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# KTORENVELOPES procedure

Gives bounds for the bivariate K function under independence (M.A. Mugglestone, S.A. Harding, B.Y.Y. Lee, P.J. Diggle & B.S. Rowlingson).

### Option

`PRINT` = string tokens What to print (`summary`, `monitoring`); default `summ`, `moni`

### 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 How many simulations of independence to use; 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 lower bound of the bivariate K function Variates to receive the values of the upper bound of the bivariate K function Seeds for the random numbers used to generate the random shifts; default 0

### 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). For independent processes, the bivariate K function 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` can be used to obtain an approximately unbiased estimate of K12(s) for two observed patterns which can be compared with the expected value under independence given by the above expression. However, the variance of the estimate under the null hypothesis cannot be expressed in closed form, and so critical values for the estimated K function cannot be obtained analytically. This problem can be overcome by repeatedly simulating from the null hypothesis and estimating the K function for each simulation. If `NSIMULATIONS` denotes the number of simulations used, then, for each value of s, the minimum (maximum) value of the estimated K function provides an approximate 100/(`NSIMULATIONS`+1) percent lower (upper) critical value for K12(s).

The established method for simulating independent patterns whilst retaining the observed degree of clustering/regularity in the univariate patterns is to perform a random toroidal shift of one observed pattern whilst holding the other fixed. This method is due to Lotwick and Silverman (1982). Random toroidal shifts can be performed using the procedure `GRTORSHIFT`.

The procedure `KTORENVELOPES` computes lower and upper bounds (envelopes) for the bivariate K function under independence. The data required by the procedure are 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`), the number of simulations to use (specified by the parameter `NSIMULATIONS`) and a vector of distances at which to estimate the K function (specified by the parameter `S`). The simulations of independence are generated by performing random toroidal shifts of the second pattern whilst holding the first pattern fixed. The `SEED` parameter allows a seed to be supplied for generating the random numbers required to generate the random shifts (thereby producing reproducible results). If this is not supplied, the default of 0 initializes the random number generator (if necessary) from the system clock. The output of the procedure consists of two vectors, the first containing the minimum value obtained for K12(s) for each distance in `S`, and the second containing the corresponding maximum values. The minimum and maximum values of the bivariate K function can be saved using the parameters `KLOWER` and `KUPPER`.

Printed output is controlled using the `PRINT` option. The settings available are `monitoring` (which prints a message to mark the start of each simulation) and `summary` (which prints the distances at which the K function is estimated under the heading `S`, together with the lower and upper bounds for the K function under the headings `KLOWER` and `KUPPER`).

Option: `PRINT`.

Parameters: `Y1`, `X1`, `Y2`, `X2`, `YPOLYGON`, `XPOLYGON`, `NSIMULATIONS`, `S`, `KLOWER`, `KUPPER`, `SEED`.

### 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`. 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 PTBOX` is used to generate a bounding box for the polygon specified by `XPOLYGON` and `YPOLYGON`. The procedures `GRTORSHIFT` and `K12HAT` are then called `NSIMULATIONS` times to calculate estimates of the bivariate K function under independence (using the bounding box as the toroidal region for the random shifts of the second pattern). Finally, the `VMINIMA` and `VMAXIMA` functions are used to calculate the minimum and maximum values of the bivariate K function for each distance in `S`.

### 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.

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

Commands for: Spatial statistics.

### Example

```CAPTION 'KTORENVELOPES example'; STYLE=meta
VARIATE 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
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
KTORENVELOPES [PRINT=monitoring] Y1=hicky; X1=hickx;\
Y2=mapley; X2=maplex; YPOLYGON=ypoly; XPOLYGON=xpoly;\
NSIMULATIONS=19; S=s; KLOWER=minindep; KUPPER=maxindep; SEED=274089
PRINT   s,k12hm,minindep,maxindep
```
Updated on March 7, 2019