87:, The Poisson point process often exhibits a type of mathematical closure such that when a point process operation is applied to some Poisson point process, then provided some conditions on the point process operation, the resulting process will be often another Poisson point process operation, hence it is often used as a mathematical model.
115:, which gives the average number of points of the point process located in some region. In other words, in the limit as the number of operations applied approaches infinity, the point process will converge in distribution (or weakly) to a Poisson point process or, if its measure is a random measure, to a
696:
These three operations are all types of independent thinning, which means the interaction between points has no effect on the where a point is removed (or kept). Another generalization involves dependent thinning where points of the point process are removed (or kept) depending on their location in
1486:
results, in the limit as the number of performed operations approaches infinity. For example, if each point in a general point process is repeatedly displaced in a certain random and independent manner, then the new point process, informally speaking, will more and more resemble a
Poisson point
110:
in general. Provided that the original point process and the point process operation meet certain mathematical conditions, then as point process operations are applied to the process, then often the resulting point process will behave stochastically more like a
Poisson point process if it has a
384:
To develop suitable models with point processes in stochastic geometry, spatial statistics and related fields, there are number of useful transformations that can be performed on point processes including: thinning, superposition, mapping (or transformation of space), clustering, and random
697:
relation to other points of the point process. Thinning can be used to create new point processes such as hard-core processes where points do not exist (due to thinning) within a certain radius of each point in the thinned point process.
1481:
A point operation performed once on some point process can be, in general, performed again and again. In the theory of point processes, results have been derived to study the behaviour of the resulting point process, via
1472:
says that if the original process is a
Poisson point process with some intensity measure, then the resulting mapped (or transformed) collection of points also forms a Poisson point process with another intensity measure.
1006:
808:
872:
1166:
760:
367:
1321:
583:
225:
1248:
1279:
922:
450:
320:
1442:
Another property that is considered useful is the ability to map a point process from one underlying space to another space. For example, a point process defined on the plane
1375:
1198:
1097:
645:
614:
1226:
1064:
946:
669:
546:
496:
419:
256:
189:
1418:
Applying random displacements or translations to point processes may be used as mathematical models for mobility of objects in, for example, ecology or wireless networks.
1346:
1040:
691:
518:
472:
281:
165:
1487:
process. Similar convergence results have been developed for the operations of thinning and superposition (with suitable rescaling of the underlying space).
68:
the underlying space of the point process into another space. Point process operations and the resulting point processes are used in the theory of
56:
or both, and are used to construct new point processes, which can be then also used as mathematical models. The operations may include removing or
1412:
1201:
1403:. If each point in the process is displaced or translated independently to other all other points in the process, then the operation forms an
1645:
954:
1431:
1391:
A mathematical model may require randomly moving points of a point process from some locations to other locations on the underlying
106:
in the 1950s and 1960s who both studied point processes on the real line, in the context of studying the arrival of phone calls and
137:
Point processes are mathematical objects that can be used to represent collections of points randomly scattered on some underlying
123:
for renewal processes, are then also used to justify the use of the
Poisson point process as a mathematical of various phenomena.
765:
1463:
709:
is used to combine two or more point processes together onto one underlying mathematical space or state space. If there is a
452:. These thinning rules may be deterministic, that is, not random, which is the case for one of the simplest rules known as
816:
1108:
1434:
displacement of points of a
Poisson point process (on the same underlying space) forms another Poisson point process.
716:
1704:
1682:
Stochastic
Geometry: Lectures given at the CIME Summer School held in Martina Franca, Italy, September 13–18, 2004
1099:. Each cluster is also a point process, but with a finite number of points. The union of all the clusters forms a
339:
1483:
370:
65:
1699:
1468:
Provided that the mapping (or transformation) adheres to some conditions, then a result sometimes known as the
1408:
1287:
551:
1623:
882:
197:
142:
132:
95:
83:
One point process that gives particularly convenient results under random point process operations is the
33:
1231:
1447:
1253:
896:
424:
289:
84:
1351:
1174:
1073:
619:
588:
1628:
1207:
1045:
927:
650:
523:
477:
400:
237:
170:
373:
73:
45:
1329:
1023:
674:
501:
455:
264:
148:
1407:
displacement or translation. It is usually assume that all the random translations have a common
1392:
878:
671:
is located on the underlying space. A further generalization is to have the thinning probability
138:
103:
91:
77:
1641:
1451:
231:
94:
as the number of random point process operations applied approaches infinity. This had led to
332:
A point process needs to be defined on an underlying mathematical space. Often this space is
1633:
49:
877:
also forms a point process. In this expression the superposition operation is denoted by a
1680:
A. Baddeley, I. Bárány, and R. Schneider. Spatial point processes and their applications.
107:
1588:}. Probability and its Applications (New York). Springer, New York, second edition, 2008.
326:
112:
1693:
710:
69:
53:
41:
397:
operation entails using some predefined rule to remove points from a point process
141:. They have a number of interpretations, which is reflected by the various types of
1386:
1622:. C&H/CRC Monographs on Statistics & Applied Probability. Vol. 100.
116:
98:
of point process operations, which have their origins in the pioneering work of
17:
99:
21:
259:
1637:
881:), which implies the random set interpretation of point processes; see
548:). This rule may be generalized by introducing a non-negative function
37:
1602:
Stochastic
Geometry and Wireless Networks, Volume II – Applications
1001:{\displaystyle \Lambda =\sum \limits _{i=1}^{\infty }\Lambda _{i}.}
52:
randomly located in space. These operations can be purely random,
1620:
Statistical
Inference and Simulation for Spatial Point Processes
616:-thinning where now the probability of a point being removed is
1665:
Stochastic
Geometry and Wireless Networks, Volume I – Theory
803:{\displaystyle \textstyle \Lambda _{1},\Lambda _{2},\dots }
1586:
An introduction to the theory of point processes. Vol. {II
1395:. This point process operation is referred to as random
498:
is independently removed (or kept) with some probability
167:
belongs to or is a member of a point process, denoted by
1517:
D. Stoyan, W. S. Kendall, J. Mecke, and L. Ruschendorf.
924:
is a
Poisson point process, then the resulting process
1513:
1511:
1509:
1507:
1505:
1503:
1501:
1499:
1355:
1333:
1257:
1235:
1211:
1178:
1077:
1049:
1027:
931:
900:
769:
720:
678:
654:
623:
592:
555:
527:
505:
481:
459:
428:
404:
343:
293:
268:
241:
201:
174:
152:
1415:
random vectors in the underlying mathematical space.
1354:
1332:
1290:
1256:
1234:
1210:
1177:
1111:
1076:
1048:
1026:
957:
930:
899:
819:
768:
719:
677:
653:
622:
591:
554:
526:
504:
480:
458:
427:
403:
342:
292:
267:
240:
200:
173:
151:
1543:
1541:
1539:
1537:
1535:
1533:
1531:
1529:
1527:
948:
is also a Poisson point process with mean intensity
867:{\displaystyle {N}=\bigcup _{i=1}^{\infty }{N}_{i},}
64:multiple point processes into one point process or
1369:
1340:
1315:
1273:
1250:, then the intensity of the cluster point process
1242:
1220:
1200:are all sets of finite points with each set being
1192:
1160:
1091:
1058:
1034:
1000:
940:
916:
866:
802:
754:
685:
663:
639:
608:
577:
540:
512:
490:
466:
444:
413:
369:, although point processes can be defined on more
361:
314:
275:
250:
219:
183:
159:
90:Point process operations have been studied in the
1161:{\displaystyle {N}_{c}=\bigcup _{x\in {N}}N^{x}.}
755:{\displaystyle \textstyle {N}_{1},{N}_{2}\dots }
1580:
1578:
1576:
1574:
1572:
1570:
1618:Moller, J.; Plenge Waagepetersen, R. (2003).
1204:. Furthermore, if the original point process
336:-dimensional Euclidean space denoted here by
230:and represents the point process as a random
8:
362:{\displaystyle \textstyle {\textbf {R}}^{d}}
1551:, volume 3. Oxford university press, 1992.
60:points from a point process, combining or
1627:
1360:
1353:
1331:
1295:
1289:
1264:
1259:
1255:
1233:
1212:
1209:
1183:
1176:
1149:
1138:
1131:
1118:
1113:
1110:
1082:
1075:
1050:
1047:
1025:
989:
979:
968:
956:
932:
929:
907:
902:
898:
855:
850:
843:
832:
820:
818:
787:
774:
767:
742:
737:
727:
722:
718:
676:
655:
652:
621:
590:
585:in order to define the located-dependent
553:
525:
503:
482:
479:
457:
435:
430:
426:
405:
402:
352:
346:
345:
341:
294:
291:
266:
242:
239:
234:. Alternatively, the number of points of
208:
199:
175:
172:
150:
1519:Stochastic geometry and its applications
1411:; hence the displacements form a set of
1348:is the mean of number of points in each
647:and is dependent on where the point of
48:of phenomena that can be represented as
1495:
1477:Convergence of point process operations
1413:independent and identically distributed
1316:{\displaystyle \lambda _{c}=c\lambda ,}
1202:independent and identically distributed
1171:Often is it assumed that the clusters
1659:
1657:
1596:
1594:
578:{\displaystyle \textstyle p(x)\leq 1}
7:
1669:Foundations and Trends in Networking
1606:Foundations and Trends in Networking
329:interpretation for point processes.
220:{\displaystyle \textstyle x\in {N},}
119:. Convergence results, such as the
1564:. Pages 173-175, Academic Pr, 1983.
1521:, volume 2. Wiley Chichester, 1995.
1381:Random displacement and translation
1243:{\displaystyle \textstyle \lambda }
965:
347:
1274:{\displaystyle \textstyle {N}_{c}}
986:
980:
958:
917:{\displaystyle \textstyle {N}_{i}}
844:
784:
771:
445:{\displaystyle \textstyle {N}_{p}}
315:{\displaystyle \textstyle {N}(B),}
14:
1663:F. Baccelli and B. Błaszczyszyn.
1600:F. Baccelli and B. Błaszczyszyn.
1430:effectively says that the random
713:or collection of point processes
1370:{\displaystyle \textstyle N^{x}}
1193:{\displaystyle \textstyle N^{x}}
1092:{\displaystyle \textstyle N^{x}}
1584:D. J. Daley and D. Vere-Jones.
1464:Mapping theorem (point process)
640:{\displaystyle \textstyle p(x)}
609:{\displaystyle \textstyle p(x)}
191:, then this can be written as:
1221:{\displaystyle \textstyle {N}}
1059:{\displaystyle \textstyle {N}}
1020:entails replacing every point
941:{\displaystyle \textstyle {N}}
664:{\displaystyle \textstyle {N}}
633:
627:
602:
596:
565:
559:
541:{\displaystyle \textstyle 1-p}
491:{\displaystyle \textstyle {N}}
414:{\displaystyle \textstyle {N}}
305:
299:
251:{\displaystyle \textstyle {N}}
184:{\displaystyle \textstyle {N}}
1:
1016:The point operation known as
1341:{\displaystyle \textstyle c}
1035:{\displaystyle \textstyle x}
686:{\displaystyle \textstyle p}
513:{\displaystyle \textstyle p}
467:{\displaystyle \textstyle p}
421:to form a new point process
276:{\displaystyle \textstyle B}
160:{\displaystyle \textstyle x}
30:point process transformation
810:, then their superposition
145:. For example, if a point
72:and related fields such as
1721:
1461:
1384:
889:Poisson point process case
474:-thinning: each point of
130:
44:, which are often used as
1228:has a constant intensity
1042:in a given point process
1446:can be transformed from
1426:The result known as the
1409:probability distribution
1671:. NoW Publishers, 2009.
1608:. NoW Publishers, 2009.
1438:Transformation of space
893:In the case where each
707:superposition operation
26:point process operation
1667:, volume 3, No 3–4 of
1604:, volume 4, No 1–2 of
1371:
1342:
1317:
1275:
1244:
1222:
1194:
1162:
1093:
1060:
1036:
1002:
984:
942:
918:
885:for more information.
883:Point process notation
868:
848:
804:
756:
687:
665:
641:
610:
579:
542:
514:
492:
468:
446:
415:
380:Examples of operations
363:
316:
277:
252:
221:
185:
161:
143:point process notation
133:Point process notation
127:Point process notation
34:mathematical operation
1638:10.1201/9780203496930
1448:Cartesian coordinates
1372:
1343:
1318:
1276:
1245:
1223:
1195:
1163:
1101:cluster point process
1094:
1061:
1037:
1003:
964:
943:
919:
869:
828:
805:
757:
688:
666:
642:
611:
580:
543:
515:
493:
469:
447:
416:
364:
317:
283:is often written as:
278:
253:
222:
186:
162:
121:Palm-Khinchin theorem
85:Poisson point process
1428:Displacement theorem
1422:Displacement theorem
1352:
1330:
1288:
1254:
1232:
1208:
1175:
1109:
1074:
1046:
1024:
955:
928:
897:
817:
766:
717:
675:
651:
620:
589:
552:
524:
502:
478:
456:
425:
401:
340:
290:
265:
238:
198:
171:
149:
96:convergence theorems
1684:, pages 1–75, 2007.
1326:where the constant
762:with mean measures
374:mathematical spaces
102:in 1940s and later
74:stochastic geometry
46:mathematical models
1547:J. F. C. Kingman.
1393:mathematical space
1367:
1366:
1338:
1337:
1313:
1271:
1270:
1240:
1239:
1218:
1217:
1190:
1189:
1158:
1144:
1089:
1088:
1056:
1055:
1032:
1031:
998:
938:
937:
914:
913:
864:
800:
799:
752:
751:
683:
682:
661:
660:
637:
636:
606:
605:
575:
574:
538:
537:
510:
509:
488:
487:
464:
463:
442:
441:
411:
410:
359:
358:
312:
311:
273:
272:
248:
247:
217:
216:
181:
180:
157:
156:
139:mathematical space
104:Aleksandr Khinchin
92:mathematical limit
78:spatial statistics
40:object known as a
1705:Spatial processes
1647:978-1-58488-265-7
1549:Poisson processes
1452:polar coordinates
1127:
349:
325:which reflects a
117:Cox point process
1712:
1685:
1678:
1672:
1661:
1652:
1651:
1631:
1615:
1609:
1598:
1589:
1582:
1565:
1558:
1552:
1545:
1522:
1515:
1376:
1374:
1373:
1368:
1365:
1364:
1347:
1345:
1344:
1339:
1322:
1320:
1319:
1314:
1300:
1299:
1280:
1278:
1277:
1272:
1269:
1268:
1263:
1249:
1247:
1246:
1241:
1227:
1225:
1224:
1219:
1216:
1199:
1197:
1196:
1191:
1188:
1187:
1167:
1165:
1164:
1159:
1154:
1153:
1143:
1142:
1123:
1122:
1117:
1098:
1096:
1095:
1090:
1087:
1086:
1065:
1063:
1062:
1057:
1054:
1041:
1039:
1038:
1033:
1007:
1005:
1004:
999:
994:
993:
983:
978:
947:
945:
944:
939:
936:
923:
921:
920:
915:
912:
911:
906:
873:
871:
870:
865:
860:
859:
854:
847:
842:
824:
809:
807:
806:
801:
792:
791:
779:
778:
761:
759:
758:
753:
747:
746:
741:
732:
731:
726:
692:
690:
689:
684:
670:
668:
667:
662:
659:
646:
644:
643:
638:
615:
613:
612:
607:
584:
582:
581:
576:
547:
545:
544:
539:
519:
517:
516:
511:
497:
495:
494:
489:
486:
473:
471:
470:
465:
451:
449:
448:
443:
440:
439:
434:
420:
418:
417:
412:
409:
368:
366:
365:
360:
357:
356:
351:
350:
321:
319:
318:
313:
298:
282:
280:
279:
274:
258:located in some
257:
255:
254:
249:
246:
226:
224:
223:
218:
212:
190:
188:
187:
182:
179:
166:
164:
163:
158:
1720:
1719:
1715:
1714:
1713:
1711:
1710:
1709:
1700:Point processes
1690:
1689:
1688:
1679:
1675:
1662:
1655:
1648:
1629:10.1.1.124.1275
1617:
1616:
1612:
1599:
1592:
1583:
1568:
1562:Random measures
1560:O. Kallenberg.
1559:
1555:
1546:
1525:
1516:
1497:
1493:
1479:
1470:Mapping theorem
1466:
1460:
1458:Mapping theorem
1440:
1424:
1389:
1383:
1356:
1350:
1349:
1328:
1327:
1291:
1286:
1285:
1258:
1252:
1251:
1230:
1229:
1206:
1205:
1179:
1173:
1172:
1145:
1112:
1107:
1106:
1078:
1072:
1071:
1044:
1043:
1022:
1021:
1014:
985:
953:
952:
926:
925:
901:
895:
894:
891:
849:
815:
814:
783:
770:
764:
763:
736:
721:
715:
714:
703:
693:random itself.
673:
672:
649:
648:
618:
617:
587:
586:
550:
549:
522:
521:
500:
499:
476:
475:
454:
453:
429:
423:
422:
399:
398:
391:
382:
344:
338:
337:
288:
287:
263:
262:
236:
235:
196:
195:
169:
168:
147:
146:
135:
129:
108:queueing theory
70:point processes
36:performed on a
12:
11:
5:
1718:
1716:
1708:
1707:
1702:
1692:
1691:
1687:
1686:
1673:
1653:
1646:
1610:
1590:
1566:
1553:
1523:
1494:
1492:
1489:
1478:
1475:
1462:Main article:
1459:
1456:
1439:
1436:
1423:
1420:
1385:Main article:
1382:
1379:
1363:
1359:
1336:
1324:
1323:
1312:
1309:
1306:
1303:
1298:
1294:
1267:
1262:
1238:
1215:
1186:
1182:
1169:
1168:
1157:
1152:
1148:
1141:
1137:
1134:
1130:
1126:
1121:
1116:
1085:
1081:
1053:
1030:
1013:
1010:
1009:
1008:
997:
992:
988:
982:
977:
974:
971:
967:
963:
960:
935:
910:
905:
890:
887:
875:
874:
863:
858:
853:
846:
841:
838:
835:
831:
827:
823:
798:
795:
790:
786:
782:
777:
773:
750:
745:
740:
735:
730:
725:
702:
699:
681:
658:
635:
632:
629:
626:
604:
601:
598:
595:
573:
570:
567:
564:
561:
558:
536:
533:
530:
508:
485:
462:
438:
433:
408:
390:
387:
385:displacement.
381:
378:
355:
327:random measure
323:
322:
310:
307:
304:
301:
297:
271:
245:
228:
227:
215:
211:
207:
204:
178:
155:
131:Main article:
128:
125:
13:
10:
9:
6:
4:
3:
2:
1717:
1706:
1703:
1701:
1698:
1697:
1695:
1683:
1677:
1674:
1670:
1666:
1660:
1658:
1654:
1649:
1643:
1639:
1635:
1630:
1625:
1621:
1614:
1611:
1607:
1603:
1597:
1595:
1591:
1587:
1581:
1579:
1577:
1575:
1573:
1571:
1567:
1563:
1557:
1554:
1550:
1544:
1542:
1540:
1538:
1536:
1534:
1532:
1530:
1528:
1524:
1520:
1514:
1512:
1510:
1508:
1506:
1504:
1502:
1500:
1496:
1490:
1488:
1485:
1476:
1474:
1471:
1465:
1457:
1455:
1453:
1449:
1445:
1437:
1435:
1433:
1429:
1421:
1419:
1416:
1414:
1410:
1406:
1402:
1398:
1394:
1388:
1380:
1378:
1361:
1357:
1334:
1310:
1307:
1304:
1301:
1296:
1292:
1284:
1283:
1282:
1265:
1260:
1236:
1213:
1203:
1184:
1180:
1155:
1150:
1146:
1139:
1135:
1132:
1128:
1124:
1119:
1114:
1105:
1104:
1103:
1102:
1083:
1079:
1069:
1051:
1028:
1019:
1011:
995:
990:
975:
972:
969:
961:
951:
950:
949:
933:
908:
903:
888:
886:
884:
880:
861:
856:
851:
839:
836:
833:
829:
825:
821:
813:
812:
811:
796:
793:
788:
780:
775:
748:
743:
738:
733:
728:
723:
712:
711:countable set
708:
701:Superposition
700:
698:
694:
679:
656:
630:
624:
599:
593:
571:
568:
562:
556:
534:
531:
528:
506:
483:
460:
436:
431:
406:
396:
388:
386:
379:
377:
375:
372:
353:
335:
330:
328:
308:
302:
295:
286:
285:
284:
269:
261:
243:
233:
213:
209:
205:
202:
194:
193:
192:
176:
153:
144:
140:
134:
126:
124:
122:
118:
114:
109:
105:
101:
97:
93:
88:
86:
81:
79:
75:
71:
67:
63:
62:superimposing
59:
55:
54:deterministic
51:
47:
43:
42:point process
39:
35:
32:is a type of
31:
27:
23:
19:
1681:
1676:
1668:
1664:
1619:
1613:
1605:
1601:
1585:
1561:
1556:
1548:
1518:
1480:
1469:
1467:
1443:
1441:
1427:
1425:
1417:
1404:
1400:
1397:displacement
1396:
1390:
1387:Nu-transform
1325:
1170:
1100:
1067:
1017:
1015:
892:
876:
706:
704:
695:
394:
392:
383:
333:
331:
324:
229:
136:
120:
113:mean measure
89:
82:
66:transforming
61:
57:
29:
25:
15:
1484:convergence
1432:independent
1405:independent
1401:translation
111:non-random
18:probability
1694:Categories
1491:References
1070:of points
1018:clustering
1012:Clustering
100:Conny Palm
22:statistics
1624:CiteSeerX
1308:λ
1293:λ
1237:λ
1136:∈
1129:⋃
987:Λ
981:∞
966:∑
959:Λ
879:set union
845:∞
830:⋃
797:…
785:Λ
772:Λ
749:…
569:≤
532:−
260:Borel set
206:∈
1281:will be
395:thinning
389:Thinning
371:abstract
58:thinning
1068:cluster
1066:with a
1644:
1626:
50:points
38:random
1642:ISBN
705:The
520:(or
393:The
76:and
24:, a
20:and
1634:doi
1450:to
1399:or
232:set
28:or
16:In
1696::
1656:^
1640:.
1632:.
1593:^
1569:^
1526:^
1498:^
1454:.
1377:.
376:.
80:.
1650:.
1636::
1444:R
1362:x
1358:N
1335:c
1311:,
1305:c
1302:=
1297:c
1266:c
1261:N
1214:N
1185:x
1181:N
1156:.
1151:x
1147:N
1140:N
1133:x
1125:=
1120:c
1115:N
1084:x
1080:N
1052:N
1029:x
996:.
991:i
976:1
973:=
970:i
962:=
934:N
909:i
904:N
862:,
857:i
852:N
840:1
837:=
834:i
826:=
822:N
794:,
789:2
781:,
776:1
744:2
739:N
734:,
729:1
724:N
680:p
657:N
634:)
631:x
628:(
625:p
603:)
600:x
597:(
594:p
572:1
566:)
563:x
560:(
557:p
535:p
529:1
507:p
484:N
461:p
437:p
432:N
407:N
354:d
348:R
334:d
309:,
306:)
303:B
300:(
296:N
270:B
244:N
214:,
210:N
203:x
177:N
154:x
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.