60:
or both. More specifically, a spherical contact distribution function is defined as probability distribution of the radius of a sphere when it first encounters or makes contact with a point in a point process. This function can be contrasted with the
65:, which is defined in relation to some point in the point process as being the probability distribution of the distance from that point to its nearest neighbouring point in the same point process.
1308:
1151:
969:
874:
1074:
764:
535:
471:
164:
1209:
are identical for the
Poisson point process can be used to statistically test if point process data appears to be that of a Poisson point process. For example, in spatial statistics the
72:, but some authors define the contact distribution function in relation to a more general set, and not simply a sphere as in the case of the spherical contact distribution function.
691:
257:
1019:
786:
614:
352:
1170:
are not equal. However, these two functions are identical for
Poisson point processes. In fact, this characteristic is due to a unique property of Poisson processes and their
731:
288:
221:
1041:
586:
557:
313:
197:
1648:
1599:
1613:
Foxall, Rob, Baddeley, Adrian (2002). "Nonparametric measures of association between a spatial point process and a random set, with geological applications".
1332:
test for whether data behaves as though it were from a
Poisson process. It is, however, thought possible to construct non-Poisson point processes for which
492:. In other words, spherical contact distribution function is the probability there are no points from the point process located in a hyper-sphere of radius
125:. Since these processes are often used to represent collections of points randomly scattered in space, time or both, the underlying space is usually
1538:
1223:
1667:
1082:
885:
794:
1442:
Stochastic
Geometry: Lectures given at the CIME Summer School held in Martina Franca, Italy, September 13--18, 2004
1363:
1167:
75:
Spherical contact distribution functions are used in the study of point processes as well as the related fields of
62:
1046:
736:
507:
384:
136:
1672:
1329:
167:
1343:=1, but such counterexamples are viewed as somewhat 'artificial' by some and exist for other statistical tests.
1368:
1351:
622:
229:
174:
116:
977:
769:
504:
The spherical contact distribution function can be generalized for sets other than the (hyper-)sphere in
1642:
1593:
707:
591:
321:
29:
1171:
361:
interpretation for point processes. These two notations are often used in parallel or interchangeably.
41:
33:
712:
269:
202:
76:
1024:
569:
540:
296:
180:
1630:
1581:
1573:
481:
122:
104:
80:
45:
263:
173:
Point processes have a number of interpretations, which is reflected by the various types of
1622:
1565:
53:
1440:
A. Baddeley, I. Bárány, and R. Schneider. Spatial point processes and their applications.
130:
1478:}. Probability and its Applications (New York). Springer, New York, second edition, 2008.
1462:. Probability and its Applications (New York). Springer, New York, second edition, 2003.
1373:
358:
1021:
denotes the volume (or more specifically, the
Lebesgue measure) of the ball of radius
1661:
1585:
37:
1634:
88:
1166:
In general, the spherical contact distribution function and the corresponding
84:
121:
Point processes are mathematical objects that are defined on some underlying
1626:
291:
1554:"A remark on the Van Lieshout and Baddeley J-function for point processes"
1577:
100:
96:
92:
49:
1569:
1534:
Stochastic
Geometry and Wireless Networks, Volume II – Applications
559:
with positive volume (or more specifically, Lebesgue measure), the
57:
1494:
Statistical inference and simulation for spatial point processes
1354:) to measure the interaction between points in a point process.
262:
and represents the point process being interpreted as a random
1553:
1514:
Stochastic
Geometry and Wireless Networks, Volume I – Theory
1476:
An introduction to the theory of point processes. Vol. {II
68:
The spherical contact function is also referred to as the
199:
belongs to or is a member of a point process, denoted by
1460:
An introduction to the theory of point processes. Vol. I
1416:
D. Stoyan, W. S. Kendall, J. Mecke, and L. Ruschendorf.
1488:
1486:
1484:
1412:
1410:
1408:
1406:
1404:
1402:
1400:
1398:
1396:
1394:
1392:
1390:
1388:
1050:
1028:
981:
773:
740:
716:
595:
573:
544:
511:
325:
300:
273:
233:
206:
184:
140:
1436:
1434:
1432:
1430:
1428:
1426:
1303:{\displaystyle J(r)={\frac {1-D_{o}(r)}{1-H_{s}(r)}}}
1226:
1085:
1049:
1027:
980:
888:
797:
772:
739:
715:
625:
594:
572:
543:
510:
387:
324:
299:
272:
232:
205:
183:
139:
1615:Journal of the Royal Statistical Society, Series C
1350:-function serves as one way (others include using
1302:
1193:The fact that the spherical distribution function
1146:{\displaystyle H_{s}(r)=1-e^{-\lambda \pi r^{2}}.}
1145:
1068:
1035:
1013:
963:
868:
780:
758:
725:
685:
608:
580:
551:
529:
465:
346:
307:
282:
251:
215:
191:
158:
964:{\displaystyle H_{s}(r)=1-e^{-\lambda |b(o,r)|},}
869:{\displaystyle H_{s}(r)=1-e^{-\Lambda (b(o,r))},}
1470:
1468:
1454:
1452:
1450:
1174:, which forms part of the result known as the
8:
1647:: CS1 maint: multiple names: authors list (
1598:: CS1 maint: multiple names: authors list (
1069:{\displaystyle \textstyle {\textbf {R}}^{2}}
759:{\displaystyle \textstyle {\textbf {R}}^{d}}
530:{\displaystyle \textstyle {\textbf {R}}^{d}}
466:{\displaystyle H_{s}(r)=1-P({N}(b(o,r))=0).}
159:{\displaystyle \textstyle {\textbf {R}}^{d}}
1282:
1255:
1242:
1225:
1132:
1118:
1090:
1084:
1059:
1053:
1052:
1048:
1026:
1005:
982:
979:
951:
928:
921:
893:
887:
830:
802:
796:
771:
749:
743:
742:
738:
717:
714:
654:
630:
624:
593:
571:
542:
520:
514:
513:
509:
422:
392:
386:
326:
323:
298:
274:
271:
266:. Alternatively, the number of points of
240:
231:
207:
204:
182:
149:
143:
142:
138:
1418:Stochastic geometry and its applications
1384:
879:which for the homogeneous case becomes
376:spherical contact distribution function
370:Spherical contact distribution function
48:of physical phenomena representable as
18:spherical contact distribution function
1640:
1591:
686:{\displaystyle H_{B}(r)=P({N}(rB)=0).}
1528:
1526:
1508:
1506:
1504:
1502:
7:
1621:(2). Wiley Online Library: 165–182.
1552:Bedford, T, Van den Berg, J (1997).
1539:Foundations and Trends in Networking
1518:Foundations and Trends in Networking
1420:, edition 2. Wiley Chichester, 1995.
252:{\displaystyle \textstyle x\in {N},}
1492:J. Moller and R. P. Waagepetersen.
1054:
1014:{\displaystyle \textstyle |b(o,r)|}
781:{\displaystyle \textstyle \Lambda }
744:
515:
144:
22:first contact distribution function
834:
774:
609:{\displaystyle \textstyle r\geq 0}
347:{\displaystyle \textstyle {N}(B),}
166:, but they can be defined on more
14:
1532:F. Baccelli and B. Błaszczyszyn.
1512:F. Baccelli and B. Błaszczyszyn.
1313:For a Poisson point process, the
16:In probability and statistics, a
1076:, this expression simplifies to
1558:Advances in Applied Probability
1474:D. J. Daley and D. Vere-Jones.
1458:D. J. Daley and D. Vere-Jones.
1201:and nearest neighbour function
1157:Relationship to other functions
223:, then this can be written as:
83:, which are applied in various
32:that is defined in relation to
1328:=1, hence why it is used as a
1294:
1288:
1267:
1261:
1236:
1230:
1102:
1096:
1006:
1002:
990:
983:
952:
948:
936:
929:
905:
899:
858:
855:
843:
837:
814:
808:
726:{\displaystyle \textstyle {N}}
677:
668:
659:
651:
642:
636:
457:
448:
445:
433:
427:
419:
404:
398:
337:
331:
283:{\displaystyle \textstyle {N}}
216:{\displaystyle \textstyle {N}}
1:
1213:-function is defined for all
561:contact distribution function
500:Contact distribution function
70:contact distribution function
1036:{\displaystyle \textstyle r}
616:is defined by the equation:
581:{\displaystyle \textstyle B}
552:{\displaystyle \textstyle B}
308:{\displaystyle \textstyle B}
192:{\displaystyle \textstyle x}
1689:
1364:Nearest neighbour function
1168:nearest neighbour function
1162:Nearest neighbour function
177:. For example, if a point
114:
63:nearest neighbour function
1352:factorial moment measures
1369:Factorial moment measure
1627:10.1111/1467-9876.00261
1542:. NoW Publishers, 2009.
1520:. NoW Publishers, 2009.
766:with intensity measure
488:centered at the origin
1536:, volume 4, No 1-2 of
1516:, volume 3, No 3-4 of
1304:
1147:
1070:
1037:
1015:
965:
870:
782:
760:
727:
687:
610:
582:
553:
531:
467:
348:
309:
284:
253:
217:
193:
175:point process notation
160:
117:Point process notation
111:Point process notation
1305:
1148:
1071:
1038:
1016:
966:
871:
783:
761:
728:
708:Poisson point process
702:Poisson point process
688:
611:
583:
554:
537:. For some Borel set
532:
468:
349:
315:is often written as:
310:
285:
254:
218:
194:
170:mathematical spaces.
161:
40:, which are types of
30:mathematical function
1668:Stochastic processes
1444:, pages 1--75, 2007.
1224:
1083:
1047:
1025:
978:
886:
795:
770:
737:
713:
623:
592:
570:
541:
508:
385:
322:
297:
270:
230:
203:
181:
137:
91:disciplines such as
42:stochastic processes
34:mathematical objects
26:empty space function
1564:(1). JSTOR: 19–25.
1317:function is simply
1217: ≥ 0 as:
77:stochastic geometry
46:mathematical models
1496:. CRC Press, 2003.
1300:
1180:Slivnyak's theorem
1172:Palm distributions
1143:
1066:
1065:
1033:
1032:
1011:
1010:
961:
866:
778:
777:
756:
755:
723:
722:
683:
606:
605:
578:
577:
549:
548:
527:
526:
463:
344:
343:
305:
304:
280:
279:
249:
248:
213:
212:
189:
188:
156:
155:
123:mathematical space
105:telecommunications
81:spatial statistics
1298:
1056:
746:
517:
357:which reflects a
146:
1680:
1673:Spatial analysis
1653:
1652:
1646:
1638:
1610:
1604:
1603:
1597:
1589:
1549:
1543:
1530:
1521:
1510:
1497:
1490:
1479:
1472:
1463:
1456:
1445:
1438:
1421:
1414:
1349:
1346:More generally,
1342:
1327:
1316:
1309:
1307:
1306:
1301:
1299:
1297:
1287:
1286:
1270:
1260:
1259:
1243:
1216:
1212:
1208:
1200:
1188:
1152:
1150:
1149:
1144:
1139:
1138:
1137:
1136:
1095:
1094:
1075:
1073:
1072:
1067:
1064:
1063:
1058:
1057:
1042:
1040:
1039:
1034:
1020:
1018:
1017:
1012:
1009:
986:
970:
968:
967:
962:
957:
956:
955:
932:
898:
897:
875:
873:
872:
867:
862:
861:
807:
806:
787:
785:
784:
779:
765:
763:
762:
757:
754:
753:
748:
747:
732:
730:
729:
724:
721:
692:
690:
689:
684:
658:
635:
634:
615:
613:
612:
607:
587:
585:
584:
579:
558:
556:
555:
550:
536:
534:
533:
528:
525:
524:
519:
518:
472:
470:
469:
464:
426:
397:
396:
353:
351:
350:
345:
330:
314:
312:
311:
306:
290:located in some
289:
287:
286:
281:
278:
258:
256:
255:
250:
244:
222:
220:
219:
214:
211:
198:
196:
195:
190:
165:
163:
162:
157:
154:
153:
148:
147:
133:denoted here by
1688:
1687:
1683:
1682:
1681:
1679:
1678:
1677:
1658:
1657:
1656:
1639:
1612:
1611:
1607:
1590:
1570:10.2307/1427858
1551:
1550:
1546:
1531:
1524:
1511:
1500:
1491:
1482:
1473:
1466:
1457:
1448:
1439:
1424:
1415:
1386:
1382:
1360:
1347:
1333:
1318:
1314:
1278:
1271:
1251:
1244:
1222:
1221:
1214:
1210:
1206:
1202:
1198:
1194:
1191:
1186:
1164:
1159:
1128:
1114:
1086:
1081:
1080:
1051:
1045:
1044:
1043:. In the plane
1023:
1022:
976:
975:
917:
889:
884:
883:
826:
798:
793:
792:
768:
767:
741:
735:
734:
711:
710:
704:
699:
626:
621:
620:
590:
589:
568:
567:
565:with respect to
539:
538:
512:
506:
505:
502:
388:
383:
382:
378:is defined as:
372:
367:
320:
319:
295:
294:
268:
267:
228:
227:
201:
200:
179:
178:
141:
135:
134:
131:Euclidean space
119:
113:
38:point processes
12:
11:
5:
1686:
1684:
1676:
1675:
1670:
1660:
1659:
1655:
1654:
1605:
1544:
1522:
1498:
1480:
1464:
1446:
1422:
1383:
1381:
1378:
1377:
1376:
1374:Moment measure
1371:
1366:
1359:
1356:
1330:non-parametric
1311:
1310:
1296:
1293:
1290:
1285:
1281:
1277:
1274:
1269:
1266:
1263:
1258:
1254:
1250:
1247:
1241:
1238:
1235:
1232:
1229:
1204:
1196:
1190:
1184:
1176:Slivnyak-Mecke
1163:
1160:
1158:
1155:
1154:
1153:
1142:
1135:
1131:
1127:
1124:
1121:
1117:
1113:
1110:
1107:
1104:
1101:
1098:
1093:
1089:
1062:
1031:
1008:
1004:
1001:
998:
995:
992:
989:
985:
972:
971:
960:
954:
950:
947:
944:
941:
938:
935:
931:
927:
924:
920:
916:
913:
910:
907:
904:
901:
896:
892:
877:
876:
865:
860:
857:
854:
851:
848:
845:
842:
839:
836:
833:
829:
825:
822:
819:
816:
813:
810:
805:
801:
776:
752:
720:
703:
700:
698:
695:
694:
693:
682:
679:
676:
673:
670:
667:
664:
661:
657:
653:
650:
647:
644:
641:
638:
633:
629:
604:
601:
598:
576:
547:
523:
501:
498:
474:
473:
462:
459:
456:
453:
450:
447:
444:
441:
438:
435:
432:
429:
425:
421:
418:
415:
412:
409:
406:
403:
400:
395:
391:
371:
368:
366:
363:
359:random measure
355:
354:
342:
339:
336:
333:
329:
303:
277:
260:
259:
247:
243:
239:
236:
210:
187:
152:
115:Main article:
112:
109:
44:often used as
13:
10:
9:
6:
4:
3:
2:
1685:
1674:
1671:
1669:
1666:
1665:
1663:
1650:
1644:
1636:
1632:
1628:
1624:
1620:
1616:
1609:
1606:
1601:
1595:
1587:
1583:
1579:
1575:
1571:
1567:
1563:
1559:
1555:
1548:
1545:
1541:
1540:
1535:
1529:
1527:
1523:
1519:
1515:
1509:
1507:
1505:
1503:
1499:
1495:
1489:
1487:
1485:
1481:
1477:
1471:
1469:
1465:
1461:
1455:
1453:
1451:
1447:
1443:
1437:
1435:
1433:
1431:
1429:
1427:
1423:
1419:
1413:
1411:
1409:
1407:
1405:
1403:
1401:
1399:
1397:
1395:
1393:
1391:
1389:
1385:
1379:
1375:
1372:
1370:
1367:
1365:
1362:
1361:
1357:
1355:
1353:
1344:
1340:
1336:
1331:
1325:
1321:
1291:
1283:
1279:
1275:
1272:
1264:
1256:
1252:
1248:
1245:
1239:
1233:
1227:
1220:
1219:
1218:
1185:
1183:
1181:
1177:
1173:
1169:
1161:
1156:
1140:
1133:
1129:
1125:
1122:
1119:
1115:
1111:
1108:
1105:
1099:
1091:
1087:
1079:
1078:
1077:
1060:
1029:
999:
996:
993:
987:
958:
945:
942:
939:
933:
925:
922:
918:
914:
911:
908:
902:
894:
890:
882:
881:
880:
863:
852:
849:
846:
840:
831:
827:
823:
820:
817:
811:
803:
799:
791:
790:
789:
788:this becomes
750:
718:
709:
701:
696:
680:
674:
671:
665:
662:
655:
648:
645:
639:
631:
627:
619:
618:
617:
602:
599:
596:
574:
566:
562:
545:
521:
499:
497:
495:
491:
487:
483:
479:
460:
454:
451:
442:
439:
436:
430:
423:
416:
413:
410:
407:
401:
393:
389:
381:
380:
379:
377:
369:
364:
362:
360:
340:
334:
327:
318:
317:
316:
301:
293:
275:
265:
245:
241:
237:
234:
226:
225:
224:
208:
185:
176:
171:
169:
150:
132:
129:-dimensional
128:
124:
118:
110:
108:
106:
102:
98:
94:
90:
86:
82:
78:
73:
71:
66:
64:
59:
55:
51:
47:
43:
39:
35:
31:
27:
23:
19:
1643:cite journal
1618:
1614:
1608:
1594:cite journal
1561:
1557:
1547:
1537:
1533:
1517:
1513:
1493:
1475:
1459:
1441:
1417:
1345:
1338:
1334:
1323:
1319:
1312:
1192:
1179:
1175:
1165:
973:
878:
705:
564:
560:
503:
493:
489:
485:
484:with radius
477:
475:
375:
373:
356:
261:
172:
126:
120:
74:
69:
67:
25:
21:
17:
15:
365:Definitions
89:engineering
52:positioned
1662:Categories
1380:References
85:scientific
1586:122029903
1276:−
1249:−
1189:-function
1126:π
1123:λ
1120:−
1112:−
926:λ
923:−
915:−
835:Λ
832:−
824:−
775:Λ
600:≥
414:−
292:Borel set
238:∈
56:in time,
36:known as
1358:See also
697:Examples
168:abstract
50:randomly
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