2030:
20:
644:
867:
The idea is to construct a matrix that accurately reflects your assumptions about the particular spatial phenomenon in question. A common approach is to give a weight of 1 if two zones are neighbors, and 0 otherwise, though the definition of 'neighbors' can vary. Another common approach might be to
888:
nearest neighbors, 0 otherwise. An alternative is to use a distance decay function for assigning weights. Sometimes the length of a shared edge is used for assigning different weights to neighbors. The selection of spatial weights matrix should be guided by theory about the phenomenon in question.
295:
2037:
Global spatial autocorrelation analysis yields only one statistic to summarize the whole study area. In other words, the global analysis assumes homogeneity. If that assumption does not hold, then having only one statistic does not make sense as the statistic should differ over space.
1584:
1191:
2201:
83:
1408:
852:. The matrix is required because, in order to address spatial autocorrelation and also model spatial interaction, we need to impose a structure to constrain the number of neighbors to be considered. This is related to
1299:
2305:
1795:
Values significantly below -1/(N-1) indicate negative spatial autocorrelation and values significantly above -1/(N-1) indicate positive spatial autocorrelation. For statistical hypothesis testing, Moran's
635:
1690:
2372:
1790:
1416:
2041:
Moreover, even if there is no global autocorrelation or no clustering, we can still find clusters at a local level using local spatial autocorrelation analysis. The fact that Moran's
977:
1895:
1852:
789:
1994:
1034:
414:
685:
502:
1961:
1928:
58:. Spatial autocorrelation is characterized by a correlation in a signal among nearby locations in space. Spatial autocorrelation is more complex than one-dimensional
850:
555:
466:
2639:
2070:
1020:
1000:
907:
886:
820:
725:
705:
525:
434:
383:
361:
341:
321:
864:
function, such that even though all observations have an influence on all other observations, after some distance threshold that influence can be neglected.
2468:
290:{\displaystyle I={\frac {N}{W}}{\frac {\sum _{i=1}^{N}\sum _{j=1}^{N}w_{ij}(x_{i}-{\bar {x}})(x_{j}-{\bar {x}})}{\sum _{i=1}^{N}(x_{i}-{\bar {x}})^{2}}}}
2049:
is exploited by the "local indicators of spatial association" (LISA) to evaluate the clustering in those individual units by calculating Local Moran's
1307:
2033:
Clusters of the estimated percent of people in poverty by county in the contiguous United States in 2020 calculated using
Anselin Local Moran's I.
1202:
2212:
2811:"Automatic delineation of geomorphological slope units with r.slopeunits v1.0 and their optimization for landslide susceptibility modeling"
2853:
2611:
de Jong, P., Sprenger, C. and van Veen, F., 1984. On extreme values of Moran's I and Geary's c. Geographical
Analysis, 16(1), pp.17-24.
2493:
853:
2483:
909:
is quite sensitive to the weights and can influence the conclusions you make about a phenomenon, especially when using distances.
560:
2431:
31:
neighbors definition. If the white squares were stacked to one half of the board and the black squares to the other, Moran's
1592:
2316:
1698:
55:
2498:
2779:
1579:{\displaystyle S_{3}={\frac {N^{-1}\sum _{i}(x_{i}-{\bar {x}})^{4}}{(N^{-1}\sum _{i}(x_{i}-{\bar {x}})^{2})^{2}}}}
2848:
2518:
927:
51:
2809:
Alvioli, M.; Marchesini, I.; Reichenbach, P.; Rossi, M.; Ardizzone, F.; Fiorucci, F.; Guzzetti, F. (2016).
1857:
1814:
2029:
742:
62:
because spatial correlation is multi-dimensional (i.e. 2 or 3 dimensions of space) and multi-directional.
2488:
1963:
are the corresponding minimum and maximum eigenvalues of the weight matrix. For a row normalised matrix
1966:
1186:{\displaystyle \operatorname {Var} (I)={\frac {NS_{4}-S_{3}S_{5}}{(N-1)(N-2)(N-3)W^{2}}}-(E(I))^{2}}
2538:
390:
2760:
2546:
2196:{\displaystyle I_{i}={\frac {x_{i}-{\bar {x}}}{m_{2}}}\sum _{j=1}^{N}w_{ij}(x_{j}-{\bar {x}})}
727:
and then row normalizing the weight matrix. Top left shows anti-correlation giving a negative
654:
471:
2622:
1933:
1900:
2822:
2791:
2750:
2740:
2707:
2674:
2585:
2530:
1025:
At large sample sizes (i.e., as N approaches infinity), the expected value approaches zero.
825:
530:
441:
1005:
59:
2755:
2728:
2712:
2695:
2679:
2662:
2473:
1801:
985:
892:
871:
861:
805:
791:). Bottom right shows an 'ink blot' or spreading pattern with positive autocorrelation.
710:
690:
510:
419:
368:
346:
326:
306:
35:
approaches +1 as N increases. A random arrangement of square colors would give Moran's
2842:
2589:
2456:
2046:
19:
2569:
2449:
648:
643:
468:
are the elements of a matrix of spatial weights with zeroes on the diagonal (i.e.,
28:
2478:
2004:
1022:
picked uniformly at random (and the expectation is over picking the permutation).
2565:
2411:
822:
can depend quite a bit on the assumptions built into the spatial weights matrix
1403:{\displaystyle S_{2}=\sum _{i}\left(\sum _{j}w_{ij}+\sum _{j}w_{ji}\right)^{2}}
2780:"A regional analysis of contraction rate in written Standard American English"
2403:
2640:"Cluster and Outlier Analysis (Anselin Local Moran's I) (Spatial Statistics)"
2576:: Testing for Spatial Dependence Based on the Spatial Autoregressive Model".
77:
is a measure of the overall clustering of the spatial data. It is defined as
2827:
2810:
2795:
2745:
2427:
2764:
2729:"Geospatial examination of lithium in drinking water and suicide mortality"
2550:
2053:
for each spatial unit and evaluating the statistical significance for each
1294:{\displaystyle S_{1}={\frac {1}{2}}\sum _{i}\sum _{j}(w_{ij}+w_{ji})^{2}}
2300:{\displaystyle m_{2}={\frac {\sum _{i=1}^{N}(x_{i}-{\bar {x}})^{2}}{N}}}
2542:
2455:
Defining an objective function for meaningful terrain segmentation for
2442:
2534:
647:
Moran's I statistic computed for different spatial patterns. Using '
2696:"The Analysis of Spatial Association by Use of Distance Statistics"
2399:
2028:
642:
18:
858:
Everything depends on everything else, but closer things more so
23:
The white and black squares are perfectly dispersed so Moran's
2014:
is a measure of global spatial autocorrelation, while Geary's
731:. Top right shows a spatial gradient giving a large positive
982:
The null distribution used for this expectation is that the
2452:
to measure the significance of regional language variation.
2438:
The analysis of geographic differences in health variables.
921:
under the null hypothesis of no spatial autocorrelation is
630:{\displaystyle W=\sum _{i=1}^{N}\sum _{j=1}^{N}{w_{ij}}}
2319:
2215:
2073:
1969:
1936:
1903:
1860:
1817:
1701:
1685:{\displaystyle S_{4}=(N^{2}-3N+3)S_{1}-NS_{2}+3W^{2}}
1595:
1419:
1310:
1205:
1037:
1008:
988:
930:
895:
874:
828:
808:
745:
713:
693:
657:
563:
533:
513:
474:
444:
422:
393:
371:
349:
329:
309:
86:
2521:(1950). "Notes on Continuous Stochastic Phenomena".
2018:
is more sensitive to local spatial autocorrelation.
2367:{\displaystyle I=\sum _{i=1}^{N}{\frac {I_{i}}{N}}}
1785:{\displaystyle S_{5}=(N^{2}-N)S_{1}-2NS_{2}+6W^{2}}
2366:
2299:
2195:
1988:
1955:
1922:
1889:
1846:
1784:
1684:
1578:
1402:
1293:
1185:
1014:
994:
971:
901:
880:
844:
814:
783:
735:. Bottom left shows random data giving a value of
719:
699:
679:
629:
549:
519:
496:
460:
428:
408:
377:
355:
335:
315:
289:
2445:concentrations in public water on mental health.
2623:"Exploring Spatial Data with GeoDa: A Workbook"
2602:Cliff and Ord (1981), Spatial Processes, London
2663:"Local Indicators of Spatial Association—LISA"
8:
2727:Helbich, M; Leitner, M; Kapusta, ND (2012).
2395:is the number of analysis units on the map.
2784:International Journal of Corpus Linguistics
2628:. Spatial Analysis Laboratory. p. 138.
2469:Concepts and Techniques in Modern Geography
860:—in other words, the law implies a spatial
323:is the number of spatial units indexed by
2826:
2754:
2744:
2711:
2678:
2353:
2347:
2341:
2330:
2318:
2285:
2270:
2269:
2260:
2247:
2236:
2229:
2220:
2214:
2179:
2178:
2169:
2153:
2143:
2132:
2120:
2104:
2103:
2094:
2087:
2078:
2072:
1970:
1968:
1941:
1935:
1908:
1902:
1875:
1861:
1859:
1832:
1818:
1816:
1776:
1760:
1741:
1722:
1706:
1700:
1676:
1660:
1644:
1616:
1600:
1594:
1567:
1557:
1542:
1541:
1532:
1519:
1506:
1491:
1476:
1475:
1466:
1453:
1440:
1433:
1424:
1418:
1394:
1380:
1370:
1354:
1344:
1328:
1315:
1309:
1285:
1272:
1256:
1243:
1233:
1219:
1210:
1204:
1177:
1146:
1089:
1079:
1066:
1056:
1036:
1007:
987:
946:
929:
894:
873:
833:
827:
807:
752:
744:
712:
692:
662:
656:
617:
612:
606:
595:
585:
574:
562:
538:
532:
512:
479:
473:
449:
443:
421:
395:
394:
392:
370:
348:
328:
308:
278:
263:
262:
253:
240:
229:
209:
208:
199:
178:
177:
168:
152:
142:
131:
121:
110:
103:
93:
85:
651:' neighbors for each grid cell, setting
2510:
2060:. From the equation of Global Moran's
972:{\displaystyle E(I)={\frac {-1}{N-1}}}
1890:{\displaystyle {\frac {N}{W}}w_{max}}
1847:{\displaystyle {\frac {N}{W}}w_{min}}
7:
1002:input is permuted by a permutation
784:{\displaystyle -1/(N-1)\simeq -0.04}
2010:, but it is not identical. Moran's
798:Defining the spatial weights matrix
2713:10.1111/j.1538-4632.1992.tb00261.x
2680:10.1111/j.1538-4632.1995.tb00338.x
2384:measuring global autocorrelation,
16:Measure of spatial autocorrelation
14:
2484:Indicators of spatial association
2590:10.1111/j.1538-4632.2007.00708.x
2426:is widely used in the fields of
1989:{\displaystyle {\frac {N}{W}}=1}
2815:Geoscientific Model Development
2494:Tobler's first law of geography
854:Tobler's first law of geography
27:would be −1 using a
2432:geographic information science
2282:
2275:
2253:
2190:
2184:
2162:
2109:
1734:
1715:
1637:
1609:
1564:
1554:
1547:
1525:
1499:
1488:
1481:
1459:
1282:
1249:
1174:
1170:
1164:
1158:
1139:
1127:
1124:
1112:
1109:
1097:
1050:
1044:
940:
934:
917:The expected value of Moran's
769:
757:
400:
275:
268:
246:
220:
214:
192:
189:
183:
161:
1:
2441:Characterising the impact of
2406:which uses the Local Moran's
2045:is a summation of individual
1800:values can be transformed to
2694:Getis, Arthur (3 Sep 2010).
385:is the variable of interest;
2398:LISAs can be calculated in
56:Patrick Alfred Pierce Moran
39:a value that is close to 0.
2870:
2854:Covariance and correlation
409:{\displaystyle {\bar {x}}}
2434:. Some examples include:
2572:(2007). "Beyond Moran's
2499:Wartenberg's coefficient
2457:geomorphological studies
2003:is inversely related to
680:{\displaystyle w_{ij}=1}
497:{\displaystyle w_{ii}=0}
2828:10.5194/gmd-9-3975-2016
2796:10.1075/ijcl.16.4.04gri
2746:10.1186/1476-072X-11-19
1956:{\displaystyle w_{max}}
1923:{\displaystyle w_{min}}
52:spatial autocorrelation
2380:is the Global Moran's
2368:
2346:
2301:
2252:
2197:
2148:
2034:
1990:
1957:
1924:
1891:
1848:
1786:
1686:
1580:
1404:
1295:
1187:
1016:
996:
973:
903:
882:
868:give a weight of 1 to
846:
845:{\displaystyle w_{ij}}
816:
792:
785:
721:
701:
681:
631:
611:
590:
551:
550:{\displaystyle w_{ij}}
521:
498:
462:
461:{\displaystyle w_{ij}}
430:
410:
379:
357:
337:
317:
291:
245:
147:
126:
40:
2778:Grieve, Jack (2011).
2700:Geographical Analysis
2667:Geographical Analysis
2661:Anselin, Luc (1995).
2621:Anselin, Luc (2005).
2578:Geographical Analysis
2489:Spatial heterogeneity
2369:
2326:
2302:
2232:
2198:
2128:
2032:
1991:
1958:
1925:
1892:
1849:
1787:
1687:
1581:
1405:
1296:
1188:
1017:
997:
974:
904:
883:
847:
817:
786:
722:
702:
682:
646:
632:
591:
570:
552:
522:
499:
463:
431:
411:
380:
358:
338:
318:
292:
225:
127:
106:
22:
2566:Calder, Catherine A.
2317:
2213:
2071:
1967:
1934:
1901:
1858:
1815:
1699:
1593:
1417:
1308:
1203:
1035:
1028:Its variance equals
1015:{\displaystyle \pi }
1006:
986:
928:
893:
872:
856:, which states that
826:
806:
743:
711:
691:
655:
561:
531:
511:
472:
442:
420:
391:
369:
347:
327:
307:
84:
2733:Int J Health Geogr
2364:
2297:
2193:
2035:
1986:
1953:
1920:
1887:
1844:
1782:
1682:
1576:
1524:
1458:
1400:
1375:
1349:
1333:
1291:
1248:
1238:
1183:
1012:
992:
969:
899:
878:
842:
812:
793:
781:
717:
697:
677:
627:
547:
527:is the sum of all
517:
494:
458:
426:
406:
375:
353:
333:
313:
287:
41:
2362:
2295:
2278:
2187:
2126:
2112:
2064:, we can obtain:
2021:
1978:
1869:
1826:
1574:
1550:
1515:
1484:
1449:
1366:
1340:
1324:
1239:
1229:
1227:
1153:
995:{\displaystyle x}
967:
902:{\displaystyle I}
881:{\displaystyle k}
815:{\displaystyle I}
720:{\displaystyle i}
700:{\displaystyle j}
520:{\displaystyle W}
429:{\displaystyle x}
403:
378:{\displaystyle x}
356:{\displaystyle j}
336:{\displaystyle i}
316:{\displaystyle N}
285:
271:
217:
186:
101:
65:
2861:
2849:Spatial analysis
2833:
2832:
2830:
2806:
2800:
2799:
2775:
2769:
2768:
2758:
2748:
2724:
2718:
2717:
2715:
2691:
2685:
2684:
2682:
2658:
2652:
2651:
2649:
2647:
2636:
2630:
2629:
2627:
2618:
2612:
2609:
2603:
2600:
2594:
2593:
2561:
2555:
2554:
2515:
2394:
2390:
2379:
2373:
2371:
2370:
2365:
2363:
2358:
2357:
2348:
2345:
2340:
2306:
2304:
2303:
2298:
2296:
2291:
2290:
2289:
2280:
2279:
2271:
2265:
2264:
2251:
2246:
2230:
2225:
2224:
2202:
2200:
2199:
2194:
2189:
2188:
2180:
2174:
2173:
2161:
2160:
2147:
2142:
2127:
2125:
2124:
2115:
2114:
2113:
2105:
2099:
2098:
2088:
2083:
2082:
2059:
1995:
1993:
1992:
1987:
1979:
1971:
1962:
1960:
1959:
1954:
1952:
1951:
1929:
1927:
1926:
1921:
1919:
1918:
1896:
1894:
1893:
1888:
1886:
1885:
1870:
1862:
1853:
1851:
1850:
1845:
1843:
1842:
1827:
1819:
1791:
1789:
1788:
1783:
1781:
1780:
1765:
1764:
1746:
1745:
1727:
1726:
1711:
1710:
1691:
1689:
1688:
1683:
1681:
1680:
1665:
1664:
1649:
1648:
1621:
1620:
1605:
1604:
1585:
1583:
1582:
1577:
1575:
1573:
1572:
1571:
1562:
1561:
1552:
1551:
1543:
1537:
1536:
1523:
1514:
1513:
1497:
1496:
1495:
1486:
1485:
1477:
1471:
1470:
1457:
1448:
1447:
1434:
1429:
1428:
1409:
1407:
1406:
1401:
1399:
1398:
1393:
1389:
1388:
1387:
1374:
1362:
1361:
1348:
1332:
1320:
1319:
1300:
1298:
1297:
1292:
1290:
1289:
1280:
1279:
1264:
1263:
1247:
1237:
1228:
1220:
1215:
1214:
1192:
1190:
1189:
1184:
1182:
1181:
1154:
1152:
1151:
1150:
1095:
1094:
1093:
1084:
1083:
1071:
1070:
1057:
1021:
1019:
1018:
1013:
1001:
999:
998:
993:
978:
976:
975:
970:
968:
966:
955:
947:
908:
906:
905:
900:
887:
885:
884:
879:
851:
849:
848:
843:
841:
840:
821:
819:
818:
813:
790:
788:
787:
782:
756:
726:
724:
723:
718:
706:
704:
703:
698:
687:for neighbours
686:
684:
683:
678:
670:
669:
636:
634:
633:
628:
626:
625:
624:
610:
605:
589:
584:
556:
554:
553:
548:
546:
545:
526:
524:
523:
518:
503:
501:
500:
495:
487:
486:
467:
465:
464:
459:
457:
456:
435:
433:
432:
427:
415:
413:
412:
407:
405:
404:
396:
384:
382:
381:
376:
362:
360:
359:
354:
342:
340:
339:
334:
322:
320:
319:
314:
296:
294:
293:
288:
286:
284:
283:
282:
273:
272:
264:
258:
257:
244:
239:
223:
219:
218:
210:
204:
203:
188:
187:
179:
173:
172:
160:
159:
146:
141:
125:
120:
104:
102:
94:
50:is a measure of
2869:
2868:
2864:
2863:
2862:
2860:
2859:
2858:
2839:
2838:
2837:
2836:
2808:
2807:
2803:
2777:
2776:
2772:
2726:
2725:
2721:
2693:
2692:
2688:
2660:
2659:
2655:
2645:
2643:
2638:
2637:
2633:
2625:
2620:
2619:
2615:
2610:
2606:
2601:
2597:
2563:
2562:
2558:
2535:10.2307/2332142
2519:Moran, P. A. P.
2517:
2516:
2512:
2507:
2465:
2420:
2392:
2389:
2385:
2377:
2349:
2315:
2314:
2281:
2256:
2231:
2216:
2211:
2210:
2165:
2149:
2116:
2090:
2089:
2074:
2069:
2068:
2058:
2054:
2027:
1965:
1964:
1937:
1932:
1931:
1904:
1899:
1898:
1871:
1856:
1855:
1828:
1813:
1812:
1772:
1756:
1737:
1718:
1702:
1697:
1696:
1672:
1656:
1640:
1612:
1596:
1591:
1590:
1563:
1553:
1528:
1502:
1498:
1487:
1462:
1436:
1435:
1420:
1415:
1414:
1376:
1350:
1339:
1335:
1334:
1311:
1306:
1305:
1281:
1268:
1252:
1206:
1201:
1200:
1173:
1142:
1096:
1085:
1075:
1062:
1058:
1033:
1032:
1004:
1003:
984:
983:
956:
948:
926:
925:
915:
891:
890:
870:
869:
829:
824:
823:
804:
803:
800:
741:
740:
709:
708:
689:
688:
658:
653:
652:
613:
559:
558:
534:
529:
528:
509:
508:
475:
470:
469:
445:
440:
439:
418:
417:
416:is the mean of
389:
388:
367:
366:
345:
344:
325:
324:
305:
304:
274:
249:
224:
195:
164:
148:
105:
82:
81:
73:Global Moran's
71:
66:Global Moran's
60:autocorrelation
43:In statistics,
17:
12:
11:
5:
2867:
2865:
2857:
2856:
2851:
2841:
2840:
2835:
2834:
2801:
2790:(4): 514–546.
2770:
2719:
2706:(3): 189–206.
2686:
2653:
2631:
2613:
2604:
2595:
2584:(4): 357–375.
2556:
2509:
2508:
2506:
2503:
2502:
2501:
2496:
2491:
2486:
2481:
2476:
2474:Distance decay
2471:
2464:
2461:
2460:
2459:
2453:
2446:
2439:
2419:
2416:
2410:, proposed by
2391:is local, and
2387:
2375:
2374:
2361:
2356:
2352:
2344:
2339:
2336:
2333:
2329:
2325:
2322:
2308:
2307:
2294:
2288:
2284:
2277:
2274:
2268:
2263:
2259:
2255:
2250:
2245:
2242:
2239:
2235:
2228:
2223:
2219:
2204:
2203:
2192:
2186:
2183:
2177:
2172:
2168:
2164:
2159:
2156:
2152:
2146:
2141:
2138:
2135:
2131:
2123:
2119:
2111:
2108:
2102:
2097:
2093:
2086:
2081:
2077:
2056:
2047:cross products
2026:
2022:Local Moran's
2020:
1985:
1982:
1977:
1974:
1950:
1947:
1944:
1940:
1917:
1914:
1911:
1907:
1884:
1881:
1878:
1874:
1868:
1865:
1841:
1838:
1835:
1831:
1825:
1822:
1811:range between
1793:
1792:
1779:
1775:
1771:
1768:
1763:
1759:
1755:
1752:
1749:
1744:
1740:
1736:
1733:
1730:
1725:
1721:
1717:
1714:
1709:
1705:
1693:
1692:
1679:
1675:
1671:
1668:
1663:
1659:
1655:
1652:
1647:
1643:
1639:
1636:
1633:
1630:
1627:
1624:
1619:
1615:
1611:
1608:
1603:
1599:
1587:
1586:
1570:
1566:
1560:
1556:
1549:
1546:
1540:
1535:
1531:
1527:
1522:
1518:
1512:
1509:
1505:
1501:
1494:
1490:
1483:
1480:
1474:
1469:
1465:
1461:
1456:
1452:
1446:
1443:
1439:
1432:
1427:
1423:
1411:
1410:
1397:
1392:
1386:
1383:
1379:
1373:
1369:
1365:
1360:
1357:
1353:
1347:
1343:
1338:
1331:
1327:
1323:
1318:
1314:
1302:
1301:
1288:
1284:
1278:
1275:
1271:
1267:
1262:
1259:
1255:
1251:
1246:
1242:
1236:
1232:
1226:
1223:
1218:
1213:
1209:
1194:
1193:
1180:
1176:
1172:
1169:
1166:
1163:
1160:
1157:
1149:
1145:
1141:
1138:
1135:
1132:
1129:
1126:
1123:
1120:
1117:
1114:
1111:
1108:
1105:
1102:
1099:
1092:
1088:
1082:
1078:
1074:
1069:
1065:
1061:
1055:
1052:
1049:
1046:
1043:
1040:
1011:
991:
980:
979:
965:
962:
959:
954:
951:
945:
942:
939:
936:
933:
914:
913:Expected value
911:
898:
877:
862:distance decay
839:
836:
832:
811:
799:
796:
795:
794:
780:
777:
774:
771:
768:
765:
762:
759:
755:
751:
748:
716:
696:
676:
673:
668:
665:
661:
639:
638:
623:
620:
616:
609:
604:
601:
598:
594:
588:
583:
580:
577:
573:
569:
566:
544:
541:
537:
516:
505:
493:
490:
485:
482:
478:
455:
452:
448:
437:
425:
402:
399:
386:
374:
364:
352:
332:
312:
298:
297:
281:
277:
270:
267:
261:
256:
252:
248:
243:
238:
235:
232:
228:
222:
216:
213:
207:
202:
198:
194:
191:
185:
182:
176:
171:
167:
163:
158:
155:
151:
145:
140:
137:
134:
130:
124:
119:
116:
113:
109:
100:
97:
92:
89:
70:
64:
15:
13:
10:
9:
6:
4:
3:
2:
2866:
2855:
2852:
2850:
2847:
2846:
2844:
2829:
2824:
2821:: 3975–3991.
2820:
2816:
2812:
2805:
2802:
2797:
2793:
2789:
2785:
2781:
2774:
2771:
2766:
2762:
2757:
2752:
2747:
2742:
2738:
2734:
2730:
2723:
2720:
2714:
2709:
2705:
2701:
2697:
2690:
2687:
2681:
2676:
2673:(2): 93–115.
2672:
2668:
2664:
2657:
2654:
2641:
2635:
2632:
2624:
2617:
2614:
2608:
2605:
2599:
2596:
2591:
2587:
2583:
2579:
2575:
2571:
2570:Cressie, Noel
2567:
2564:Li, Hongfei;
2560:
2557:
2552:
2548:
2544:
2540:
2536:
2532:
2528:
2524:
2520:
2514:
2511:
2504:
2500:
2497:
2495:
2492:
2490:
2487:
2485:
2482:
2480:
2477:
2475:
2472:
2470:
2467:
2466:
2462:
2458:
2454:
2451:
2447:
2444:
2440:
2437:
2436:
2435:
2433:
2429:
2425:
2417:
2415:
2413:
2409:
2405:
2401:
2396:
2383:
2359:
2354:
2350:
2342:
2337:
2334:
2331:
2327:
2323:
2320:
2313:
2312:
2311:
2292:
2286:
2272:
2266:
2261:
2257:
2248:
2243:
2240:
2237:
2233:
2226:
2221:
2217:
2209:
2208:
2207:
2181:
2175:
2170:
2166:
2157:
2154:
2150:
2144:
2139:
2136:
2133:
2129:
2121:
2117:
2106:
2100:
2095:
2091:
2084:
2079:
2075:
2067:
2066:
2065:
2063:
2052:
2048:
2044:
2039:
2031:
2025:
2019:
2017:
2013:
2009:
2008:
2002:
1997:
1983:
1980:
1975:
1972:
1948:
1945:
1942:
1938:
1915:
1912:
1909:
1905:
1882:
1879:
1876:
1872:
1866:
1863:
1839:
1836:
1833:
1829:
1823:
1820:
1810:
1805:
1803:
1799:
1777:
1773:
1769:
1766:
1761:
1757:
1753:
1750:
1747:
1742:
1738:
1731:
1728:
1723:
1719:
1712:
1707:
1703:
1695:
1694:
1677:
1673:
1669:
1666:
1661:
1657:
1653:
1650:
1645:
1641:
1634:
1631:
1628:
1625:
1622:
1617:
1613:
1606:
1601:
1597:
1589:
1588:
1568:
1558:
1544:
1538:
1533:
1529:
1520:
1516:
1510:
1507:
1503:
1492:
1478:
1472:
1467:
1463:
1454:
1450:
1444:
1441:
1437:
1430:
1425:
1421:
1413:
1412:
1395:
1390:
1384:
1381:
1377:
1371:
1367:
1363:
1358:
1355:
1351:
1345:
1341:
1336:
1329:
1325:
1321:
1316:
1312:
1304:
1303:
1286:
1276:
1273:
1269:
1265:
1260:
1257:
1253:
1244:
1240:
1234:
1230:
1224:
1221:
1216:
1211:
1207:
1199:
1198:
1197:
1178:
1167:
1161:
1155:
1147:
1143:
1136:
1133:
1130:
1121:
1118:
1115:
1106:
1103:
1100:
1090:
1086:
1080:
1076:
1072:
1067:
1063:
1059:
1053:
1047:
1041:
1038:
1031:
1030:
1029:
1026:
1023:
1009:
989:
963:
960:
957:
952:
949:
943:
937:
931:
924:
923:
922:
920:
912:
910:
896:
889:The value of
875:
865:
863:
859:
855:
837:
834:
830:
809:
802:The value of
797:
778:
775:
772:
766:
763:
760:
753:
749:
746:
738:
734:
730:
714:
694:
674:
671:
666:
663:
659:
650:
645:
641:
640:
621:
618:
614:
607:
602:
599:
596:
592:
586:
581:
578:
575:
571:
567:
564:
542:
539:
535:
514:
506:
491:
488:
483:
480:
476:
453:
450:
446:
438:
423:
397:
387:
372:
365:
350:
330:
310:
303:
302:
301:
279:
265:
259:
254:
250:
241:
236:
233:
230:
226:
211:
205:
200:
196:
180:
174:
169:
165:
156:
153:
149:
143:
138:
135:
132:
128:
122:
117:
114:
111:
107:
98:
95:
90:
87:
80:
79:
78:
76:
69:
63:
61:
57:
54:developed by
53:
49:
48:
38:
34:
30:
26:
21:
2818:
2814:
2804:
2787:
2783:
2773:
2736:
2732:
2722:
2703:
2699:
2689:
2670:
2666:
2656:
2644:. Retrieved
2634:
2616:
2607:
2598:
2581:
2577:
2573:
2559:
2529:(1): 17–23.
2526:
2522:
2513:
2450:dialectology
2423:
2421:
2407:
2397:
2381:
2376:
2309:
2205:
2061:
2050:
2042:
2040:
2036:
2023:
2015:
2011:
2006:
2000:
1998:
1808:
1806:
1797:
1794:
1195:
1027:
1024:
981:
918:
916:
866:
857:
801:
736:
732:
728:
299:
74:
72:
67:
46:
44:
42:
36:
32:
24:
2412:Luc Anselin
739:near 0 (or
2843:Categories
2523:Biometrika
2505:References
2404:ArcGIS Pro
1807:Values of
2739:(1): 19.
2479:Geary's C
2428:geography
2414:in 1995.
2328:∑
2276:¯
2267:−
2234:∑
2185:¯
2176:−
2130:∑
2110:¯
2101:−
1748:−
1729:−
1651:−
1623:−
1548:¯
1539:−
1517:∑
1508:−
1482:¯
1473:−
1451:∑
1442:−
1368:∑
1342:∑
1326:∑
1241:∑
1231:∑
1156:−
1134:−
1119:−
1104:−
1073:−
1042:
1010:π
961:−
950:−
776:−
773:≃
764:−
747:−
593:∑
572:∑
401:¯
269:¯
260:−
227:∑
215:¯
206:−
184:¯
175:−
129:∑
108:∑
2765:22695110
2551:15420245
2463:See also
2422:Moran's
2005:Geary's
1999:Moran's
1802:z-scores
45:Moran's
2756:3441892
2543:2332142
2443:lithium
2206:where:
2763:
2753:
2646:28 May
2642:. ESRI
2549:
2541:
2310:then,
1897:where
1196:where
557:(i.e.
300:where
2626:(PDF)
2539:JSTOR
2400:GeoDa
2761:PMID
2648:2024
2547:PMID
2430:and
2418:Uses
2402:and
1930:and
1854:and
779:0.04
649:rook
507:and
343:and
29:Rook
2823:doi
2792:doi
2751:PMC
2741:doi
2708:doi
2675:doi
2586:doi
2531:doi
2448:In
1039:Var
707:of
2845::
2817:.
2813:.
2788:16
2786:.
2782:.
2759:.
2749:.
2737:11
2735:.
2731:.
2704:24
2702:.
2698:.
2671:27
2669:.
2665:.
2582:39
2580:.
2568:;
2545:.
2537:.
2527:37
2525:.
1996:.
1804:.
637:).
504:);
2831:.
2825::
2819:9
2798:.
2794::
2767:.
2743::
2716:.
2710::
2683:.
2677::
2650:.
2592:.
2588::
2574:I
2553:.
2533::
2424:I
2408:I
2393:N
2388:i
2386:I
2382:I
2378:I
2360:N
2355:i
2351:I
2343:N
2338:1
2335:=
2332:i
2324:=
2321:I
2293:N
2287:2
2283:)
2273:x
2262:i
2258:x
2254:(
2249:N
2244:1
2241:=
2238:i
2227:=
2222:2
2218:m
2191:)
2182:x
2171:j
2167:x
2163:(
2158:j
2155:i
2151:w
2145:N
2140:1
2137:=
2134:j
2122:2
2118:m
2107:x
2096:i
2092:x
2085:=
2080:i
2076:I
2062:I
2057:i
2055:I
2051:I
2043:I
2024:I
2016:C
2012:I
2007:C
2001:I
1984:1
1981:=
1976:W
1973:N
1949:x
1946:a
1943:m
1939:w
1916:n
1913:i
1910:m
1906:w
1883:x
1880:a
1877:m
1873:w
1867:W
1864:N
1840:n
1837:i
1834:m
1830:w
1824:W
1821:N
1809:I
1798:I
1778:2
1774:W
1770:6
1767:+
1762:2
1758:S
1754:N
1751:2
1743:1
1739:S
1735:)
1732:N
1724:2
1720:N
1716:(
1713:=
1708:5
1704:S
1678:2
1674:W
1670:3
1667:+
1662:2
1658:S
1654:N
1646:1
1642:S
1638:)
1635:3
1632:+
1629:N
1626:3
1618:2
1614:N
1610:(
1607:=
1602:4
1598:S
1569:2
1565:)
1559:2
1555:)
1545:x
1534:i
1530:x
1526:(
1521:i
1511:1
1504:N
1500:(
1493:4
1489:)
1479:x
1468:i
1464:x
1460:(
1455:i
1445:1
1438:N
1431:=
1426:3
1422:S
1396:2
1391:)
1385:i
1382:j
1378:w
1372:j
1364:+
1359:j
1356:i
1352:w
1346:j
1337:(
1330:i
1322:=
1317:2
1313:S
1287:2
1283:)
1277:i
1274:j
1270:w
1266:+
1261:j
1258:i
1254:w
1250:(
1245:j
1235:i
1225:2
1222:1
1217:=
1212:1
1208:S
1179:2
1175:)
1171:)
1168:I
1165:(
1162:E
1159:(
1148:2
1144:W
1140:)
1137:3
1131:N
1128:(
1125:)
1122:2
1116:N
1113:(
1110:)
1107:1
1101:N
1098:(
1091:5
1087:S
1081:3
1077:S
1068:4
1064:S
1060:N
1054:=
1051:)
1048:I
1045:(
990:x
964:1
958:N
953:1
944:=
941:)
938:I
935:(
932:E
919:I
897:I
876:k
838:j
835:i
831:w
810:I
770:)
767:1
761:N
758:(
754:/
750:1
737:I
733:I
729:I
715:i
695:j
675:1
672:=
667:j
664:i
660:w
622:j
619:i
615:w
608:N
603:1
600:=
597:j
587:N
582:1
579:=
576:i
568:=
565:W
543:j
540:i
536:w
515:W
492:0
489:=
484:i
481:i
477:w
454:j
451:i
447:w
436:;
424:x
398:x
373:x
363:;
351:j
331:i
311:N
280:2
276:)
266:x
255:i
251:x
247:(
242:N
237:1
234:=
231:i
221:)
212:x
201:j
197:x
193:(
190:)
181:x
170:i
166:x
162:(
157:j
154:i
150:w
144:N
139:1
136:=
133:j
123:N
118:1
115:=
112:i
99:W
96:N
91:=
88:I
75:I
68:I
47:I
37:I
33:I
25:I
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
