 0.424 0.951 0.399 0.881 0.394
0.916 0.396 0.922 0.380 0.927 0.399 0.892 0.417 0.885 0.425
0.872 0.406 0.872 0.427 0.908 0.430 0.921 0.501 0.908 0.479
0.877 0.465 0.972 0.444 0.977 0.445 0.953 0.496 0.977 0.534
0.950 0.557 0.890 0.515 0.939 0.511 0.918 0.544 0.905 0.621
0.897 0.607 0.924 0.591 0.955 0.663 0.969 0.657 0.945 0.692
0.968 0.607 0.982 0.604 0.978 0.628 0.966 0.697 0.882 0.662
0.890 0.665 0.890 0.661 0.889 0.652 0.890 0.647 0.927 0.654
0.946 0.660 0.932 0.701 0.903 0.712 0.902 0.688 0.885 0.696
0.902 0.697 0.895 0.747 0.926 0.775 0.930 0.763 0.905 0.760
0.894 0.774 0.895 0.781 0.896 0.782 0.889 0.749 0.892 0.762
0.952 0.736 0.960 0.722 0.973 0.722 0.996 0.736 0.950 0.760
0.950 0.747 0.938 0.748 0.936 0.771 0.969 0.737 0.953 0.791
0.957 0.805 0.959 0.800 0.996 0.802 0.972 0.837 0.974 0.827
0.944 0.826 0.883 0.789 0.874 0.794 0.919 0.792 0.916 0.828
0.908 0.854 0.900 0.841 0.881 0.855 0.868 0.857 0.921 0.884
0.923 0.921 0.934 0.856 0.963 0.904 0.936 0.922 0.968 0.902
0.984 0.931 0.998 0.975 0.987 0.977 0.992 0.986 0.963 0.985
0.944 0.977 0.930 0.277 0.957 0.939 0.994 0.985 0.883 0.958
0.909 0.963 0.925 0.989 0.879 0.985 0.896 0.961 0.891 0.991
0.858 0.984 0.391 0.402 0.421 0.383 :
VARIATE   maplex,mapley
READ      [SETNVALUES=yes] maplex,mapley
0.121 0.041 0.097 0.064 0.135 0.064 0.131 0.106 0.116 0.079
0.067 0.121 0.058 0.083 0.009 0.147 0.116 0.157 0.137 0.169
0.119 0.200 0.096 0.194 0.081 0.208 0.094 0.205 0.106 0.156
0.140 0.184 0.096 0.195 0.095 0.190 0.077 0.218 0.081 0.242
0.131 0.266 0.132 0.267 0.128 0.276 0.104 0.261 0.092 0.254
0.087 0.258 0.070 0.223 0.140 0.334 0.066 0.732 0.227 0.865
0.249 0.945 0.146 0.721 0.260 0.744 0.195 0.583 0.275 0.616
0.285 0.571 0.275 0.570 0.284 0.683 0.290 0.697 0.274 0.648
0.200 0.455 0.250 0.466 0.248 0.478 0.255 0.567 0.170 0.294
0.251 0.301 0.225 0.292 0.265 0.413 0.182 0.372 0.159 0.169
0.186 0.175 0.198 0.212 0.202 0.180 0.206 0.206 0.196 0.170
0.188 0.147 0.173 0.146 0.170 0.161 0.226 0.260 0.255 0.266
0.265 0.228 0.194 0.282 0.197 0.266 0.187 0.253 0.152 0.224
0.143 0.031 0.200 0.042 0.207 0.006 0.176 0.028 0.177 0.024
0.269 0.067 0.274 0.050 0.228 0.094 0.150 0.083 0.158 0.132
0.170 0.126 0.176 0.114 0.301 0.173 0.318 0.143 0.329 0.156
0.339 0.201 0.314 0.188 0.313 0.187 0.310 0.209 0.376 0.166
0.406 0.167 0.425 0.202 0.373 0.174 0.372 0.173 0.372 0.237
0.398 0.228 0.425 0.271 0.334 0.271 0.213 0.096 0.147 0.139
0.179 0.083 0.175 0.103 0.353 0.029 0.334 0.042 0.303 0.042
0.295 0.068 0.351 0.049 0.358 0.002 0.377 0.009 0.410 0.003
0.373 0.063 0.402 0.026 0.380 0.094 0.408 0.120 0.425 0.122
0.372 0.077 0.331 0.137 0.343 0.462 0.310 0.489 0.287 0.461
0.399 0.481 0.392 0.504 0.416 0.534 0.400 0.548 0.372 0.564
0.402 0.531 0.318 0.517 0.347 0.517 0.346 0.518 0.344 0.537
0.354 0.519 0.335 0.568 0.331 0.574 0.314 0.568 0.305 0.552
0.350 0.629 0.376 0.578 0.391 0.584 0.425 0.585 0.432 0.578
0.429 0.596 0.429 0.597 0.430 0.622 0.430 0.639 0.415 0.621
0.413 0.622 0.398 0.626 0.396 0.627 0.369 0.632 0.368 0.631
0.385 0.630 0.368 0.662 0.369 0.667 0.402 0.649 0.410 0.649
0.417 0.666 0.402 0.670 0.412 0.681 0.416 0.684 0.421 0.697
0.406 0.708 0.398 0.690 0.379 0.695 0.422 0.661 0.346 0.663
0.354 0.655 0.356 0.655 0.351 0.709 0.325 0.698 0.294 0.689
0.312 0.699 0.354 0.718 0.424 0.971 0.318 0.978 0.494 0.870
0.531 0.874 0.529 0.860 0.508 0.948 0.463 0.734 0.511 0.746
0.532 0.749 0.548 0.733 0.517 0.731 0.516 0.716 0.486 0.801
0.439 0.632 0.439 0.585 0.439 0.584 0.435 0.574 0.522 0.634
0.552 0.632 0.545 0.569 0.524 0.629 0.499 0.712 0.519 0.696
0.516 0.682 0.537 0.639 0.436 0.682 0.499 0.478 0.521 0.439
0.503 0.430 0.450 0.463 0.459 0.434 0.435 0.460 0.474 0.421
0.571 0.470 0.552 0.541 0.568 0.540 0.537 0.530 0.543 0.505
0.524 0.521 0.455 0.555 0.478 0.526 0.463 0.522 0.456 0.498
0.483 0.327 0.502 0.290 0.466 0.289 0.465 0.289 0.494 0.340
0.515 0.320 0.566 0.298 0.545 0.335 0.524 0.403 0.570 0.379
0.534 0.372 0.512 0.399 0.642 0.429 0.573 0.396 0.526 0.359
0.466 0.405 0.447 0.202 0.479 0.186 0.484 0.144 0.462 0.162
0.460 0.147 0.503 0.208 0.549 0.210 0.555 0.192 0.534 0.174
0.526 0.161 0.514 0.229 0.504 0.254 0.500 0.231 0.499 0.232
0.485 0.234 0.484 0.234 0.463 0.225 0.458 0.037 0.476 0.045
0.494 0.021 0.484 0.009 0.475 0.021 0.460 0.008 0.450 0.025
0.531 0.061 0.571 0.084 0.561 0.076 0.531 0.084 0.529 0.090
0.497 0.140 0.482 0.092 0.444 0.085 0.577 0.016 0.613 0.039
0.620 0.017 0.585 0.027 0.665 0.005 0.679 0.022 0.701 0.065
0.673 0.062 0.685 0.014 0.649 0.110 0.584 0.092 0.590 0.096
0.633 0.073 0.576 0.157 0.590 0.152 0.592 0.150 0.593 0.153
0.607 0.153 0.608 0.153 0.621 0.168 0.627 0.158 0.605 0.189
0.587 0.161 0.643 0.152 0.598 0.201 0.650 0.156 0.711 0.153
0.673 0.190 0.700 0.209 0.698 0.212 0.682 0.209 0.657 0.249
0.676 0.237 0.705 0.227 0.715 0.227 0.714 0.232 0.705 0.253
0.703 0.263 0.681 0.256 0.714 0.247 0.615 0.244 0.618 0.226
0.619 0.226 0.622 0.240 0.646 0.221 0.616 0.267 0.576 0.269
0.616 0.268 0.593 0.292 0.609 0.311 0.655 0.295 0.666 0.304
0.663 0.317 0.688 0.308 0.688 0.360 0.662 0.370 0.658 0.389
0.626 0.390 0.578 0.460 0.620 0.432 0.630 0.482 0.608 0.475
0.592 0.496 0.607 0.469 0.595 0.466 0.596 0.482 0.624 0.496
0.634 0.588 0.645 0.630 0.640 0.643 0.626 0.629 0.615 0.644
0.576 0.618 0.681 0.608 0.660 0.618 0.689 0.644 0.712 0.649
0.685 0.631 0.653 0.685 0.668 0.686 0.666 0.656 0.675 0.657
0.692 0.680 0.692 0.701 0.679 0.681 0.677 0.709 0.676 0.709
0.620 0.656 0.626 0.676 0.632 0.676 0.640 0.685 0.609 0.693
0.597 0.733 0.606 0.744 0.607 0.744 0.578 0.781 0.579 0.742
0.715 0.737 0.715 0.738 0.661 0.830 0.574 0.785 0.600 0.785
0.621 0.792 0.583 0.805 0.577 0.805 0.578 0.826 0.613 0.905
0.598 0.887 0.593 0.891 0.576 0.923 0.592 0.892 0.583 0.906
0.595 0.925 0.672 0.891 0.574 0.937 0.594 0.950 0.590 0.953
0.627 0.978 0.780 0.868 0.787 0.856 0.735 0.929 0.763 0.971
0.759 0.789 0.782 0.737 0.782 0.738 0.778 0.746 0.806 0.713
0.722 0.820 0.750 0.633 0.858 0.584 0.872 0.642 0.726 0.666
0.755 0.685 0.839 0.494 0.848 0.509 0.812 0.508 0.807 0.515
0.858 0.530 0.765 0.532 0.768 0.537 0.787 0.514 0.738 0.515
0.869 0.380 0.866 0.364 0.733 0.337 0.752 0.331 0.779 0.320
0.793 0.320 0.763 0.285 0.765 0.308 0.756 0.294 0.752 0.304
0.725 0.285 0.814 0.292 0.727 0.382 0.724 0.190 0.772 0.144
0.751 0.169 0.841 0.193 0.810 0.281 0.824 0.267 0.819 0.252
0.831 0.267 0.845 0.237 0.791 0.218 0.750 0.258 0.754 0.273
0.766 0.274 0.772 0.260 0.767 0.240 0.779 0.229 0.736 0.054
0.737 0.054 0.740 0.025 0.715 0.0   0.739 0.024 0.826 0.026
0.790 0.022 0.844 0.137 0.828 0.093 0.808 0.103 0.839 0.096
0.752 0.116 0.764 0.076 0.727 0.094 0.718 0.073 0.879 0.032
0.907 0.009 0.921 0.029 0.895 0.067 0.880 0.049 0.861 0.050
0.656 0.062 0.935 0.016 0.945 0.027 0.946 0.028 0.973 0.017
0.985 0.062 0.946 0.058 0.945 0.057 0.968 0.025 0.932 0.084
0.931 0.107 0.961 0.106 0.968 0.098 0.973 0.107 0.969 0.118
0.868 0.074 0.867 0.078 0.908 0.079 0.917 0.141 0.909 0.132
0.900 0.117 0.915 0.193 0.916 0.185 0.911 0.186 0.910 0.185
0.898 0.194 0.899 0.182 0.886 0.176 0.890 0.205 0.879 0.201
0.867 0.202 0.934 0.152 0.970 0.150 0.930 0.192 0.951 0.219
0.899 0.241 0.906 0.232 0.905 0.231 0.925 0.494 0.924 0.494
0.963 0.460 0.978 0.537 0.979 0.551 0.986 0.561 0.963 0.548
0.935 0.568 0.897 0.564 0.884 0.557 0.878 0.563 0.878 0.564
0.913 0.584 0.933 0.620 0.880 0.609 0.916 0.630 0.950 0.631
0.951 0.668 0.985 0.591 0.992 0.606 0.994 0.608 0.969 0.628
0.907 0.652 0.978 0.791 0.867 0.977 0.315 0.074 :
VARIATE   xpoly; VALUES=!(0,1,1,0)
&         ypoly; VALUES=!(0,0,1,1)
&         s; VALUES=!(0.01,0.02...0.1)
K12HAT    [PRINT=*] Y1=hicky; X1=hickx; Y2=mapley; X2=maplex;\
YPOLYGON=ypoly; XPOLYGON=xpoly; S=s; KVALUES=k12hm
CALCULATE k12indep = CONSTANTS('pi')*(s**2)
PRINT     s,k12hm,k12indep
```
Updated on March 7, 2019