The Local Geary is a local adaptation of Geary's C statistic of spatial autocorrelation. The Local Geary uses squared differences to measure dissimilarity unlike the Local Moran. Low values of the Local Geary indicate positive spatial autocorrelation and large refers to negative spatial autocorrelation. Inference for the Local Geary is based on a permutation approach which compares the observed value to the reference distribution under spatial randomness. The Local Geary creates a pseudo p-value. This is not an analytical p-value and is based on the number of permutations and as such should be used with care.
Arguments
- x
a numeric vector, or list of numeric vectors of equal length.
- nb
a neighbor list
- wt
a weights list
- ...
other arguments passed to
spdep::localC_perm(), e.g.zero.policy = TRUEto allow for zones without neighbors.- nsim
The number of simulations used to generate reference distribution.
- alternative
A character defining the alternative hypothesis. Must be one of "two.sided", "less" or "greater".
Value
a data.frame with columns
ci: Local Geary statistice_ci: expected value of the Local Geary based on permutationsz_ci: standard deviation based on permutationsvar_ci: variance based on permutationsp_ci: p-value based on permutation sample standard deviation and meansp_ci_sim: p-value based on rank of observed statisticp_folded_sim: p-value based on the implementation of Pysal which always assumes a two-sided test taking the minimum possible p-valueskewness: sample skewnesskurtosis: sample kurtosis
Details
Overview
The Local Geary can be extended to a multivariate context. When x is a numeric vector, the univariate Local Geary will be calculated. To calculate the multivariate Local Moran provide either a list or a matrix. When x is a list, each element must be a numeric vector of the same length and of the same length as the neighbours in listw. In the case that x is a matrix the number of rows must be the same as the length of the neighbours in listw.
While not required in the univariate context, the standardized Local Geary is calculated. The multivariate Local Geary is always standardized.
The univariate Local Geary is calculated as \(c_i = \sum_j w_{ij}(x_i - x_j)^2\) and the multivariate Local Geary is calculated as \(c_{k,i} = \sum_{v=1}^{k} c_{v,i}\) as described in Anselin (2019).
Implementation
These functions are based on the implementations of the local Geary statistic in the development version of spdep. They are based on spdep::localC and spdep::localC_perm.
spdep::localC_perm and thus local_c_perm utilize a conditional permutation approach to approximate a reference distribution where each observation i is held fixed, randomly samples neighbors, and calculated the local C statistic for that tuple (ci). This is repeated nsim times. From the simulations 3 different types of p-values are calculated—all of which have their potential flaws. So be extra judicious with using p-values to make conclusions.
p_ci: utilizes the sample mean and standard deviation. The p-value is then calculated usingpnorm()--assuming a normal distribution which isn't always true.p_ci_sim: uses the rank of the observed statistic.p_folded_sim: follows the pysal implementation where p-values are in the range of [0, 0.5]. This excludes 1/2 of all p-values and should be used with caution.
References
Anselin, L. (1995), Local Indicators of Spatial Association—LISA. Geographical Analysis, 27: 93-115. doi:10.1111/j.1538-4632.1995.tb00338.x
Anselin, L. (2019), A Local Indicator of Multivariate Spatial Association: Extending Geary's c. Geogr Anal, 51: 133-150. doi:10.1111/gean.12164
Author
Josiah Parry, josiah.parry@gmail.com
Examples
local_c_perm(guerry_nb$crime_pers, guerry_nb$nb, guerry_nb$wt)
#> ci cluster e_ci var_ci z_ci p_ci
#> 1 0.987725319 High-High 2.5302015 1.4992710 -1.25973265 0.20776582
#> 2 0.838345145 High-High 1.7386999 0.5914383 -1.17073588 0.24170496
#> 3 0.702801925 High-High 1.8621051 0.6528794 -1.43476423 0.15135429
#> 4 0.100402310 Low-Low 1.9210607 1.5619953 -1.45676196 0.14518210
#> 5 0.244977901 Low-Low 1.1500848 0.6981356 -1.08325313 0.27869608
#> 6 1.353310326 Low-Low 3.1855816 1.7139859 -1.39954272 0.16165030
#> 7 3.623634688 High-High 5.4245539 5.7201046 -0.75299527 0.45145276
#> 8 1.536196129 Low-Low 4.5135154 5.5528101 -1.26348184 0.20641607
#> 9 0.861013426 Other Positive 1.0012085 0.2591610 -0.27538989 0.78301671
#> 10 0.725268063 Low-Low 1.3100696 0.6449975 -0.72816444 0.46651293
#> 11 0.571428163 Low-Low 3.6245484 1.7457675 -2.31073804 0.02084733
#> 12 0.001151306 Low-Low 1.7714760 1.6741020 -1.36823901 0.17123728
#> 13 1.912088100 Other Positive 1.1447576 0.7037120 0.91471332 0.36034215
#> 14 1.140906230 Low-Low 1.1218034 0.3459832 0.03247657 0.97409200
#> 15 1.007912947 Negative 1.4407788 0.4940950 -0.61581194 0.53801864
#> 16 0.303683489 Other Positive 1.0317763 0.3147637 -1.29776023 0.19436974
#> 17 1.214635190 High-High 1.0545683 0.2342039 0.33075358 0.74083063
#> 18 1.920535316 Low-Low 1.4312576 0.5977627 0.63283578 0.52684089
#> 19 1.610359805 High-High 3.9023805 1.9477387 -1.64230264 0.10052730
#> 20 0.446973853 High-High 2.3870300 2.0097301 -1.36850197 0.17115501
#> 21 5.156798989 High-High 6.5389013 3.2889894 -0.76209473 0.44600347
#> 22 0.919215466 Negative 1.0350553 0.2066213 -0.25484169 0.79884537
#> 23 3.107530093 Other Positive 2.4380062 4.1283634 0.32951649 0.74176533
#> 24 0.230533117 Low-Low 1.6975244 1.0389532 -1.43922777 0.15008600
#> 25 1.349244911 Other Positive 1.4953061 0.6777055 -0.17742478 0.85917474
#> 26 1.067920103 High-High 1.0564530 0.2362492 0.02359218 0.98117791
#> 27 0.418377023 High-High 2.9531117 4.0422928 -1.26071996 0.20740976
#> 28 0.208541175 Low-Low 1.9141620 0.9707897 -1.73109097 0.08343554
#> 29 0.788634381 Low-Low 1.0204786 0.2824414 -0.43624673 0.66265773
#> 30 0.248262048 Low-Low 1.0549304 0.2957507 -1.48331061 0.13799190
#> 31 0.364299667 High-High 1.2898347 0.5986834 -1.19617372 0.23162878
#> 32 0.136623420 Low-Low 1.8892490 1.2783310 -1.55012774 0.12111086
#> 33 0.708692856 High-High 1.0896611 0.2620231 -0.74425029 0.45672507
#> 34 3.108893756 High-High 4.0051591 1.8712519 -0.65519482 0.51234233
#> 35 1.981523756 Other Positive 0.9766742 0.2972077 1.84319391 0.06530070
#> 36 0.922565985 Low-Low 1.0544186 0.2830862 -0.24781600 0.80427677
#> 37 1.044173246 High-High 1.8300265 0.8287265 -0.86324856 0.38800082
#> 38 0.332298593 Other Positive 1.0495041 0.4374009 -1.08443555 0.27817173
#> 39 0.940330887 High-High 1.0556557 0.2135067 -0.24958421 0.80290891
#> 40 1.904654615 Negative 2.1141878 0.6952451 -0.25129496 0.80158607
#> 41 0.936064027 Low-Low 1.2483153 0.5664930 -0.41486514 0.67824063
#> 42 0.628686757 Other Positive 0.9959316 0.3536202 -0.61757145 0.53685787
#> 43 0.390598000 Other Positive 1.0738022 0.2806409 -1.28965948 0.19716891
#> 44 2.689756388 Low-Low 4.9666815 3.1791078 -1.27701534 0.20159683
#> 45 1.278289774 Negative 1.1920025 0.2896560 0.16032663 0.87262379
#> 46 0.793948353 Low-Low 3.9557159 2.6032180 -1.95963269 0.05003874
#> 47 1.683658729 High-High 2.8175537 0.8708148 -1.21509414 0.22433019
#> 48 1.301187948 High-High 3.3126397 2.3573068 -1.31009130 0.19016495
#> 49 2.342671020 Other Positive 1.3741101 0.5623938 1.29153640 0.19651774
#> 50 0.832225668 High-High 1.7702214 0.6524277 -1.16127392 0.24553052
#> 51 0.296215805 High-High 2.4036991 1.2170990 -1.91029826 0.05609482
#> 52 2.268056379 Negative 1.9133776 0.9858641 0.35721252 0.72093271
#> 53 0.876680843 High-High 1.4065244 0.5125325 -0.74009411 0.45924289
#> 54 0.414701945 High-High 1.2260675 0.4932709 -1.15524434 0.24799038
#> 55 1.979054136 Low-Low 2.2719459 3.9061689 -0.14819433 0.88218940
#> 56 0.561171709 High-High 1.4808781 0.4657015 -1.34770660 0.17775277
#> 57 0.378238452 High-High 1.8229805 1.3437253 -1.24633578 0.21264115
#> 58 1.849165484 High-High 2.2955249 1.0050703 -0.44523210 0.65615203
#> 59 1.207382342 High-High 2.2373789 0.8762710 -1.10031376 0.27119544
#> 60 1.157149740 High-High 1.1457056 0.8938569 0.01210458 0.99034218
#> 61 1.848857476 Other Positive 1.1352981 0.3566800 1.19478778 0.23216995
#> 62 0.155525055 Low-Low 1.1814654 0.8093517 -1.14039021 0.25412377
#> 63 0.642205848 Low-Low 2.1269621 2.1891453 -1.00350112 0.31561914
#> 64 0.749419937 Low-Low 4.3791252 7.3750254 -1.33656350 0.18136516
#> 65 1.283938081 Low-Low 2.1890394 1.6726563 -0.69983210 0.48403216
#> 66 1.470817990 Low-Low 4.0503016 7.5556343 -0.93842013 0.34802855
#> 67 1.263739703 Other Positive 1.0391647 0.3884136 0.36034149 0.71859178
#> 68 0.964844188 High-High 1.1122214 0.2975052 -0.27019862 0.78700745
#> 69 0.338990398 High-High 2.3442681 0.8920578 -2.12313696 0.03374238
#> 70 1.924872243 High-High 4.6855852 2.1742435 -1.87226433 0.06117004
#> 71 0.659890869 Low-Low 1.6835069 2.4823600 -0.64968777 0.51589392
#> 72 2.135743520 Other Positive 1.0592632 0.6162752 1.37125668 0.17029496
#> 73 0.710008255 Negative 1.0754975 0.1966501 -0.82419029 0.40983144
#> 74 1.424041460 Low-Low 2.0722928 0.9946043 -0.65000737 0.51568746
#> 75 0.706001903 Other Positive 1.0876162 0.3633600 -0.63307633 0.52668380
#> 76 1.774528303 High-High 4.5140731 2.5103004 -1.72908188 0.08379444
#> 77 0.242422537 Low-Low 1.8883149 1.1128984 -1.56017630 0.11871823
#> 78 0.694451670 Low-Low 1.5172115 0.6462868 -1.02343536 0.30610205
#> 79 0.001874143 Low-Low 1.9442535 1.9214462 -1.40126388 0.16113518
#> 80 0.055356016 Low-Low 1.7597653 0.8905756 -1.80608633 0.07090486
#> 81 0.428128655 High-High 0.9913836 0.3489998 -0.95343790 0.34036825
#> 82 2.024633135 Other Positive 1.5417557 0.7493661 0.55781460 0.57697099
#> 83 2.497611820 Other Positive 1.2637478 0.5126094 1.72335218 0.08482486
#> 84 1.048741881 Low-Low 1.0220117 0.2484388 0.05362805 0.95723151
#> 85 1.026405574 Other Positive 1.0738224 0.3646807 -0.07851924 0.93741503
#> p_ci_sim p_folded_sim skewness kurtosis
#> 1 0.176 0.088 0.5547185 -0.06279701
#> 2 0.188 0.094 0.5697743 0.03835036
#> 3 0.100 0.050 0.6337871 0.12030182
#> 4 0.008 0.004 0.8616578 0.30289555
#> 5 0.260 0.130 0.9581043 0.62918986
#> 6 0.116 0.058 0.5466322 0.20462670
#> 7 0.480 0.240 0.6296362 0.30378009
#> 8 0.136 0.068 0.8460502 0.66053909
#> 9 0.848 0.424 0.6365611 0.32384195
#> 10 0.496 0.248 0.9797378 1.38660001
#> 11 0.004 0.002 0.5788117 0.36588430
#> 12 0.004 0.002 0.7452809 -0.12068707
#> 13 0.368 0.184 0.9226400 0.43294904
#> 14 0.852 0.426 0.8140157 0.64142620
#> 15 0.592 0.296 0.7932099 0.58557136
#> 16 0.104 0.052 0.8545307 0.62920291
#> 17 0.700 0.350 0.5160029 -0.09860068
#> 18 0.524 0.262 0.6378879 -0.05971532
#> 19 0.052 0.026 0.4700876 -0.12102364
#> 20 0.140 0.070 0.7323359 0.46913854
#> 21 0.468 0.234 0.1104821 -0.22012809
#> 22 0.896 0.448 0.6131696 0.14063447
#> 23 0.628 0.314 0.9267789 0.20482444
#> 24 0.056 0.028 0.7988173 0.49499575
#> 25 0.968 0.484 0.8479828 0.72304437
#> 26 0.916 0.458 0.5901119 0.17261793
#> 27 0.144 0.072 0.7101465 0.05155841
#> 28 0.016 0.008 0.5881690 -0.21887991
#> 29 0.712 0.356 0.9897033 1.92269644
#> 30 0.052 0.026 0.7349385 0.13319099
#> 31 0.160 0.080 0.9134813 0.37217986
#> 32 0.008 0.004 0.6904486 -0.00560892
#> 33 0.524 0.262 0.6366330 0.31613341
#> 34 0.516 0.258 0.3993656 -0.06970491
#> 35 0.108 0.054 0.7745696 0.73151659
#> 36 0.908 0.454 0.8831444 1.30637241
#> 37 0.444 0.222 0.5864434 0.02285063
#> 38 0.276 0.138 0.7474730 0.17878478
#> 39 0.880 0.440 0.4282149 -0.42555541
#> 40 0.900 0.450 0.5599459 0.36536696
#> 41 0.820 0.410 0.9630359 0.91240120
#> 42 0.648 0.324 0.7166774 0.09156859
#> 43 0.144 0.072 0.6529311 0.23912892
#> 44 0.184 0.092 0.3986520 -0.22797854
#> 45 0.764 0.382 0.6814098 0.52975140
#> 46 0.012 0.006 0.4229850 -0.10871401
#> 47 0.208 0.104 0.4047752 0.07551500
#> 48 0.184 0.092 0.4229555 -0.02720629
#> 49 0.232 0.116 0.6741496 0.20991262
#> 50 0.200 0.100 0.7983370 0.68739202
#> 51 0.012 0.006 0.5790616 0.45552295
#> 52 0.664 0.332 0.6017639 0.04004606
#> 53 0.536 0.268 0.7582262 0.16082852
#> 54 0.216 0.108 0.7021160 -0.05614549
#> 55 0.928 0.464 1.0922717 0.74880114
#> 56 0.108 0.054 0.8511583 0.76400897
#> 57 0.120 0.060 0.8576174 0.33192122
#> 58 0.728 0.364 0.4988748 0.03415802
#> 59 0.276 0.138 0.5153176 0.55570840
#> 60 0.764 0.382 1.2334326 1.40118120
#> 61 0.268 0.134 0.5571515 -0.20651072
#> 62 0.176 0.088 1.1422501 1.43899246
#> 63 0.344 0.172 0.8431191 0.42036098
#> 64 0.100 0.050 0.8697918 0.96687956
#> 65 0.548 0.274 0.7055437 0.59846921
#> 66 0.400 0.200 0.7951423 0.12010944
#> 67 0.644 0.322 0.6578707 -0.03568370
#> 68 0.892 0.446 0.5268665 -0.21384765
#> 69 0.008 0.004 0.6113943 0.03335488
#> 70 0.044 0.022 0.2990709 0.07142324
#> 71 0.664 0.332 1.3168408 1.53523412
#> 72 0.204 0.102 1.1830468 2.06570055
#> 73 0.480 0.240 0.6516212 0.13604109
#> 74 0.608 0.304 0.3311255 -0.49036773
#> 75 0.596 0.298 0.8049037 0.77875110
#> 76 0.068 0.034 0.2864614 -0.01220005
#> 77 0.036 0.018 0.7397824 0.59545091
#> 78 0.316 0.158 0.5288488 -0.39998419
#> 79 0.008 0.004 0.7591347 -0.08177678
#> 80 0.004 0.002 0.6974548 0.54730854
#> 81 0.352 0.176 0.8392758 0.47595712
#> 82 0.528 0.264 0.7824387 0.71186709
#> 83 0.136 0.068 0.7182784 0.06797578
#> 84 0.828 0.414 0.6337278 0.07381572
#> 85 0.956 0.478 0.8143583 0.60778857