33:
1159:
1740:
A. Fotiou (2018) "A Discussion on Least
Squares Adjustment with Worked Examples" In: Fotiou A., D. Rossikopoulos, eds. (2018): “Quod erat demonstrandum. In quest for the ultimate geodetic insight.” Special issue for Professor Emeritus Athanasios Dermanis. Publication of the School of Rural and
974:
567:
747:
430:
1039:
873:
806:
1679:
Harvey, Bruce R., "Practical least squares and statistics for surveyors", Monograph 13, Third
Edition, School of Surveying and Spatial Information Systems, University of New South Wales, 2006
673:
626:
488:
459:
1116:
1087:
367:
338:
1503:
1525:
1682:
Huaan Fan, "Theory of Errors and Least
Squares Adjustment", Royal Institute of Technology (KTH), Division of Geodesy and Geoinformatics, Stockholm, Sweden, 2010,
497:
1304:
is encountered, it can often be rectified by the inclusion of additional equations imposing constraints on the parameters and/or observations, leading to
1726:
Erik
Grafarend and Joseph Awange, "Applications of Linear and Nonlinear Models: Fixed Effects, Random Effects, and Total Least Squares", Springer, 2012
1782:
1746:
1711:
1655:
1630:
1375:
1342:
678:
896:
979:
54:
1440:
Neitzel, Frank (2010-09-17). "Generalization of total least-squares on example of unweighted and weighted 2D similarity transformation".
1753:
John
Olusegun Ogundare (2018), "Understanding Least Squares Estimation and Geomatics Data Analysis", John Wiley & Sons, 720 pages,
1603:
878:
1758:
1734:
1687:
1647:
76:
631:
584:
574:
1595:
1325:
Kotz, Samuel; Read, Campbell B.; Balakrishnan, N.; Vidakovic, Brani; Johnson, Norman L. (2004-07-15). "Gauss-Helmert Model".
1723:
Charles D. Ghilani and Paul R. Wolf, "Elementary
Surveying: An Introduction to Geomatics", 13th Edition, Prentice Hall, 2011
372:
1541:
819:
752:
1518:
47:
41:
1130:
Given the matrices and vectors above, their solution is found via standard least-squares methods; e.g., forming the
1060:
least squares problem into an unconstrained one (albeit a larger one). In any case, their manipulation leads to the
1274:
1729:
Alfred Leick, Lev
Rapoport, and Dmitry Tatarnikov, "GPS Satellite Surveying", 4th Edition, John Wiley & Sons,
1548:
1529:
1305:
58:
1569:
1270:
1057:
1609:
Edward M. Mikhail, Friedrich E. Ackermann, "Observations and least squares", University Press of
America, 1982
1507:
1540:, January 1994. First edition April 1983, Reprinted with corrections January 1990. (Original Working Papers,
291:, respectively. Yet the special cases warrant simpler solutions, as detailed below. Often in the literature,
1822:
1289:
570:
1252:
1807:
1802:
1676:
Edward M. Mikhail, James S. Bethel, J. Chris McGlone, "Introduction to Modern
Photogrammetry", Wiley, 2001
1235:
1135:
1042:
1749:
1670:
Karl-Rudolf Koch, "Parameter
Estimation and Hypothesis Testing in Linear Models", 2a ed., Springer, 2000
1587:
1552:
101:
93:
1449:
1266:
1199:
1053:
1278:
1248:
1667:
Paul Wolf and Bon DeWitt, "Elements of Photogrammetry with Applications in GIS", McGraw-Hill, 2000
1639:
1362:. Geometry and Computing. Vol. 11. Cham: Springer International Publishing. pp. 75–190.
464:
435:
242:
Clearly, parametric and conditional adjustments correspond to the more general combined case when
1817:
1720:
Charles D. Ghilani, "Adjustment Computations: Spatial Data Analysis", John Wiley & Sons, 2011
1473:
1537:
1092:
1063:
343:
314:
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1699:
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1363:
1330:
1230:
1203:
1139:
1301:
17:
1590:, "Adjustments by least squares in geodesy and photogrammetry", Ungar, New York. 261 p.,
1401:"Total Least-Squares regularization of Tykhonov type and an ancient racetrack in Corinth"
1453:
1737:; Chapter 2, "Least-Squares Adjustments", pp. 11–79, doi:10.1002/9781119018612.ch2
1661:
113:
1658:; chap. 12, "Least-squares solution of overdetermined models", pp. 202–213, 1986.
1549:
Applications of Parameter Estimation and Hypothesis Testing to GPS Network Adjustments
1158:
1796:
1477:
1217:
1213:
1131:
578:
97:
1334:
1673:
P.J.G. Teunissen, "Adjustment theory, an introduction", Delft Academic Press, 2000
1622:
1774:
1703:
1367:
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and E.J. Krakiwsky, "Geodesy: The Concepts." Amsterdam: Elsevier. (third ed.):
1461:
1417:
1400:
1469:
1426:
1385:
1765:
Shen, Yunzhong; Xu, Guochang (2012-07-31). "Regularization and Adjustment".
117:
105:
1664:
and Kai Borre, "Linear Algebra, Geodesy, and GPS", SIAM, 624 pages, 1997.
1613:
Wolf, Paul R. (1995). "Survey Measurement Adjustments by Least Squares".
1195:
893:, and the misclosure vector can be interpreted as the pre-fit residuals,
1741:
Surveying Engineering, Aristotle University of Thessaloniki, 405 pages.
1514:
885:
In the parametric adjustment, the second design matrix is an identity,
109:
1517:, Lecture Notes #42, Department of Geodesy and Geomatics Engineering,
1056:
are introduced to relate the two Jacobian matrices, and transform the
969:{\displaystyle {\tilde {y}}={\tilde {w}}=h({\tilde {X}})-{\tilde {Y}}}
562:{\displaystyle {\tilde {w}}=f\left({\tilde {X}},{\tilde {Y}}\right).}
1769:. Berlin, Heidelberg: Springer Berlin Heidelberg. pp. 293–337.
1045:. In the conditional adjustment, the first design matrix is null,
1694:
Gielsdorf, F.; Hillmann, T. (2011). "Mathematics and Statistics".
1515:"A synthesis of recent advances in the method of least squares"
1581:
1574:
Die Ausgleichsrechnung nach der Methode der kleinsten Quadrate
1358:
Förstner, Wolfgang; Wrobel, Bernhard P. (2016). "Estimation".
1226:
1153:
1118:
vectors as well as the respective parameters and observations
26:
311:
The equalities above only hold for the estimated parameters
1578:
Adjustment computation based on the method of least squares
1448:(12). Springer Science and Business Media LLC: 751–762.
1170:
742:{\displaystyle {\tilde {w}}+A{\hat {x}}+B{\hat {y}}=0,}
1095:
1066:
982:
899:
822:
755:
681:
634:
587:
500:
467:
438:
425:{\displaystyle f\left({\hat {X}},{\hat {Y}}\right)=0}
375:
346:
317:
1034:{\displaystyle A{\hat {x}}={\hat {y}}-{\tilde {y}},}
198:(leading to the B-model below) — with no parameters
1526:"Advanced least squares applied to position-fixing"
128:There are three forms of least squares adjustment:
1110:
1081:
1033:
968:
868:{\displaystyle {\hat {y}}={\hat {Y}}-{\tilde {Y}}}
867:
801:{\displaystyle {\hat {x}}={\hat {X}}-{\tilde {X}}}
800:
741:
667:
620:
561:
482:
453:
424:
361:
332:
223:are involved implicitly in a mixed-model equation
1329:. Hoboken, NJ, USA: John Wiley & Sons, Inc.
104:. It is used extensively in the disciplines of
1265:(named after German mathematicians/geodesists
181:, there exists a condition equation which is
8:
1532:, School of Surveying, Working Paper No. 6,
1247:Parametric adjustment is similar to most of
668:{\displaystyle B=\partial {f}/\partial {Y}.}
621:{\displaystyle A=\partial {f}/\partial {X};}
1696:Springer Handbook of Geographic Information
1416:
1097:
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319:
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316:
77:Learn how and when to remove this message
1399:Schaffrin, Burkhard; Snow, Kyle (2010).
40:This article includes a list of general
1317:
1288:parameter covariance matrix is akin to
1257:Combined adjustment, also known as the
573:of the equations, which results in the
148:, one can find an observation equation
96:of equations based on the principle of
1544:, Dept. of Surveying, 205 pp., 1983.)
432:. In contrast, measured observations
7:
1327:Encyclopedia of Statistical Sciences
1494:Lecture notes and technical reports
1405:Linear Algebra and Its Applications
654:
641:
607:
594:
168:explicitly in terms of parameters
92:is a model for the solution of an
46:it lacks sufficient corresponding
25:
1142:directly to the Jacobian matrix,
675:The linearized model then reads:
1551:, Division of Geodetic Science,
1502:Nico Sneeuw and Friedhelm Krum,
1157:
31:
1580:). Leipzig: Teubner, 1872. <
1360:Photogrammetric Computer Vision
1335:10.1002/0471667196.ess0854.pub2
976:, so the system simplifies to:
174:(leading to the A-model below).
1102:
1073:
1052:. For the more general cases,
1022:
1007:
992:
960:
948:
942:
933:
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906:
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829:
792:
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724:
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688:
545:
530:
507:
474:
445:
405:
390:
353:
324:
1:
1542:North East London Polytechnic
1411:(8). Elsevier BV: 2061–2076.
1146:for very large systems, etc.
1623:10.1007/978-1-4615-2067-2_16
483:{\displaystyle {\tilde {X}}}
454:{\displaystyle {\tilde {Y}}}
192:involving only observations
1775:10.1007/978-3-642-28000-9_6
1704:10.1007/978-3-540-72680-7_2
1582:http://eudml.org/doc/203764
1519:University of New Brunswick
1368:10.1007/978-3-319-11550-4_4
461:and approximate parameters
1839:
1275:errors-in-variables models
1111:{\displaystyle {\hat {Y}}}
1082:{\displaystyle {\hat {X}}}
362:{\displaystyle {\hat {Y}}}
333:{\displaystyle {\hat {X}}}
18:Adjustment of observations
1530:University of East London
1506:, Geodätisches Institut,
1462:10.1007/s00190-010-0408-0
1418:10.1016/j.laa.2009.09.014
1306:constrained least squares
1767:Sciences of Geodesy - II
1570:Friedrich Robert Helmert
1041:which is in the form of
90:Least-squares adjustment
1290:Tikhonov regularization
1251:and coincides with the
571:Taylor series expansion
61:more precise citations.
1615:The Surveying Handbook
1236:Helmert transformation
1136:Cholesky decomposition
1112:
1083:
1043:ordinary least squares
1035:
970:
869:
802:
743:
669:
622:
563:
484:
455:
426:
363:
334:
179:conditional adjustment
162:relating observations
1588:Reino Antero Hirvonen
1553:Ohio State University
1508:Universität Stuttgart
1273:), is related to the
1122:covariance matrices.
1113:
1084:
1036:
971:
870:
810:parameter corrections
803:
744:
670:
623:
564:
485:
456:
427:
364:
335:
146:parametric adjustment
102:observation residuals
94:overdetermined system
1617:. pp. 383–413.
1093:
1064:
1054:Lagrange multipliers
980:
897:
820:
753:
679:
632:
628:and the second one,
585:
498:
465:
436:
373:
344:
315:
1504:"Adjustment theory"
1454:2010JGeod..84..751N
1279:total least squares
1261:Gauss–Helmert model
1249:regression analysis
1150:Worked-out examples
569:One can proceed to
209:combined adjustment
1561:Books and chapters
1442:Journal of Geodesy
1253:Gauss–Markov model
1169:. You can help by
1108:
1079:
1031:
966:
865:
798:
739:
665:
618:
559:
490:produce a nonzero
480:
451:
422:
359:
330:
211:, both parameters
1784:978-3-642-27999-7
1747:978-960-89704-4-1
1713:978-3-540-72678-4
1698:. pp. 7–10.
1656:978-0-444-87777-2
1632:978-1-4613-5858-9
1377:978-3-319-11549-8
1344:978-0-471-66719-3
1222:Triangulateration
1209:Bundle adjustment
1187:
1186:
1144:iterative methods
1105:
1076:
1025:
1010:
995:
963:
945:
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909:
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795:
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581:: the first one,
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533:
510:
477:
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408:
393:
356:
340:and observations
327:
217:and observations
87:
86:
79:
16:(Redirected from
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1322:
1263:
1262:
1242:Related concepts
1231:GNSS positioning
1204:control networks
1182:
1179:
1161:
1154:
1140:QR factorization
1117:
1115:
1114:
1109:
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57:this article by
48:inline citations
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1547:Snow, Kyle B.,
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1397:
1393:
1378:
1357:
1356:
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1319:
1314:
1302:rank deficiency
1298:
1260:
1259:
1244:
1192:
1183:
1177:
1174:
1167:needs expansion
1152:
1138:, applying the
1128:
1091:
1090:
1062:
1061:
1046:
978:
977:
895:
894:
818:
817:
751:
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677:
676:
630:
629:
583:
582:
579:design matrices
523:
519:
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463:
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434:
433:
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342:
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53:Please help to
52:
36:
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12:
11:
5:
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1823:Photogrammetry
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1665:
1662:Gilbert Strang
1659:
1637:
1631:
1610:
1607:
1604:978-0804443975
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905:
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846:
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837:
831:
828:
808:are estimated
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207:Finally, in a
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116:—the field of
114:photogrammetry
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84:
39:
37:
30:
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2:
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1809:
1808:Least squares
1806:
1804:
1803:Curve fitting
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1800:
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1763:
1760:
1759:9781119501404
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1735:9781119018612
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1688:91-7170-200-8
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1648:0-444-87777-0
1645:
1641:
1640:Peter VanĂÄŤek
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1250:
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1245:
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1237:
1234:
1232:
1228:
1225:
1223:
1219:
1218:Trilateration
1215:
1214:Triangulation
1212:
1210:
1207:
1205:
1201:
1197:
1194:
1193:
1189:
1181:
1172:
1168:
1165:This section
1163:
1160:
1156:
1155:
1149:
1147:
1145:
1141:
1137:
1134:and applying
1133:
1132:normal matrix
1125:
1123:
1121:
1099:
1070:
1059:
1055:
1049:
1044:
1028:
1019:
1013:
1004:
998:
989:
983:
957:
951:
939:
930:
927:
918:
912:
903:
892:
888:
883:
881:
880:
875:are post-fit
856:
850:
841:
835:
826:
815:
811:
789:
783:
774:
768:
759:
736:
733:
730:
721:
715:
712:
703:
697:
694:
685:
662:
658:
650:
645:
638:
635:
615:
611:
603:
598:
591:
588:
580:
576:
572:
556:
552:
542:
536:
527:
520:
516:
513:
504:
493:
471:
442:
419:
416:
412:
402:
396:
387:
380:
376:
350:
321:
306:
304:
301:
295:
288:
284:
280:
276:
272:
266:
262:
258:
254:
250:
246:
235:
231:
227:
221:
215:
210:
206:
202:
196:
189:
185:
180:
176:
172:
166:
160:
156:
152:
147:
143:
142:
141:
139:
135:
131:
123:
121:
119:
115:
111:
107:
103:
99:
98:least squares
95:
91:
81:
78:
70:
60:
56:
50:
49:
43:
38:
29:
28:
19:
1766:
1695:
1614:
1577:
1573:
1524:Cross, P.A.
1489:Bibliography
1445:
1441:
1435:
1408:
1404:
1394:
1359:
1353:
1326:
1320:
1299:
1285:
1271:F.R. Helmert
1258:
1190:Applications
1175:
1171:adding to it
1166:
1129:
1120:a posteriori
1119:
1047:
890:
886:
884:
877:observation
876:
816:values, and
813:
809:
491:
310:
299:
293:
286:
282:
278:
274:
270:
264:
260:
256:
252:
248:
244:
241:
233:
229:
225:
219:
213:
208:
200:
194:
187:
183:
178:
170:
164:
158:
154:
150:
145:
137:
133:
129:
127:
89:
88:
73:
64:
45:
1513:Krakiwsky,
1284:The use of
1126:Computation
1058:constrained
134:conditional
124:Formulation
59:introducing
1797:Categories
1596:0804443971
1312:References
1296:Extensions
1267:C.F. Gauss
492:misclosure
130:parametric
42:references
1818:Surveying
1538:0260-9142
1478:123207786
1470:0949-7714
1427:0024-3795
1386:1866-6795
1178:June 2014
1103:^
1074:^
1023:~
1014:−
1008:^
993:^
961:~
952:−
943:~
922:~
907:~
879:residuals
860:~
851:−
845:^
830:^
793:~
784:−
778:^
763:^
725:^
707:^
689:~
655:∂
642:∂
608:∂
595:∂
575:Jacobians
546:~
531:~
508:~
475:~
446:~
406:^
391:^
354:^
325:^
118:geomatics
106:surveying
67:June 2014
1286:a priori
1200:traverse
1196:Leveling
814:a priori
307:Solution
138:combined
1813:Geodesy
1606:, 1971.
1450:Bibcode
812:to the
369:, thus
204:at all.
110:geodesy
55:improve
1781:
1757:
1745:
1733:
1710:
1686:
1654:
1646:
1629:
1602:
1594:
1555:, 2002
1536:
1521:, 1975
1510:, 2014
1476:
1468:
1425:
1384:
1374:
1341:
1202:, and
749:where
136:, and
112:, and
44:, but
1584:>.
1474:S2CID
236:) = 0
190:) = 0
1779:ISBN
1755:ISBN
1743:ISBN
1731:ISBN
1708:ISBN
1684:ISBN
1652:ISBN
1644:ISBN
1627:ISBN
1600:ISBN
1592:ISBN
1534:ISSN
1466:ISSN
1423:ISSN
1382:ISSN
1372:ISBN
1339:ISBN
1277:and
1269:and
1089:and
281:) =
268:and
263:) -
255:) =
157:) =
1771:doi
1700:doi
1619:doi
1458:doi
1413:doi
1409:432
1364:doi
1331:doi
1300:If
1227:GPS
1173:.
1050:= 0
577:or
177:In
144:In
100:of
1799::
1777:.
1706:.
1650:,
1625:.
1598:,
1572:.
1528:,
1472:.
1464:.
1456:.
1446:84
1444:.
1421:.
1407:.
1403:.
1380:.
1370:.
1337:.
1308:.
1220:,
1216:,
1198:,
889:=-
882:.
494::
303:.
277:,
232:,
140::
132:,
108:,
1787:.
1773::
1761:.
1716:.
1702::
1690:.
1635:.
1621::
1576:(
1480:.
1460::
1452::
1429:.
1415::
1388:.
1366::
1347:.
1333::
1281:.
1229:/
1180:)
1176:(
1100:Y
1071:X
1048:A
1029:,
1020:y
1005:y
999:=
990:x
984:A
958:Y
949:)
940:X
934:(
931:h
928:=
919:w
913:=
904:y
891:I
887:B
857:Y
842:Y
836:=
827:y
790:X
775:X
769:=
760:x
737:,
734:0
731:=
722:y
716:B
713:+
704:x
698:A
695:+
686:w
663:.
659:Y
651:/
646:f
639:=
636:B
616:;
612:X
604:/
599:f
592:=
589:A
557:.
553:)
543:Y
537:,
528:X
521:(
517:f
514:=
505:w
472:X
443:Y
420:0
417:=
413:)
403:Y
397:,
388:X
381:(
377:f
351:Y
322:X
300:L
294:Y
289:)
287:Y
285:(
283:g
279:Y
275:X
273:(
271:f
265:Y
261:X
259:(
257:h
253:Y
251:,
249:X
247:(
245:f
238:.
234:Y
230:X
228:(
226:f
220:Y
214:X
201:X
195:Y
188:Y
186:(
184:g
171:X
165:Y
159:Y
155:X
153:(
151:h
80:)
74:(
69:)
65:(
51:.
20:)
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