MSEKERNEL2D procedure

Estimates the mean square error for a kernel smoothing (M.A. Mugglestone, S.A. Harding, B.Y.Y. Lee, P.J. Diggle & B.S. Rowlingson).

Option

Parameters

Description

This procedure calculates an estimate of the mean square error for a kernel smoothing given a particular kernel width. The method used is that of Berman & Diggle (1989). The data required by the procedure are the coordinates of a spatial point pattern (specified using the parameters X and Y), the coordinates of a polygon within which smoothing is to be performed (specified using the parameters XPOLYGON and YPOLYGON), the number of values of the kernel width at which to estimate the mean square error (specified using the parameter NSTEP), and the maximum kernel width to use (specified using the parameter HMAX). The output of the procedure is a variate containing a sequence of NSTEP equally-spaced values of the kernel width parameter from HMAX/NSTEP up to HMAX, and a corresponding vector containing the mean square error for each kernel width. The values of the kernel width and the corresponding mean square error estimates can be saved using the parameters HVALUES and MSE.

Printed output is controlled using the PRINT option. The default setting of summary prints the values of the kernel width and the corresponding mean square error estimates under the headings HVALUES and MSE.

The output of the procedure may be used to select the optimum kernel width to use with the procedure PTKERNEL2D. Note that the estimated mean square errors returned by the procedure are, in fact, scaled estimates. The scaling simplifies the calculations but it can produce negative estimates of mean square errors. The scaling is, however, independent of the kernel width, so that the true mean square error has its minimum at the same kernel width as the scaled version.

Option: PRINT.

Parameters: Y, X, YPOLYGON, XPOLYGON, NSTEP, HMAX, HVALUES, MSE.

Method

A procedure PTCHECKXY is called to check that X and Y have identical restrictions. A similar check is made on XPOLYGON and YPOLYGON. The procedure then calculates a sequence of NSTEP equally-spaced values for the kernel width, starting at HMAX/NSTEP and finishing at HMAX. It then calls a procedure PTPASS to call a Fortran program to calculate the estimated mean square error associated with each value of the kernel width.

Action with RESTRICT

If X and Y are restricted, only the subset of values specified by the restriction will be included in the calculations. XPOLYGON and YPOLYGON may also be restricted, as long as the same restrictions apply to both parameters.

Reference

Berman, M. & Diggle, P.J. (1989). Estimating weighted integrals of the second-order intensity of a spatial point process. Journal of the Royal Statistical Society, Series B, 51, 81-92.

See also

Procedures: KERNELDENSITY, PTKERNEL2D, PTK3D.

Commands for: Spatial statistics.

Example

CAPTION     'MSEKERNEL2D 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     xpoly; VALUES=!(0,1,1,0)
&           ypoly; VALUES=!(0,0,1,1)
MSEKERNEL2D Y=hicky; X=hickx; YPOLYGON=ypoly; XPOLYGON=xpoly;\ 
            NSTEP=10; HMAX=0.2