4768:
adding quantities measured in different units, which is meaningless. Secondly, if we rescale one of the variables e.g., measure in grams rather than kilograms, then we shall end up with different results (a different line). To avoid these problems it is sometimes suggested that we convert to dimensionless variables—this may be called normalization or standardization. However, there are various ways of doing this, and these lead to fitted models which are not equivalent to each other. One approach is to normalize by known (or estimated) measurement precision thereby minimizing the
6233:
405:
4787:. For a meaningful model we require this property to hold. A way forward is to realise that residuals (distances) measured in different units can be combined if multiplication is used instead of addition. Consider fitting a line: for each data point the product of the vertical and horizontal residuals equals twice the area of the triangle formed by the residual lines and the fitted line. We choose the line which minimizes the sum of these areas. Nobel laureate
2985:
449:
2621:
3744:
1671:
2980:{\displaystyle ={\begin{bmatrix}\Sigma _{X}&0\\0&\Sigma _{Y}\end{bmatrix}}{\begin{bmatrix}V_{XX}&V_{XY}\\V_{YX}&V_{YY}\end{bmatrix}}^{*}={\begin{bmatrix}\Sigma _{X}&0\\0&\Sigma _{Y}\end{bmatrix}}{\begin{bmatrix}V_{XX}^{*}&V_{YX}^{*}\\V_{XY}^{*}&V_{YY}^{*}\end{bmatrix}}}
5253:, Documented Fortran 77 programs of the extended classical total least squares algorithm, the partial singular value decomposition algorithm and the partial total least squares algorithm, Internal Report ESAT-KUL 88/1, ESAT Lab., Dept. of Electrical Engineering, Katholieke Universiteit Leuven, 1988.
4746:
When the independent variable is error-free a residual represents the "vertical" distance between the observed data point and the fitted curve (or surface). In total least squares a residual represents the distance between a data point and the fitted curve measured along some direction. In fact, if
3504:
4791:
proved in 1942 that, in two dimensions, it is the only line expressible solely in terms of the ratios of standard deviations and the correlation coefficient which (1) fits the correct equation when the observations fall on a straight line, (2) exhibits scale invariance, and (3) exhibits invariance
3293:
1278:
4767:
A serious difficulty arises if the variables are not measured in the same units. First consider measuring distance between a data point and the line: what are the measurement units for this distance? If we consider measuring distance based on
Pythagoras' Theorem then it is clear that we shall be
4204:
4955:
3519:
4823:
Tofallis (2015, 2023) has extended this approach to deal with multiple variables. The calculations are simpler than for total least squares as they only require knowledge of covariances, and can be computed using standard spreadsheet functions.
1996:
4695:
1380:
1477:
3304:
3075:
1149:
1009:
1465:
2017:
As was shown in 1980 by Golub and Van Loan, the TLS problem does not have a solution in general. The following considers the simple case where a unique solution exists without making any particular assumptions.
1848:
2228:
815:
1137:
3864:
4982:
5688:
2412:
667:
3976:
5031:
1754:
1857:
th point is determined by the variances of both independent and dependent variables and by the model being used to fit the data. The expression may be generalized by noting that the parameter
2312:, the square root of the sum of the squares of all entries in a matrix and so equivalently the square root of the sum of squares of the lengths of the rows or columns of the matrix.
5057:
5004:
622:
4891:
4273:
581:
5274:
M. Plešinger, The Total Least
Squares Problem and Reduction of Data in AX ≈ B. Doctoral Thesis, TU of Liberec and Institute of Computer Science, AS CR Prague, 2008. Ph.D. Thesis
3739:{\displaystyle =-U_{Y}\Sigma _{Y}{\begin{bmatrix}V_{XY}\\V_{YY}\end{bmatrix}}^{*}=-{\begin{bmatrix}V_{XY}\\V_{YY}\end{bmatrix}}{\begin{bmatrix}V_{XY}\\V_{YY}\end{bmatrix}}^{*}.}
1070:
1041:
889:
860:
746:
5681:
2306:
752:. The independent variables are assumed to be error-free. The parameter estimates are found by setting the gradient equations to zero, which results in the normal equations
4720:
3968:
2055:
2468:
4508:
3927:
3897:
1883:
1875:
2442:
1080:
respectively. Clearly these residuals cannot be independent of each other, but they must be constrained by some kind of relationship. Writing the model function as
2583:
5750:
5674:
1666:{\displaystyle \mathbf {M=K_{x}M_{x}K_{x}^{T}+K_{y}M_{y}K_{y}^{T};\ K_{x}=-{\frac {\partial f}{\partial r_{x}}},\ K_{y}=-{\frac {\partial f}{\partial r_{y}}}} .}
4588:
4564:
4544:
3067:
3047:
3027:
5075:
is used here to reflect the notation used in the earlier part of the article. In the computational literature the problem has been more commonly presented as
4612:
2613:
2528:
2498:
2261:
1297:
3499:{\displaystyle =-{\begin{bmatrix}0_{n\times n}&0\\0&\Sigma _{Y}\end{bmatrix}}{\begin{bmatrix}V_{XX}&V_{XY}\\V_{YX}&V_{YY}\end{bmatrix}}^{*}.}
3288:{\displaystyle ={\begin{bmatrix}\Sigma _{X}&0\\0&0_{k\times k}\end{bmatrix}}{\begin{bmatrix}V_{XX}&V_{XY}\\V_{YX}&V_{YY}\end{bmatrix}}^{*}}
1273:{\displaystyle \mathbf {F=\Delta y-{\frac {\partial f}{\partial r_{x}}}r_{x}-{\frac {\partial f}{\partial r_{y}}}r_{y}-X\Delta {\boldsymbol {\beta }}=0} .}
5287:, The total least squares problem in AX ≈ B. A new classification with the relationship to the classical works. SIMAX vol. 32 issue 3 (2011), pp. 748–770.
5759:
897:
5764:
492:
data modeling technique in which observational errors on both dependent and independent variables are taken into account. It is a generalization of
435:
6197:
345:
1388:
2002:
4526:, see also. All modern implementations based, for example, on solving a sequence of ordinary least squares problems, approximate the matrix
6092:
5732:
335:
4751:, that is, the residual vector is perpendicular to the tangent of the curve. For this reason, this type of regression is sometimes called
1765:
2117:
758:
6072:
5722:
5477:
5223:
5200:
1083:
599:
591:
3763:
5992:
5156:
4960:
5798:
299:
2321:
4748:
4723:
4199:{\displaystyle {\begin{bmatrix}-V_{XY}V_{YY}^{-1}\\-V_{YY}V_{YY}^{-1}\end{bmatrix}}={\begin{bmatrix}B\\-I_{k}\end{bmatrix}}=0,}
350:
288:
108:
83:
4747:
both variables are measured in the same units and the errors on both variables are the same, then the residual represents the
6034:
4869:
630:
485:
210:
5398:
Warton, David I.; Wright, Ian J.; Falster, Daniel S.; Westoby, Mark (2006). "Bivariate line-fitting methods for allometry".
169:
5009:
1695:
6268:
6220:
6120:
6110:
6029:
5974:
4864:
2022:
428:
4792:
under interchange of variables. This solution has been rediscovered in different disciplines and is variously known as
6247:
6062:
371:
5547:
5178:
G. H. Golub and C. F. Van Loan, An analysis of the total least squares problem. Numer. Anal., 17, 1980, pp. 883–893.
6283:
5742:
717:
340:
309:
236:
6087:
5914:
5878:
5847:
4844:
330:
319:
283:
190:
6067:
5622:, in Proc. IEEE Int. Conf. Acoust., Speech, Signal Process. (ICASSP’87), Apr. 1987, vol. 12, pp. 1485–1488.
3006:
391:
5945:
5909:
5837:
5727:
5709:
4603:
262:
185:
78:
57:
5558:
5464:
Tofallis, Chris (2002). "Model
Fitting for Multiple Variables by Minimising the Geometric Mean Deviation". In
4950:{\displaystyle \mathbf {X^{T}WX{\boldsymbol {\Delta }}{\boldsymbol {\beta }}=X^{T}W{\boldsymbol {\Delta }}y} }
5544:
The total least squares problem in AX ≈ B. A new classification with the relationship to the classical works.
5808:
5040:
4987:
605:
421:
314:
4215:
542:
6278:
6273:
6161:
5987:
5852:
5842:
5793:
5407:
508:
278:
273:
215:
6242:
6202:
6166:
6151:
6102:
6046:
5873:
5309:
4741:
497:
366:
62:
6232:
404:
5265:, The extended classical total least squares algorithm, J. Comput. Appl. Math., 25, pp. 111–119, 1989.
1046:
1017:
865:
836:
722:
6207:
6146:
6133:
6082:
5982:
5904:
5883:
5857:
5146:
4777:
4769:
1288:
386:
376:
257:
225:
180:
159:
67:
5412:
2278:
452:
The bivariate (Deming regression) case of total least squares. The red lines show the error in both
6225:
6156:
6041:
6008:
5960:
5950:
5929:
5924:
5785:
5770:
5701:
5192:
4849:
4833:
460:. This is different from the traditional least squares method which measures error parallel to the
205:
200:
154:
103:
93:
38:
4783:
In short, total least squares does not have the property of units-invariance—i.e. it is not
4703:
3932:
602:
the model contains equations which are linear in the parameters appearing in the parameter vector
6237:
6141:
6130:
5955:
5433:
5353:
4773:
2031:
529:
477:
409:
138:
123:
503:
The total least squares approximation of the data is generically equivalent to the best, in the
5501:
Tofallis, Chris (2015). "Fitting
Equations to Data with the Perfect Correlation Relationship".
2447:
1991:{\displaystyle M_{ii}=\sigma _{y,i}^{2}+\left({\frac {dy}{dx}}\right)_{i}^{2}\sigma _{x,i}^{2}}
6192:
5919:
5829:
5820:
5601:
5577:
5539:
5506:
5502:
5487:
5483:
5473:
5465:
5425:
5284:
5262:
5250:
5234:
5219:
5152:
5116:
4854:
4838:
4567:
493:
195:
98:
52:
5582:
Total Least
Squares and Errors-in-Variables Modeling: Analysis, Algorithms and Applications
17:
5888:
5755:
5641:
5609:
5568:
5417:
5380:
5345:
4784:
4483:
3902:
3872:
2264:
1860:
220:
149:
5661:
5470:
Total Least
Squares and Errors-in-Variables Modeling: Analysis, Algorithms and Applications
4606:
similar reasoning shows that the normal equations for an iteration cycle can be written as
2420:
6171:
6077:
6013:
5322:
2537:
381:
88:
6018:
5519:
Tofallis, C. (2023). Fitting an
Equation to Data Impartially. Mathematics, 11(18), 3957.
4690:{\displaystyle \mathbf {J^{T}M^{-1}J\Delta {\boldsymbol {\beta }}=J^{T}M^{-1}\Delta y} ,}
1375:{\displaystyle \mathbf {X^{T}M^{-1}X\Delta {\boldsymbol {\beta }}=X^{T}M^{-1}\Delta y} ,}
5666:
5124:
1471:
is the variance-covariance matrix relative to both independent and dependent variables.
6115:
4788:
4573:
4549:
4529:
3052:
3032:
3012:
2309:
504:
133:
5557:
Doctoral Thesis, TU of
Liberec and Institute of Computer Science, AS CR Prague, 2008.
2588:
2503:
2473:
2236:
708:
matrix whose elements are either constants or functions of the independent variables,
464:
axis. The case shown, with deviations measured perpendicularly, arises when errors in
6262:
6187:
5717:
5697:
5188:
4859:
4735:
525:
489:
252:
128:
5437:
448:
118:
3929:
is singular is not well understood yet), we can then right multiply both sides by
3899:
is nonsingular, which is not always the case (note that the behavior of TLS when
5775:
5629:, in Nieuw Archief voor Wiskunde, Vierde serie, deel 14, 1996, pp. 237–253
164:
113:
4480:
The way described above of solving the problem, which requires that the matrix
4400:% Take the block of V consisting of the first n rows and the n+1 to last column
3970:
to bring the bottom block of the right matrix to the negative identity, giving
1004:{\displaystyle S=\mathbf {r_{x}^{T}M_{x}^{-1}r_{x}+r_{y}^{T}M_{y}^{-1}r_{y}} ,}
5421:
4279:
5650:
4523:
5613:
4522:
The standard implementation of classical TLS algorithm is available through
5591:
IEEE Trans. Signal
Process., vol. 53, no. 6, pp. 2112–2123, Jun. 2005.
5429:
3009:, the approximation minimising the norm of the error is such that matrices
1460:{\displaystyle \mathbf {X^{T}M^{-1}X{\boldsymbol {\beta }}=X^{T}M^{-1}y} ,}
5371:
Ricker, W. E. (1975). "A note concerning
Professor Jolicoeur's Comments".
5630:
5598:
IEEE Trans. Signal Process., vol. 41, no. 1, pp. 407–411, Jan. 1993.
5589:
Consistent normalized least mean square filtering with noisy data matrix.
5298:
5658:
Weighted total least squares formulated by standard least squares theory
5523:
5357:
1283:
Thus, the problem is to minimize the objective function subject to the
5572:
5606:
The Total Least Squares Problems: Computational Aspects and Analysis.
5645:
5520:
5384:
5349:
5239:
The Total Least Squares Problems: Computational Aspects and Analysis
1843:{\displaystyle M_{ii}=\sigma _{y,i}^{2}+\beta ^{2}\sigma _{x,i}^{2}}
891:
respectively. In this case the objective function can be written as
2223:{\displaystyle \mathrm {argmin} _{B,E,F}\|\|_{F},\qquad (X+E)B=Y+F}
4736:
Curve fitting § Algebraic fit versus geometric fit for curves
447:
2025:(SVD) is described in standard texts. We can solve the equation
833:
are observed subject to error, with variance-covariance matrices
810:{\displaystyle \mathbf {X^{T}WX{\boldsymbol {\beta }}=X^{T}Wy} .}
5555:
The Total Least Squares Problem and Reduction of Data in AX ≈ B.
5546:
SIMAX vol. 32 issue 3 (2011), pp. 748–770. Available as a
5336:
Samuelson, Paul A. (1942). "A Note on Alternative Regressions".
1132:{\displaystyle \mathbf {f(r_{x},r_{y},{\boldsymbol {\beta }})} }
5670:
5584:. Dordrecht, The Netherlands: Kluwer Academic Publishers, 2002.
3859:{\displaystyle {\begin{bmatrix}V_{XY}\\V_{YY}\end{bmatrix}}=0.}
2585:
to be the singular value decomposition of the augmented matrix
1689:
are diagonal. Then, take the example of straight line fitting.
1291:. After some algebraic manipulations, the result is obtained.
4977:{\displaystyle {\boldsymbol {\Delta }}{\boldsymbol {\beta }}}
4749:
shortest distance between the data point and the fitted curve
4841:, a special case with two predictors and independent errors.
3069:
singular values are replaced with zeroes. That is, we want
500:, and can be applied to both linear and non-linear models.
4510:
is nonsingular, can be slightly extended by the so-called
2407:{\displaystyle {\begin{bmatrix}B\\-I_{k}\end{bmatrix}}=0.}
2009:
translates to a large error on y when the slope is large.
5538:
I. Hnětynková, M. Plešinger, D. M. Sima, Z. Strakoš, and
5283:
I. Hnětynková, M. Plešinger, D. M. Sima, Z. Strakoš, and
2994:
is partitioned into blocks corresponding to the shape of
5596:
The data least squares problem and channel equalization.
5135:
W.E. Deming, Statistical Adjustment of Data, Wiley, 1943
5660:, in Journal of Geodetic Science, 2 (2): 113–124, 2012
5567:
SIAM J. Matrix Anal. Appl. 27, 2006, pp. 861–875.
662:{\displaystyle \mathbf {r=y-X{\boldsymbol {\beta }}} .}
4984:
is the parameter shift from some starting estimate of
4156:
4022:
3809:
3686:
3639:
3568:
3416:
3359:
3208:
3151:
2886:
2836:
2724:
2673:
2367:
5640:
SIAM J. on Numer. Anal., 17, 1980, pp. 883–893.
5043:
5037:
and the value calculated using the starting value of
5012:
4990:
4963:
4894:
4706:
4615:
4576:
4552:
4532:
4486:
4218:
3979:
3935:
3905:
3875:
3766:
3522:
3307:
3078:
3055:
3035:
3015:
2624:
2591:
2540:
2506:
2476:
2450:
2423:
2324:
2281:
2239:
2120:
2034:
1886:
1863:
1768:
1698:
1480:
1391:
1300:
1152:
1086:
1049:
1020:
900:
868:
839:
761:
725:
633:
608:
545:
4776:
solution; the unknown precisions could be found via
1681:
When the data errors are uncorrelated, all matrices
6180:
6129:
6101:
6055:
6001:
5973:
5938:
5897:
5866:
5828:
5817:
5784:
5741:
5708:
5026:{\displaystyle {\boldsymbol {\Delta }}\mathbf {y} }
1749:{\displaystyle f(x_{i},\beta )=\alpha +\beta x_{i}}
5218:, Society for Industrial and Applied Mathematics.
5051:
5025:
4998:
4976:
4949:
4714:
4689:
4582:
4558:
4538:
4502:
4267:
4198:
3962:
3921:
3891:
3858:
3738:
3498:
3287:
3061:
3041:
3021:
2979:
2607:
2577:
2522:
2492:
2462:
2436:
2406:
2300:
2255:
2222:
2049:
1990:
1869:
1842:
1748:
1665:
1459:
1374:
1272:
1131:
1064:
1035:
1003:
883:
854:
809:
740:
661:
616:
575:
5373:Journal of the Fisheries Research Board of Canada
5123:Signal Processing, vol. 87, pp. 2283–2302, 2007.
1125:
1091:
5638:An analysis of the total least squares problem.
4570:and Vandewalle. It is worth noting, that this
2001:An expression of this type is used in fitting
5682:
429:
8:
5216:Numerical Methods for Least Squares Problems
2289:
2282:
2177:
2160:
821:Allowing observation errors in all variables
5825:
5689:
5675:
5667:
5608:SIAM Publications, Philadelphia PA, 1991.
5565:Core problems in linear algebraic systems.
5095:matrix of unknown regression coefficients.
4132:
3998:
3785:
3627:
3529:
3340:
3314:
3132:
3097:
2817:
2654:
2631:
2598:
2513:
2483:
2470:identity matrix. The goal is then to find
2343:
2246:
2169:
436:
422:
29:
5411:
5044:
5042:
5018:
5013:
5011:
4991:
4989:
4969:
4964:
4962:
4938:
4929:
4917:
4912:
4900:
4895:
4893:
4772:from the points to the line, providing a
4707:
4705:
4668:
4658:
4646:
4631:
4621:
4616:
4614:
4575:
4551:
4531:
4491:
4485:
4253:
4245:
4232:
4217:
4173:
4151:
4094:
4086:
4073:
4053:
4045:
4032:
4017:
3978:
3951:
3943:
3934:
3910:
3904:
3880:
3874:
3833:
3816:
3804:
3765:
3727:
3710:
3693:
3681:
3663:
3646:
3634:
3609:
3592:
3575:
3563:
3556:
3546:
3521:
3487:
3470:
3455:
3438:
3423:
3411:
3396:
3366:
3354:
3345:
3334:
3306:
3279:
3262:
3247:
3230:
3215:
3203:
3182:
3158:
3146:
3137:
3126:
3077:
3054:
3034:
3014:
2963:
2955:
2943:
2935:
2921:
2913:
2901:
2893:
2881:
2867:
2843:
2831:
2822:
2811:
2795:
2778:
2763:
2746:
2731:
2719:
2704:
2680:
2668:
2659:
2648:
2623:
2590:
2569:
2539:
2505:
2475:
2449:
2428:
2422:
2384:
2362:
2323:
2292:
2280:
2238:
2180:
2142:
2122:
2119:
2033:
1982:
1971:
1961:
1956:
1932:
1918:
1907:
1891:
1885:
1862:
1834:
1823:
1813:
1800:
1789:
1773:
1767:
1740:
1709:
1697:
1650:
1632:
1620:
1601:
1583:
1571:
1555:
1550:
1540:
1530:
1517:
1512:
1502:
1492:
1481:
1479:
1441:
1431:
1419:
1407:
1397:
1392:
1390:
1353:
1343:
1331:
1316:
1306:
1301:
1299:
1255:
1240:
1227:
1209:
1200:
1187:
1169:
1153:
1151:
1120:
1111:
1098:
1087:
1085:
1056:
1051:
1048:
1027:
1022:
1019:
991:
978:
973:
963:
958:
945:
932:
927:
917:
912:
907:
899:
875:
870:
867:
846:
841:
838:
791:
779:
767:
762:
760:
732:
727:
724:
650:
634:
632:
609:
607:
557:
552:
544:
5121:Overview of total least squares methods.
4796:(Ricker 1975, Warton et al., 2006), the
1287:constraints. It is solved by the use of
6198:Numerical smoothing and differentiation
5108:
5045:
4992:
4970:
4918:
4881:
4647:
1420:
1332:
1256:
1121:
780:
651:
610:
358:
244:
44:
37:
5627:An introduction to total least squares
5318:
5307:
5299:"Two Dimensional Euclidean Regression"
5052:{\displaystyle {\boldsymbol {\beta }}}
4999:{\displaystyle {\boldsymbol {\beta }}}
4802:geometric mean functional relationship
617:{\displaystyle {\boldsymbol {\beta }}}
5241:. SIAM Publications, Philadelphia PA.
5148:Data Fitting in the Chemical Sciences
4566:in the literature), as introduced by
4268:{\displaystyle B=-V_{XY}V_{YY}^{-1}.}
576:{\displaystyle S=\mathbf {r^{T}Wr} ,}
7:
5733:Iteratively reweighted least squares
5656:A. R. Amiri-Simkooei and S. Jazaeri
5524:https://doi.org/10.3390/math11183957
4753:two dimensional Euclidean regression
5472:. Dordrecht: Kluwer Academic Publ.
4451:% Take the bottom-right block of V.
4319:% n is the width of X (X is m by n)
3509:We can then remove blocks from the
1613:
1564:
1139:, the constraints are expressed by
5751:Pearson product-moment correlation
5651:Perpendicular Regression Of A Line
5201:The Johns Hopkins University Press
3553:
3393:
3155:
3049:are unchanged, while the smallest
2864:
2840:
2701:
2677:
2553:
2138:
2135:
2132:
2129:
2126:
2123:
1643:
1635:
1594:
1586:
1220:
1212:
1180:
1172:
25:
5594:R. D. DeGroat and E. M. Dowling,
5521:https://ssrn.com/abstract=4556739
2021:The computation of the TLS using
6231:
5636:G. H. Golub and C. F. Van Loan,
5019:
5014:
4965:
4943:
4939:
4935:
4930:
4926:
4922:
4913:
4909:
4906:
4901:
4897:
4708:
4680:
4677:
4672:
4669:
4665:
4659:
4655:
4651:
4643:
4640:
4635:
4632:
4628:
4622:
4618:
1853:showing how the variance at the
1651:
1647:
1638:
1629:
1626:
1621:
1617:
1610:
1602:
1598:
1589:
1580:
1577:
1572:
1568:
1561:
1556:
1551:
1547:
1541:
1537:
1531:
1527:
1523:
1518:
1513:
1509:
1503:
1499:
1493:
1489:
1485:
1482:
1450:
1445:
1442:
1438:
1432:
1428:
1424:
1416:
1411:
1408:
1404:
1398:
1394:
1365:
1362:
1357:
1354:
1350:
1344:
1340:
1336:
1328:
1325:
1320:
1317:
1313:
1307:
1303:
1263:
1260:
1252:
1249:
1246:
1241:
1237:
1228:
1224:
1215:
1206:
1201:
1197:
1188:
1184:
1175:
1166:
1163:
1160:
1157:
1154:
1117:
1112:
1108:
1104:
1099:
1095:
1088:
1065:{\displaystyle \mathbf {r} _{y}}
1052:
1036:{\displaystyle \mathbf {r} _{x}}
1023:
992:
988:
982:
979:
974:
970:
964:
959:
955:
951:
946:
942:
936:
933:
928:
924:
918:
913:
909:
884:{\displaystyle \mathbf {M} _{y}}
871:
855:{\displaystyle \mathbf {M} _{x}}
842:
800:
797:
792:
788:
784:
776:
773:
768:
764:
741:{\displaystyle \mathbf {M} _{y}}
728:
716:is, ideally, the inverse of the
647:
644:
641:
638:
635:
624:, so the residuals are given by
566:
563:
558:
554:
403:
5620:Constrained total least squares
3513:and Σ matrices, simplifying to
2189:
351:Least-squares spectral analysis
289:Generalized estimating equation
109:Multinomial logistic regression
84:Vector generalized linear model
5454:, 3rd edition, pp. 92–96. 1998
4870:Principal component regression
4148:
4145:
4133:
4129:
4117:
4114:
4014:
4011:
3999:
3995:
3983:
3980:
3801:
3798:
3786:
3782:
3770:
3767:
3631:
3621:
3533:
3523:
3351:
3327:
3318:
3308:
3143:
3119:
3113:
3110:
3098:
3094:
3082:
3079:
2828:
2804:
2665:
2641:
2635:
2625:
2602:
2592:
2566:
2559:
2556:
2550:
2547:
2541:
2517:
2507:
2487:
2477:
2359:
2356:
2344:
2340:
2328:
2325:
2301:{\displaystyle \|\cdot \|_{F}}
2250:
2240:
2202:
2190:
2173:
2163:
2095:that minimizes error matrices
1721:
1702:
486:errors-in-variables regression
1:
5618:T. Abatzoglou and J. Mendel,
528:method of data modeling, the
170:Nonlinear mixed-effects model
18:Reduced major axis regression
6221:Regression analysis category
6111:Response surface methodology
4865:Principal component analysis
4715:{\displaystyle \mathbf {J} }
3963:{\displaystyle -V_{YY}^{-1}}
2023:singular value decomposition
6093:Frisch–Waugh–Lovell theorem
6063:Mean and predicted response
5452:Applied Regression Analysis
4814:line of organic correlation
4282:implementation of this is:
2050:{\displaystyle XB\approx Y}
372:Mean and predicted response
6300:
5743:Correlation and dependence
5033:is the difference between
4804:(Draper and Smith, 1998),
4739:
4733:
4730:Geometrical interpretation
4331:% Z is X augmented with Y.
1877:is the slope of the line.
718:variance-covariance matrix
598:is a weighting matrix. In
165:Linear mixed-effects model
6216:
6088:Minimum mean-square error
5975:Decomposition of variance
5879:Growth curve (statistics)
5848:Generalized least squares
5563:C. C. Paige, Z. Strakoš,
5468:; Lemmerling, P. (eds.).
5450:Draper, NR and Smith, H.
5422:10.1017/S1464793106007007
5237:and J. Vandewalle (1991)
4845:Errors-in-variables model
4806:least products regression
2500:that reduces the rank of
2463:{\displaystyle k\times k}
2315:This can be rewritten as
2091:That is, we seek to find
331:Least absolute deviations
5946:Generalized linear model
5838:Simple linear regression
5728:Non-linear least squares
5710:Computational statistics
4284:
79:Generalized linear model
5614:10.1137/1.9781611971002
5083:, i.e. with the letter
4888:An alternative form is
4794:standardised major axis
4763:Scale invariant methods
4512:classical TLS algorithm
2111:respectively. That is,
2013:Algebraic point of view
2005:where a small error on
825:Now, suppose that both
6238:Mathematics portal
6162:Orthogonal polynomials
5988:Analysis of covariance
5853:Weighted least squares
5843:Ordinary least squares
5794:Ordinary least squares
5317:Cite journal requires
5053:
5027:
5000:
4978:
4951:
4716:
4691:
4584:
4560:
4540:
4504:
4503:{\displaystyle V_{YY}}
4269:
4200:
3964:
3923:
3922:{\displaystyle V_{YY}}
3893:
3892:{\displaystyle V_{YY}}
3860:
3740:
3500:
3289:
3063:
3043:
3023:
2981:
2609:
2579:
2524:
2494:
2464:
2438:
2408:
2302:
2257:
2224:
2051:
1992:
1871:
1870:{\displaystyle \beta }
1844:
1750:
1667:
1461:
1376:
1274:
1133:
1066:
1037:
1005:
885:
856:
811:
742:
663:
618:
577:
509:low-rank approximation
473:
410:Mathematics portal
336:Iteratively reweighted
6203:System identification
6167:Chebyshev polynomials
6152:Numerical integration
6103:Design of experiments
6047:Regression validation
5874:Polynomial regression
5799:Partial least squares
5587:S. Jo and S. W. Kim,
5054:
5028:
5001:
4979:
4952:
4757:orthogonal regression
4742:Orthogonal regression
4740:Further information:
4717:
4692:
4585:
4561:
4541:
4505:
4270:
4201:
3965:
3924:
3894:
3861:
3741:
3501:
3290:
3064:
3044:
3024:
2982:
2610:
2580:
2525:
2495:
2465:
2439:
2437:{\displaystyle I_{k}}
2409:
2303:
2258:
2225:
2052:
1993:
1872:
1845:
1751:
1668:
1462:
1377:
1275:
1143:condition equations.
1134:
1072:are the residuals in
1067:
1038:
1006:
886:
857:
812:
743:
664:
619:
578:
498:orthogonal regression
472:have equal variances.
451:
367:Regression validation
346:Bayesian multivariate
63:Polynomial regression
27:Statistical technique
6208:Moving least squares
6147:Approximation theory
6083:Studentized residual
6073:Errors and residuals
6068:Gauss–Markov theorem
5983:Analysis of variance
5905:Nonlinear regression
5884:Segmented regression
5858:General linear model
5776:Confounding variable
5723:Linear least squares
5193:Van Loan, Charles F.
5145:Gans, Peter (1992).
5041:
5010:
4988:
4961:
4892:
4778:analysis of variance
4770:Mahalanobis distance
4704:
4613:
4592:not the TLS solution
4574:
4550:
4530:
4484:
4355:% find the SVD of Z.
4216:
3977:
3933:
3903:
3873:
3764:
3520:
3305:
3076:
3053:
3033:
3013:
3007:Eckart–Young theorem
2622:
2589:
2578:{\displaystyle ^{*}}
2538:
2504:
2474:
2448:
2421:
2322:
2279:
2237:
2118:
2032:
1884:
1861:
1766:
1696:
1478:
1389:
1298:
1289:Lagrange multipliers
1150:
1084:
1047:
1018:
898:
866:
837:
759:
748:of the observations
723:
712:. The weight matrix
631:
606:
600:linear least squares
586:is minimized, where
543:
511:of the data matrix.
392:Gauss–Markov theorem
387:Studentized residual
377:Errors and residuals
211:Principal components
181:Nonlinear regression
68:General linear model
6269:Applied mathematics
6226:Statistics category
6157:Gaussian quadrature
6042:Model specification
6009:Stepwise regression
5867:Predictor structure
5804:Total least squares
5786:Regression analysis
5771:Partial correlation
5702:regression analysis
5604:and J. Vandewalle,
5580:and P. Lemmerling,
5214:Bjõrck, Ake (1996)
5197:Matrix Computations
4850:Gauss-Helmert model
4834:Regression dilution
4820:(Tofallis, 2002).
4810:diagonal regression
4261:
4102:
4061:
3959:
2968:
2948:
2926:
2906:
1987:
1966:
1923:
1839:
1805:
1560:
1522:
986:
968:
940:
922:
482:total least squares
237:Errors-in-variables
104:Logistic regression
94:Binomial regression
39:Regression analysis
33:Part of a series on
6243:Statistics outline
6142:Numerical analysis
5466:Van Huffel, Sabine
5400:Biological Reviews
5049:
5023:
4996:
4974:
4947:
4798:reduced major axis
4774:maximum-likelihood
4712:
4687:
4604:non-linear systems
4580:
4556:
4536:
4500:
4265:
4241:
4196:
4181:
4105:
4082:
4041:
3960:
3939:
3919:
3889:
3856:
3844:
3736:
3721:
3674:
3603:
3496:
3481:
3404:
3285:
3273:
3196:
3059:
3039:
3019:
2977:
2971:
2951:
2931:
2909:
2889:
2875:
2789:
2712:
2605:
2575:
2520:
2490:
2460:
2434:
2404:
2392:
2298:
2253:
2220:
2047:
1988:
1967:
1927:
1903:
1867:
1840:
1819:
1785:
1746:
1663:
1546:
1508:
1457:
1372:
1270:
1129:
1062:
1033:
1001:
969:
954:
923:
908:
881:
852:
807:
738:
659:
614:
573:
530:objective function
478:applied statistics
474:
124:Multinomial probit
6284:Regression models
6256:
6255:
6248:Statistics topics
6193:Calibration curve
6002:Model exploration
5969:
5968:
5939:Non-normal errors
5830:Linear regression
5821:statistical model
5573:10.1137/040616991
5297:Stein, Yaakov J.
5115:I. Markovsky and
4855:Linear regression
4839:Deming regression
4755:(Stein, 1983) or
4583:{\displaystyle B}
4559:{\displaystyle X}
4539:{\displaystyle B}
3298:so by linearity,
3062:{\displaystyle k}
3042:{\displaystyle V}
3022:{\displaystyle U}
2275:side by side and
2003:pH titration data
1950:
1657:
1615:
1608:
1566:
1385:or alternatively
1234:
1194:
590:is the vector of
494:Deming regression
446:
445:
99:Binary regression
58:Simple regression
53:Linear regression
16:(Redirected from
6291:
6236:
6235:
5993:Multivariate AOV
5889:Local regression
5826:
5818:Regression as a
5809:Ridge regression
5756:Rank correlation
5691:
5684:
5677:
5668:
5526:
5517:
5511:
5510:
5498:
5492:
5491:
5461:
5455:
5448:
5442:
5441:
5415:
5395:
5389:
5388:
5379:(8): 1494–1498.
5368:
5362:
5361:
5333:
5327:
5326:
5320:
5315:
5313:
5305:
5303:
5294:
5288:
5281:
5275:
5272:
5266:
5260:
5254:
5248:
5242:
5232:
5226:
5212:
5206:
5204:
5199:(3rd ed.).
5185:
5179:
5176:
5170:
5169:
5167:
5165:
5142:
5136:
5133:
5127:
5113:
5096:
5065:
5059:
5058:
5056:
5055:
5050:
5048:
5032:
5030:
5029:
5024:
5022:
5017:
5005:
5003:
5002:
4997:
4995:
4983:
4981:
4980:
4975:
4973:
4968:
4956:
4954:
4953:
4948:
4946:
4942:
4934:
4933:
4921:
4916:
4905:
4904:
4886:
4818:least areas line
4721:
4719:
4718:
4713:
4711:
4696:
4694:
4693:
4688:
4683:
4676:
4675:
4663:
4662:
4650:
4639:
4638:
4626:
4625:
4598:Non-linear model
4589:
4587:
4586:
4581:
4565:
4563:
4562:
4557:
4545:
4543:
4542:
4537:
4509:
4507:
4506:
4501:
4499:
4498:
4476:
4473:
4470:
4467:
4464:
4461:
4458:
4455:
4452:
4449:
4446:
4443:
4440:
4437:
4434:
4431:
4428:
4425:
4422:
4419:
4416:
4413:
4410:
4407:
4404:
4401:
4398:
4395:
4392:
4389:
4386:
4383:
4380:
4377:
4374:
4371:
4368:
4365:
4362:
4359:
4356:
4353:
4350:
4347:
4344:
4341:
4338:
4335:
4332:
4329:
4326:
4323:
4320:
4317:
4314:
4311:
4308:
4305:
4302:
4298:
4295:
4292:
4288:
4274:
4272:
4271:
4266:
4260:
4252:
4240:
4239:
4205:
4203:
4202:
4197:
4186:
4185:
4178:
4177:
4110:
4109:
4101:
4093:
4081:
4080:
4060:
4052:
4040:
4039:
3969:
3967:
3966:
3961:
3958:
3950:
3928:
3926:
3925:
3920:
3918:
3917:
3898:
3896:
3895:
3890:
3888:
3887:
3865:
3863:
3862:
3857:
3849:
3848:
3841:
3840:
3824:
3823:
3745:
3743:
3742:
3737:
3732:
3731:
3726:
3725:
3718:
3717:
3701:
3700:
3679:
3678:
3671:
3670:
3654:
3653:
3614:
3613:
3608:
3607:
3600:
3599:
3583:
3582:
3561:
3560:
3551:
3550:
3505:
3503:
3502:
3497:
3492:
3491:
3486:
3485:
3478:
3477:
3463:
3462:
3446:
3445:
3431:
3430:
3409:
3408:
3401:
3400:
3377:
3376:
3350:
3349:
3339:
3338:
3294:
3292:
3291:
3286:
3284:
3283:
3278:
3277:
3270:
3269:
3255:
3254:
3238:
3237:
3223:
3222:
3201:
3200:
3193:
3192:
3163:
3162:
3142:
3141:
3131:
3130:
3068:
3066:
3065:
3060:
3048:
3046:
3045:
3040:
3028:
3026:
3025:
3020:
2986:
2984:
2983:
2978:
2976:
2975:
2967:
2962:
2947:
2942:
2925:
2920:
2905:
2900:
2880:
2879:
2872:
2871:
2848:
2847:
2827:
2826:
2816:
2815:
2800:
2799:
2794:
2793:
2786:
2785:
2771:
2770:
2754:
2753:
2739:
2738:
2717:
2716:
2709:
2708:
2685:
2684:
2664:
2663:
2653:
2652:
2614:
2612:
2611:
2608:{\displaystyle }
2606:
2584:
2582:
2581:
2576:
2574:
2573:
2529:
2527:
2526:
2523:{\displaystyle }
2521:
2499:
2497:
2496:
2493:{\displaystyle }
2491:
2469:
2467:
2466:
2461:
2443:
2441:
2440:
2435:
2433:
2432:
2413:
2411:
2410:
2405:
2397:
2396:
2389:
2388:
2307:
2305:
2304:
2299:
2297:
2296:
2265:augmented matrix
2262:
2260:
2259:
2256:{\displaystyle }
2254:
2229:
2227:
2226:
2221:
2185:
2184:
2159:
2158:
2141:
2056:
2054:
2053:
2048:
1997:
1995:
1994:
1989:
1986:
1981:
1965:
1960:
1955:
1951:
1949:
1941:
1933:
1922:
1917:
1899:
1898:
1876:
1874:
1873:
1868:
1849:
1847:
1846:
1841:
1838:
1833:
1818:
1817:
1804:
1799:
1781:
1780:
1755:
1753:
1752:
1747:
1745:
1744:
1714:
1713:
1672:
1670:
1669:
1664:
1659:
1658:
1656:
1655:
1654:
1641:
1633:
1625:
1624:
1609:
1607:
1606:
1605:
1592:
1584:
1576:
1575:
1559:
1554:
1545:
1544:
1535:
1534:
1521:
1516:
1507:
1506:
1497:
1496:
1466:
1464:
1463:
1458:
1453:
1449:
1448:
1436:
1435:
1423:
1415:
1414:
1402:
1401:
1381:
1379:
1378:
1373:
1368:
1361:
1360:
1348:
1347:
1335:
1324:
1323:
1311:
1310:
1279:
1277:
1276:
1271:
1266:
1259:
1245:
1244:
1235:
1233:
1232:
1231:
1218:
1210:
1205:
1204:
1195:
1193:
1192:
1191:
1178:
1170:
1138:
1136:
1135:
1130:
1128:
1124:
1116:
1115:
1103:
1102:
1071:
1069:
1068:
1063:
1061:
1060:
1055:
1042:
1040:
1039:
1034:
1032:
1031:
1026:
1010:
1008:
1007:
1002:
997:
996:
995:
985:
977:
967:
962:
950:
949:
939:
931:
921:
916:
890:
888:
887:
882:
880:
879:
874:
861:
859:
858:
853:
851:
850:
845:
816:
814:
813:
808:
803:
796:
795:
783:
772:
771:
747:
745:
744:
739:
737:
736:
731:
676:observations in
668:
666:
665:
660:
655:
654:
623:
621:
620:
615:
613:
582:
580:
579:
574:
569:
562:
561:
438:
431:
424:
408:
407:
315:Ridge regression
150:Multilevel model
30:
21:
6299:
6298:
6294:
6293:
6292:
6290:
6289:
6288:
6259:
6258:
6257:
6252:
6230:
6212:
6176:
6172:Chebyshev nodes
6125:
6121:Bayesian design
6097:
6078:Goodness of fit
6051:
6024:
6014:Model selection
5997:
5965:
5934:
5893:
5862:
5819:
5813:
5780:
5737:
5704:
5695:
5646:10.1137/0717073
5535:
5530:
5529:
5518:
5514:
5500:
5499:
5495:
5480:
5463:
5462:
5458:
5449:
5445:
5413:10.1.1.461.9154
5397:
5396:
5392:
5385:10.1139/f75-172
5370:
5369:
5365:
5350:10.2307/1907024
5335:
5334:
5330:
5316:
5306:
5301:
5296:
5295:
5291:
5282:
5278:
5273:
5269:
5261:
5257:
5249:
5245:
5233:
5229:
5213:
5209:
5187:
5186:
5182:
5177:
5173:
5163:
5161:
5159:
5144:
5143:
5139:
5134:
5130:
5114:
5110:
5105:
5100:
5099:
5066:
5062:
5039:
5038:
5008:
5007:
4986:
4985:
4959:
4958:
4925:
4896:
4890:
4889:
4887:
4883:
4878:
4830:
4785:scale invariant
4765:
4744:
4738:
4732:
4724:Jacobian matrix
4702:
4701:
4664:
4654:
4627:
4617:
4611:
4610:
4600:
4594:in many cases.
4572:
4571:
4548:
4547:
4528:
4527:
4520:
4487:
4482:
4481:
4478:
4477:
4474:
4471:
4468:
4465:
4462:
4459:
4456:
4453:
4450:
4447:
4444:
4441:
4438:
4435:
4432:
4429:
4426:
4423:
4420:
4417:
4414:
4411:
4408:
4405:
4402:
4399:
4396:
4393:
4390:
4387:
4384:
4381:
4378:
4375:
4372:
4369:
4366:
4363:
4360:
4357:
4354:
4351:
4348:
4345:
4342:
4339:
4336:
4333:
4330:
4327:
4324:
4321:
4318:
4315:
4312:
4309:
4306:
4303:
4300:
4296:
4293:
4290:
4286:
4228:
4214:
4213:
4180:
4179:
4169:
4163:
4162:
4152:
4104:
4103:
4069:
4063:
4062:
4028:
4018:
3975:
3974:
3931:
3930:
3906:
3901:
3900:
3876:
3871:
3870:
3843:
3842:
3829:
3826:
3825:
3812:
3805:
3762:
3761:
3720:
3719:
3706:
3703:
3702:
3689:
3682:
3680:
3673:
3672:
3659:
3656:
3655:
3642:
3635:
3602:
3601:
3588:
3585:
3584:
3571:
3564:
3562:
3552:
3542:
3518:
3517:
3480:
3479:
3466:
3464:
3451:
3448:
3447:
3434:
3432:
3419:
3412:
3410:
3403:
3402:
3392:
3390:
3384:
3383:
3378:
3362:
3355:
3341:
3330:
3303:
3302:
3272:
3271:
3258:
3256:
3243:
3240:
3239:
3226:
3224:
3211:
3204:
3202:
3195:
3194:
3178:
3176:
3170:
3169:
3164:
3154:
3147:
3133:
3122:
3074:
3073:
3051:
3050:
3031:
3030:
3011:
3010:
2970:
2969:
2949:
2928:
2927:
2907:
2882:
2874:
2873:
2863:
2861:
2855:
2854:
2849:
2839:
2832:
2818:
2807:
2788:
2787:
2774:
2772:
2759:
2756:
2755:
2742:
2740:
2727:
2720:
2718:
2711:
2710:
2700:
2698:
2692:
2691:
2686:
2676:
2669:
2655:
2644:
2620:
2619:
2587:
2586:
2565:
2536:
2535:
2502:
2501:
2472:
2471:
2446:
2445:
2424:
2419:
2418:
2391:
2390:
2380:
2374:
2373:
2363:
2320:
2319:
2288:
2277:
2276:
2235:
2234:
2176:
2121:
2116:
2115:
2030:
2029:
2015:
1942:
1934:
1928:
1887:
1882:
1881:
1859:
1858:
1809:
1769:
1764:
1763:
1736:
1705:
1694:
1693:
1679:
1646:
1642:
1634:
1616:
1597:
1593:
1585:
1567:
1536:
1526:
1498:
1488:
1476:
1475:
1437:
1427:
1403:
1393:
1387:
1386:
1349:
1339:
1312:
1302:
1296:
1295:
1236:
1223:
1219:
1211:
1196:
1183:
1179:
1171:
1148:
1147:
1107:
1094:
1082:
1081:
1050:
1045:
1044:
1021:
1016:
1015:
987:
941:
896:
895:
869:
864:
863:
840:
835:
834:
823:
787:
763:
757:
756:
726:
721:
720:
629:
628:
604:
603:
553:
541:
540:
522:
517:
442:
402:
382:Goodness of fit
89:Discrete choice
28:
23:
22:
15:
12:
11:
5:
6297:
6295:
6287:
6286:
6281:
6276:
6271:
6261:
6260:
6254:
6253:
6251:
6250:
6245:
6240:
6228:
6223:
6217:
6214:
6213:
6211:
6210:
6205:
6200:
6195:
6190:
6184:
6182:
6178:
6177:
6175:
6174:
6169:
6164:
6159:
6154:
6149:
6144:
6138:
6136:
6127:
6126:
6124:
6123:
6118:
6116:Optimal design
6113:
6107:
6105:
6099:
6098:
6096:
6095:
6090:
6085:
6080:
6075:
6070:
6065:
6059:
6057:
6053:
6052:
6050:
6049:
6044:
6039:
6038:
6037:
6032:
6027:
6022:
6011:
6005:
6003:
5999:
5998:
5996:
5995:
5990:
5985:
5979:
5977:
5971:
5970:
5967:
5966:
5964:
5963:
5958:
5953:
5948:
5942:
5940:
5936:
5935:
5933:
5932:
5927:
5922:
5917:
5915:Semiparametric
5912:
5907:
5901:
5899:
5895:
5894:
5892:
5891:
5886:
5881:
5876:
5870:
5868:
5864:
5863:
5861:
5860:
5855:
5850:
5845:
5840:
5834:
5832:
5823:
5815:
5814:
5812:
5811:
5806:
5801:
5796:
5790:
5788:
5782:
5781:
5779:
5778:
5773:
5768:
5762:
5760:Spearman's rho
5753:
5747:
5745:
5739:
5738:
5736:
5735:
5730:
5725:
5720:
5714:
5712:
5706:
5705:
5696:
5694:
5693:
5686:
5679:
5671:
5665:
5664:
5654:
5648:
5634:
5623:
5616:
5599:
5592:
5585:
5575:
5561:
5553:M. Plešinger,
5551:
5534:
5531:
5528:
5527:
5512:
5493:
5479:978-1402004766
5478:
5456:
5443:
5406:(2): 259–291.
5390:
5363:
5328:
5319:|journal=
5289:
5276:
5267:
5255:
5243:
5227:
5224:978-0898713602
5207:
5189:Golub, Gene H.
5180:
5171:
5157:
5137:
5128:
5107:
5106:
5104:
5101:
5098:
5097:
5060:
5047:
5021:
5016:
4994:
4972:
4967:
4945:
4941:
4937:
4932:
4928:
4924:
4920:
4915:
4911:
4908:
4903:
4899:
4880:
4879:
4877:
4874:
4873:
4872:
4867:
4862:
4857:
4852:
4847:
4842:
4836:
4829:
4826:
4789:Paul Samuelson
4764:
4761:
4734:Main article:
4731:
4728:
4710:
4698:
4697:
4686:
4682:
4679:
4674:
4671:
4667:
4661:
4657:
4653:
4649:
4645:
4642:
4637:
4634:
4630:
4624:
4620:
4599:
4596:
4579:
4555:
4535:
4519:
4516:
4497:
4494:
4490:
4285:
4276:
4275:
4264:
4259:
4256:
4251:
4248:
4244:
4238:
4235:
4231:
4227:
4224:
4221:
4207:
4206:
4195:
4192:
4189:
4184:
4176:
4172:
4168:
4165:
4164:
4161:
4158:
4157:
4155:
4150:
4147:
4144:
4141:
4138:
4135:
4131:
4128:
4125:
4122:
4119:
4116:
4113:
4108:
4100:
4097:
4092:
4089:
4085:
4079:
4076:
4072:
4068:
4065:
4064:
4059:
4056:
4051:
4048:
4044:
4038:
4035:
4031:
4027:
4024:
4023:
4021:
4016:
4013:
4010:
4007:
4004:
4001:
3997:
3994:
3991:
3988:
3985:
3982:
3957:
3954:
3949:
3946:
3942:
3938:
3916:
3913:
3909:
3886:
3883:
3879:
3867:
3866:
3855:
3852:
3847:
3839:
3836:
3832:
3828:
3827:
3822:
3819:
3815:
3811:
3810:
3808:
3803:
3800:
3797:
3794:
3791:
3788:
3784:
3781:
3778:
3775:
3772:
3769:
3749:This provides
3747:
3746:
3735:
3730:
3724:
3716:
3713:
3709:
3705:
3704:
3699:
3696:
3692:
3688:
3687:
3685:
3677:
3669:
3666:
3662:
3658:
3657:
3652:
3649:
3645:
3641:
3640:
3638:
3633:
3630:
3626:
3623:
3620:
3617:
3612:
3606:
3598:
3595:
3591:
3587:
3586:
3581:
3578:
3574:
3570:
3569:
3567:
3559:
3555:
3549:
3545:
3541:
3538:
3535:
3532:
3528:
3525:
3507:
3506:
3495:
3490:
3484:
3476:
3473:
3469:
3465:
3461:
3458:
3454:
3450:
3449:
3444:
3441:
3437:
3433:
3429:
3426:
3422:
3418:
3417:
3415:
3407:
3399:
3395:
3391:
3389:
3386:
3385:
3382:
3379:
3375:
3372:
3369:
3365:
3361:
3360:
3358:
3353:
3348:
3344:
3337:
3333:
3329:
3326:
3323:
3320:
3317:
3313:
3310:
3296:
3295:
3282:
3276:
3268:
3265:
3261:
3257:
3253:
3250:
3246:
3242:
3241:
3236:
3233:
3229:
3225:
3221:
3218:
3214:
3210:
3209:
3207:
3199:
3191:
3188:
3185:
3181:
3177:
3175:
3172:
3171:
3168:
3165:
3161:
3157:
3153:
3152:
3150:
3145:
3140:
3136:
3129:
3125:
3121:
3118:
3115:
3112:
3109:
3106:
3103:
3100:
3096:
3093:
3090:
3087:
3084:
3081:
3058:
3038:
3018:
2988:
2987:
2974:
2966:
2961:
2958:
2954:
2950:
2946:
2941:
2938:
2934:
2930:
2929:
2924:
2919:
2916:
2912:
2908:
2904:
2899:
2896:
2892:
2888:
2887:
2885:
2878:
2870:
2866:
2862:
2860:
2857:
2856:
2853:
2850:
2846:
2842:
2838:
2837:
2835:
2830:
2825:
2821:
2814:
2810:
2806:
2803:
2798:
2792:
2784:
2781:
2777:
2773:
2769:
2766:
2762:
2758:
2757:
2752:
2749:
2745:
2741:
2737:
2734:
2730:
2726:
2725:
2723:
2715:
2707:
2703:
2699:
2697:
2694:
2693:
2690:
2687:
2683:
2679:
2675:
2674:
2672:
2667:
2662:
2658:
2651:
2647:
2643:
2640:
2637:
2634:
2630:
2627:
2604:
2601:
2597:
2594:
2572:
2568:
2564:
2561:
2558:
2555:
2552:
2549:
2546:
2543:
2519:
2516:
2512:
2509:
2489:
2486:
2482:
2479:
2459:
2456:
2453:
2431:
2427:
2415:
2414:
2403:
2400:
2395:
2387:
2383:
2379:
2376:
2375:
2372:
2369:
2368:
2366:
2361:
2358:
2355:
2352:
2349:
2346:
2342:
2339:
2336:
2333:
2330:
2327:
2310:Frobenius norm
2295:
2291:
2287:
2284:
2252:
2249:
2245:
2242:
2231:
2230:
2219:
2216:
2213:
2210:
2207:
2204:
2201:
2198:
2195:
2192:
2188:
2183:
2179:
2175:
2172:
2168:
2165:
2162:
2157:
2154:
2151:
2148:
2145:
2140:
2137:
2134:
2131:
2128:
2125:
2058:
2057:
2046:
2043:
2040:
2037:
2014:
2011:
1999:
1998:
1985:
1980:
1977:
1974:
1970:
1964:
1959:
1954:
1948:
1945:
1940:
1937:
1931:
1926:
1921:
1916:
1913:
1910:
1906:
1902:
1897:
1894:
1890:
1866:
1851:
1850:
1837:
1832:
1829:
1826:
1822:
1816:
1812:
1808:
1803:
1798:
1795:
1792:
1788:
1784:
1779:
1776:
1772:
1757:
1756:
1743:
1739:
1735:
1732:
1729:
1726:
1723:
1720:
1717:
1712:
1708:
1704:
1701:
1678:
1675:
1674:
1673:
1662:
1653:
1649:
1645:
1640:
1637:
1631:
1628:
1623:
1619:
1612:
1604:
1600:
1596:
1591:
1588:
1582:
1579:
1574:
1570:
1563:
1558:
1553:
1549:
1543:
1539:
1533:
1529:
1525:
1520:
1515:
1511:
1505:
1501:
1495:
1491:
1487:
1484:
1456:
1452:
1447:
1444:
1440:
1434:
1430:
1426:
1422:
1418:
1413:
1410:
1406:
1400:
1396:
1383:
1382:
1371:
1367:
1364:
1359:
1356:
1352:
1346:
1342:
1338:
1334:
1330:
1327:
1322:
1319:
1315:
1309:
1305:
1281:
1280:
1269:
1265:
1262:
1258:
1254:
1251:
1248:
1243:
1239:
1230:
1226:
1222:
1217:
1214:
1208:
1203:
1199:
1190:
1186:
1182:
1177:
1174:
1168:
1165:
1162:
1159:
1156:
1127:
1123:
1119:
1114:
1110:
1106:
1101:
1097:
1093:
1090:
1059:
1054:
1030:
1025:
1012:
1011:
1000:
994:
990:
984:
981:
976:
972:
966:
961:
957:
953:
948:
944:
938:
935:
930:
926:
920:
915:
911:
906:
903:
878:
873:
849:
844:
822:
819:
818:
817:
806:
802:
799:
794:
790:
786:
782:
778:
775:
770:
766:
735:
730:
684:parameters in
670:
669:
658:
653:
649:
646:
643:
640:
637:
612:
584:
583:
572:
568:
565:
560:
556:
551:
548:
521:
518:
516:
513:
505:Frobenius norm
444:
443:
441:
440:
433:
426:
418:
415:
414:
413:
412:
397:
396:
395:
394:
389:
384:
379:
374:
369:
361:
360:
356:
355:
354:
353:
348:
343:
338:
333:
325:
324:
323:
322:
317:
312:
307:
302:
294:
293:
292:
291:
286:
281:
276:
268:
267:
266:
265:
260:
255:
247:
246:
242:
241:
240:
239:
231:
230:
229:
228:
223:
218:
213:
208:
203:
198:
193:
191:Semiparametric
188:
183:
175:
174:
173:
172:
167:
162:
160:Random effects
157:
152:
144:
143:
142:
141:
136:
134:Ordered probit
131:
126:
121:
116:
111:
106:
101:
96:
91:
86:
81:
73:
72:
71:
70:
65:
60:
55:
47:
46:
42:
41:
35:
34:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
6296:
6285:
6282:
6280:
6279:Least squares
6277:
6275:
6274:Curve fitting
6272:
6270:
6267:
6266:
6264:
6249:
6246:
6244:
6241:
6239:
6234:
6229:
6227:
6224:
6222:
6219:
6218:
6215:
6209:
6206:
6204:
6201:
6199:
6196:
6194:
6191:
6189:
6188:Curve fitting
6186:
6185:
6183:
6179:
6173:
6170:
6168:
6165:
6163:
6160:
6158:
6155:
6153:
6150:
6148:
6145:
6143:
6140:
6139:
6137:
6135:
6134:approximation
6132:
6128:
6122:
6119:
6117:
6114:
6112:
6109:
6108:
6106:
6104:
6100:
6094:
6091:
6089:
6086:
6084:
6081:
6079:
6076:
6074:
6071:
6069:
6066:
6064:
6061:
6060:
6058:
6054:
6048:
6045:
6043:
6040:
6036:
6033:
6031:
6028:
6026:
6025:
6017:
6016:
6015:
6012:
6010:
6007:
6006:
6004:
6000:
5994:
5991:
5989:
5986:
5984:
5981:
5980:
5978:
5976:
5972:
5962:
5959:
5957:
5954:
5952:
5949:
5947:
5944:
5943:
5941:
5937:
5931:
5928:
5926:
5923:
5921:
5918:
5916:
5913:
5911:
5910:Nonparametric
5908:
5906:
5903:
5902:
5900:
5896:
5890:
5887:
5885:
5882:
5880:
5877:
5875:
5872:
5871:
5869:
5865:
5859:
5856:
5854:
5851:
5849:
5846:
5844:
5841:
5839:
5836:
5835:
5833:
5831:
5827:
5824:
5822:
5816:
5810:
5807:
5805:
5802:
5800:
5797:
5795:
5792:
5791:
5789:
5787:
5783:
5777:
5774:
5772:
5769:
5766:
5765:Kendall's tau
5763:
5761:
5757:
5754:
5752:
5749:
5748:
5746:
5744:
5740:
5734:
5731:
5729:
5726:
5724:
5721:
5719:
5718:Least squares
5716:
5715:
5713:
5711:
5707:
5703:
5699:
5698:Least squares
5692:
5687:
5685:
5680:
5678:
5673:
5672:
5669:
5662:
5659:
5655:
5652:
5649:
5647:
5643:
5639:
5635:
5632:
5628:
5624:
5621:
5617:
5615:
5611:
5607:
5603:
5602:S. Van Huffel
5600:
5597:
5593:
5590:
5586:
5583:
5579:
5578:S. Van Huffel
5576:
5574:
5570:
5566:
5562:
5560:
5556:
5552:
5549:
5545:
5541:
5540:S. Van Huffel
5537:
5536:
5532:
5525:
5522:
5516:
5513:
5508:
5504:
5497:
5494:
5489:
5485:
5481:
5475:
5471:
5467:
5460:
5457:
5453:
5447:
5444:
5439:
5435:
5431:
5427:
5423:
5419:
5414:
5409:
5405:
5401:
5394:
5391:
5386:
5382:
5378:
5374:
5367:
5364:
5359:
5355:
5351:
5347:
5343:
5339:
5332:
5329:
5324:
5311:
5300:
5293:
5290:
5286:
5285:S. Van Huffel
5280:
5277:
5271:
5268:
5264:
5263:S. Van Huffel
5259:
5256:
5252:
5251:S. Van Huffel
5247:
5244:
5240:
5236:
5235:S. Van Huffel
5231:
5228:
5225:
5221:
5217:
5211:
5208:
5202:
5198:
5194:
5190:
5184:
5181:
5175:
5172:
5160:
5158:9780471934127
5154:
5150:
5149:
5141:
5138:
5132:
5129:
5126:
5122:
5118:
5117:S. Van Huffel
5112:
5109:
5102:
5094:
5090:
5087:used for the
5086:
5082:
5079: ≈
5078:
5074:
5071: ≈
5070:
5067:The notation
5064:
5061:
5036:
4885:
4882:
4875:
4871:
4868:
4866:
4863:
4861:
4860:Least squares
4858:
4856:
4853:
4851:
4848:
4846:
4843:
4840:
4837:
4835:
4832:
4831:
4827:
4825:
4821:
4819:
4815:
4811:
4807:
4803:
4799:
4795:
4790:
4786:
4781:
4779:
4775:
4771:
4762:
4760:
4758:
4754:
4750:
4743:
4737:
4729:
4727:
4725:
4684:
4609:
4608:
4607:
4605:
4597:
4595:
4593:
4590:is, however,
4577:
4569:
4553:
4533:
4525:
4517:
4515:
4513:
4495:
4492:
4488:
4283:
4281:
4262:
4257:
4254:
4249:
4246:
4242:
4236:
4233:
4229:
4225:
4222:
4219:
4212:
4211:
4210:
4193:
4190:
4187:
4182:
4174:
4170:
4166:
4159:
4153:
4142:
4139:
4136:
4126:
4123:
4120:
4111:
4106:
4098:
4095:
4090:
4087:
4083:
4077:
4074:
4070:
4066:
4057:
4054:
4049:
4046:
4042:
4036:
4033:
4029:
4025:
4019:
4008:
4005:
4002:
3992:
3989:
3986:
3973:
3972:
3971:
3955:
3952:
3947:
3944:
3940:
3936:
3914:
3911:
3907:
3884:
3881:
3877:
3853:
3850:
3845:
3837:
3834:
3830:
3820:
3817:
3813:
3806:
3795:
3792:
3789:
3779:
3776:
3773:
3760:
3759:
3758:
3756:
3752:
3733:
3728:
3722:
3714:
3711:
3707:
3697:
3694:
3690:
3683:
3675:
3667:
3664:
3660:
3650:
3647:
3643:
3636:
3628:
3624:
3618:
3615:
3610:
3604:
3596:
3593:
3589:
3579:
3576:
3572:
3565:
3557:
3547:
3543:
3539:
3536:
3530:
3526:
3516:
3515:
3514:
3512:
3493:
3488:
3482:
3474:
3471:
3467:
3459:
3456:
3452:
3442:
3439:
3435:
3427:
3424:
3420:
3413:
3405:
3397:
3387:
3380:
3373:
3370:
3367:
3363:
3356:
3346:
3342:
3335:
3331:
3324:
3321:
3315:
3311:
3301:
3300:
3299:
3280:
3274:
3266:
3263:
3259:
3251:
3248:
3244:
3234:
3231:
3227:
3219:
3216:
3212:
3205:
3197:
3189:
3186:
3183:
3179:
3173:
3166:
3159:
3148:
3138:
3134:
3127:
3123:
3116:
3107:
3104:
3101:
3091:
3088:
3085:
3072:
3071:
3070:
3056:
3036:
3016:
3008:
3003:
3001:
2997:
2993:
2972:
2964:
2959:
2956:
2952:
2944:
2939:
2936:
2932:
2922:
2917:
2914:
2910:
2902:
2897:
2894:
2890:
2883:
2876:
2868:
2858:
2851:
2844:
2833:
2823:
2819:
2812:
2808:
2801:
2796:
2790:
2782:
2779:
2775:
2767:
2764:
2760:
2750:
2747:
2743:
2735:
2732:
2728:
2721:
2713:
2705:
2695:
2688:
2681:
2670:
2660:
2656:
2649:
2645:
2638:
2632:
2628:
2618:
2617:
2616:
2599:
2595:
2570:
2562:
2544:
2533:
2514:
2510:
2484:
2480:
2457:
2454:
2451:
2429:
2425:
2401:
2398:
2393:
2385:
2381:
2377:
2370:
2364:
2353:
2350:
2347:
2337:
2334:
2331:
2318:
2317:
2316:
2313:
2311:
2293:
2285:
2274:
2270:
2266:
2247:
2243:
2217:
2214:
2211:
2208:
2205:
2199:
2196:
2193:
2186:
2181:
2170:
2166:
2155:
2152:
2149:
2146:
2143:
2114:
2113:
2112:
2110:
2106:
2102:
2098:
2094:
2089:
2087:
2083:
2079:
2075:
2071:
2067:
2063:
2044:
2041:
2038:
2035:
2028:
2027:
2026:
2024:
2019:
2012:
2010:
2008:
2004:
1983:
1978:
1975:
1972:
1968:
1962:
1957:
1952:
1946:
1943:
1938:
1935:
1929:
1924:
1919:
1914:
1911:
1908:
1904:
1900:
1895:
1892:
1888:
1880:
1879:
1878:
1864:
1856:
1835:
1830:
1827:
1824:
1820:
1814:
1810:
1806:
1801:
1796:
1793:
1790:
1786:
1782:
1777:
1774:
1770:
1762:
1761:
1760:
1759:in this case
1741:
1737:
1733:
1730:
1727:
1724:
1718:
1715:
1710:
1706:
1699:
1692:
1691:
1690:
1688:
1684:
1676:
1660:
1474:
1473:
1472:
1470:
1454:
1369:
1294:
1293:
1292:
1290:
1286:
1267:
1146:
1145:
1144:
1142:
1079:
1075:
1057:
1028:
998:
904:
901:
894:
893:
892:
876:
847:
832:
828:
820:
804:
755:
754:
753:
751:
733:
719:
715:
711:
707:
703:
699:
695:
691:
687:
683:
679:
675:
656:
627:
626:
625:
601:
597:
593:
589:
570:
549:
546:
539:
538:
537:
535:
531:
527:
526:least squares
519:
514:
512:
510:
506:
501:
499:
495:
491:
490:least squares
487:
484:is a type of
483:
479:
471:
467:
463:
459:
455:
450:
439:
434:
432:
427:
425:
420:
419:
417:
416:
411:
406:
401:
400:
399:
398:
393:
390:
388:
385:
383:
380:
378:
375:
373:
370:
368:
365:
364:
363:
362:
357:
352:
349:
347:
344:
342:
339:
337:
334:
332:
329:
328:
327:
326:
321:
318:
316:
313:
311:
308:
306:
303:
301:
298:
297:
296:
295:
290:
287:
285:
282:
280:
277:
275:
272:
271:
270:
269:
264:
261:
259:
256:
254:
253:Least squares
251:
250:
249:
248:
243:
238:
235:
234:
233:
232:
227:
224:
222:
219:
217:
214:
212:
209:
207:
204:
202:
199:
197:
194:
192:
189:
187:
186:Nonparametric
184:
182:
179:
178:
177:
176:
171:
168:
166:
163:
161:
158:
156:
155:Fixed effects
153:
151:
148:
147:
146:
145:
140:
137:
135:
132:
130:
129:Ordered logit
127:
125:
122:
120:
117:
115:
112:
110:
107:
105:
102:
100:
97:
95:
92:
90:
87:
85:
82:
80:
77:
76:
75:
74:
69:
66:
64:
61:
59:
56:
54:
51:
50:
49:
48:
43:
40:
36:
32:
31:
19:
6181:Applications
6020:
5898:Non-standard
5803:
5657:
5653:at MathPages
5637:
5626:
5625:P. de Groen
5619:
5605:
5595:
5588:
5581:
5564:
5559:Ph.D. Thesis
5554:
5543:
5515:
5496:
5469:
5459:
5451:
5446:
5403:
5399:
5393:
5376:
5372:
5366:
5344:(1): 80–83.
5341:
5338:Econometrica
5337:
5331:
5310:cite journal
5292:
5279:
5270:
5258:
5246:
5238:
5230:
5215:
5210:
5196:
5183:
5174:
5162:. Retrieved
5147:
5140:
5131:
5120:
5111:
5092:
5088:
5084:
5080:
5076:
5072:
5068:
5063:
5034:
4884:
4822:
4817:
4813:
4809:
4805:
4801:
4797:
4793:
4782:
4766:
4756:
4752:
4745:
4699:
4601:
4591:
4521:
4511:
4479:
4277:
4208:
3868:
3754:
3750:
3748:
3510:
3508:
3297:
3004:
2999:
2995:
2991:
2989:
2531:
2416:
2314:
2272:
2268:
2232:
2108:
2104:
2100:
2096:
2092:
2090:
2085:
2081:
2077:
2073:
2069:
2065:
2061:
2059:
2020:
2016:
2006:
2000:
1854:
1852:
1758:
1686:
1682:
1680:
1468:
1384:
1284:
1282:
1140:
1077:
1073:
1013:
830:
826:
824:
749:
713:
709:
705:
701:
697:
693:
689:
685:
681:
677:
673:
671:
595:
587:
585:
533:
523:
515:Linear model
502:
496:and also of
481:
475:
469:
465:
461:
457:
453:
310:Non-negative
304:
4518:Computation
320:Regularized
284:Generalized
216:Least angle
114:Mixed logit
6263:Categories
6056:Background
6019:Mallows's
5164:4 December
5103:References
4816:, and the
4568:Van Huffel
4280:GNU Octave
3005:Using the
672:There are
520:Background
359:Background
263:Non-linear
245:Estimation
6131:Numerical
5631:arxiv.org
5408:CiteSeerX
5151:. Wiley.
5046:β
5015:Δ
4993:β
4971:β
4966:Δ
4940:Δ
4919:β
4914:Δ
4678:Δ
4670:−
4648:β
4644:Δ
4633:−
4546:(denoted
4255:−
4226:−
4167:−
4096:−
4067:−
4055:−
4026:−
3953:−
3937:−
3757:so that
3729:∗
3619:−
3611:∗
3554:Σ
3540:−
3489:∗
3394:Σ
3371:×
3325:−
3281:∗
3187:×
3156:Σ
2965:∗
2945:∗
2923:∗
2903:∗
2865:Σ
2841:Σ
2797:∗
2702:Σ
2678:Σ
2571:∗
2554:Σ
2534:. Define
2455:×
2378:−
2290:‖
2286:⋅
2283:‖
2178:‖
2161:‖
2042:≈
1969:σ
1905:σ
1865:β
1821:σ
1811:β
1787:σ
1734:β
1728:α
1719:β
1644:∂
1636:∂
1630:−
1595:∂
1587:∂
1581:−
1443:−
1421:β
1409:−
1363:Δ
1355:−
1333:β
1329:Δ
1318:−
1257:β
1253:Δ
1247:−
1221:∂
1213:∂
1207:−
1181:∂
1173:∂
1167:−
1161:Δ
1122:β
980:−
934:−
781:β
652:β
645:−
611:β
592:residuals
226:Segmented
5961:Logistic
5951:Binomial
5930:Isotonic
5925:Quantile
5548:preprint
5438:16462731
5430:16573844
5195:(1996).
5125:preprint
4957:, where
4828:See also
4287:function
4278:A naive
341:Bayesian
279:Weighted
274:Ordinary
206:Isotonic
201:Quantile
5956:Poisson
5507:2707593
5488:1077322
5358:1907024
5205:pp 596.
4722:is the
4209:and so
3869:Now if
2444:is the
2308:is the
2263:is the
1677:Example
524:In the
300:Partial
139:Poisson
5920:Robust
5533:Others
5505:
5486:
5476:
5436:
5428:
5410:
5356:
5222:
5155:
4800:, the
4700:where
4524:Netlib
2990:where
2417:where
2233:where
2064:where
1614:
1565:
1467:where
1014:where
258:Linear
196:Robust
119:Probit
45:Models
5434:S2CID
5354:JSTOR
5302:(PDF)
4876:Notes
2267:with
700:is a
688:with
305:Total
221:Local
5700:and
5503:SSRN
5484:SSRN
5474:ISBN
5426:PMID
5323:help
5220:ISBN
5166:2012
5153:ISBN
5091:-by-
5006:and
4602:For
4307:size
4299:X, Y
3753:and
3029:and
2998:and
2271:and
2107:and
2103:for
2099:and
2084:-by-
2076:and
2072:-by-
2060:for
1685:and
1076:and
1043:and
862:and
829:and
692:>
680:and
594:and
488:, a
468:and
456:and
6035:BIC
6030:AIC
5642:doi
5610:doi
5569:doi
5418:doi
5381:doi
5346:doi
4475:end
4469:VYY
4463:VXY
4445:end
4427:end
4403:VYY
4394:end
4358:VXY
4337:svd
4294:tls
2530:by
2088:.
2080:is
2068:is
476:In
6265::
5542:,
5482:.
5432:.
5424:.
5416:.
5404:81
5402:.
5377:32
5375:.
5352:.
5342:10
5340:.
5314::
5312:}}
5308:{{
5191:;
5119:,
5077:AX
5069:XB
4812:,
4808:,
4780:.
4759:.
4726:.
4514:.
4448:);
4397:);
4352:);
4316:);
3854:0.
3002:.
2615:.
2402:0.
696:.
536:,
532:,
507:,
480:,
6023:p
6021:C
5767:)
5758:(
5690:e
5683:t
5676:v
5663:.
5644::
5633:.
5612::
5571::
5550:.
5509:.
5490:.
5440:.
5420::
5387:.
5383::
5360:.
5348::
5325:)
5321:(
5304:.
5203:.
5168:.
5093:k
5089:n
5085:X
5081:B
5073:Y
5035:y
5020:y
4944:y
4936:W
4931:T
4927:X
4923:=
4910:X
4907:W
4902:T
4898:X
4709:J
4685:,
4681:y
4673:1
4666:M
4660:T
4656:J
4652:=
4641:J
4636:1
4629:M
4623:T
4619:J
4578:B
4554:X
4534:B
4496:Y
4493:Y
4489:V
4472:;
4466:/
4460:-
4457:=
4454:B
4442::
4439:n
4436:+
4433:1
4430:,
4424::
4421:n
4418:+
4415:1
4412:(
4409:V
4406:=
4391::
4388:n
4385:+
4382:1
4379:,
4376:n
4373::
4370:1
4367:(
4364:V
4361:=
4349:0
4346:,
4343:Z
4340:(
4334:=
4328:;
4325:=
4322:Z
4313:X
4310:(
4304:=
4301:)
4297:(
4291:=
4289:B
4263:.
4258:1
4250:Y
4247:Y
4243:V
4237:Y
4234:X
4230:V
4223:=
4220:B
4194:,
4191:0
4188:=
4183:]
4175:k
4171:I
4160:B
4154:[
4149:]
4146:)
4143:F
4140:+
4137:Y
4134:(
4130:)
4127:E
4124:+
4121:X
4118:(
4115:[
4112:=
4107:]
4099:1
4091:Y
4088:Y
4084:V
4078:Y
4075:Y
4071:V
4058:1
4050:Y
4047:Y
4043:V
4037:Y
4034:X
4030:V
4020:[
4015:]
4012:)
4009:F
4006:+
4003:Y
4000:(
3996:)
3993:E
3990:+
3987:X
3984:(
3981:[
3956:1
3948:Y
3945:Y
3941:V
3915:Y
3912:Y
3908:V
3885:Y
3882:Y
3878:V
3851:=
3846:]
3838:Y
3835:Y
3831:V
3821:Y
3818:X
3814:V
3807:[
3802:]
3799:)
3796:F
3793:+
3790:Y
3787:(
3783:)
3780:E
3777:+
3774:X
3771:(
3768:[
3755:F
3751:E
3734:.
3723:]
3715:Y
3712:Y
3708:V
3698:Y
3695:X
3691:V
3684:[
3676:]
3668:Y
3665:Y
3661:V
3651:Y
3648:X
3644:V
3637:[
3632:]
3629:Y
3625:X
3622:[
3616:=
3605:]
3597:Y
3594:Y
3590:V
3580:Y
3577:X
3573:V
3566:[
3558:Y
3548:Y
3544:U
3537:=
3534:]
3531:F
3527:E
3524:[
3511:U
3494:.
3483:]
3475:Y
3472:Y
3468:V
3460:X
3457:Y
3453:V
3443:Y
3440:X
3436:V
3428:X
3425:X
3421:V
3414:[
3406:]
3398:Y
3388:0
3381:0
3374:n
3368:n
3364:0
3357:[
3352:]
3347:Y
3343:U
3336:X
3332:U
3328:[
3322:=
3319:]
3316:F
3312:E
3309:[
3275:]
3267:Y
3264:Y
3260:V
3252:X
3249:Y
3245:V
3235:Y
3232:X
3228:V
3220:X
3217:X
3213:V
3206:[
3198:]
3190:k
3184:k
3180:0
3174:0
3167:0
3160:X
3149:[
3144:]
3139:Y
3135:U
3128:X
3124:U
3120:[
3117:=
3114:]
3111:)
3108:F
3105:+
3102:Y
3099:(
3095:)
3092:E
3089:+
3086:X
3083:(
3080:[
3057:k
3037:V
3017:U
3000:Y
2996:X
2992:V
2973:]
2960:Y
2957:Y
2953:V
2940:Y
2937:X
2933:V
2918:X
2915:Y
2911:V
2898:X
2895:X
2891:V
2884:[
2877:]
2869:Y
2859:0
2852:0
2845:X
2834:[
2829:]
2824:Y
2820:U
2813:X
2809:U
2805:[
2802:=
2791:]
2783:Y
2780:Y
2776:V
2768:X
2765:Y
2761:V
2751:Y
2748:X
2744:V
2736:X
2733:X
2729:V
2722:[
2714:]
2706:Y
2696:0
2689:0
2682:X
2671:[
2666:]
2661:Y
2657:U
2650:X
2646:U
2642:[
2639:=
2636:]
2633:Y
2629:X
2626:[
2603:]
2600:Y
2596:X
2593:[
2567:]
2563:V
2560:[
2557:]
2551:[
2548:]
2545:U
2542:[
2532:k
2518:]
2515:Y
2511:X
2508:[
2488:]
2485:F
2481:E
2478:[
2458:k
2452:k
2430:k
2426:I
2399:=
2394:]
2386:k
2382:I
2371:B
2365:[
2360:]
2357:)
2354:F
2351:+
2348:Y
2345:(
2341:)
2338:E
2335:+
2332:X
2329:(
2326:[
2294:F
2273:F
2269:E
2251:]
2248:F
2244:E
2241:[
2218:F
2215:+
2212:Y
2209:=
2206:B
2203:)
2200:E
2197:+
2194:X
2191:(
2187:,
2182:F
2174:]
2171:F
2167:E
2164:[
2156:F
2153:,
2150:E
2147:,
2144:B
2139:n
2136:i
2133:m
2130:g
2127:r
2124:a
2109:Y
2105:X
2101:F
2097:E
2093:B
2086:k
2082:m
2078:Y
2074:n
2070:m
2066:X
2062:B
2045:Y
2039:B
2036:X
2007:x
1984:2
1979:i
1976:,
1973:x
1963:2
1958:i
1953:)
1947:x
1944:d
1939:y
1936:d
1930:(
1925:+
1920:2
1915:i
1912:,
1909:y
1901:=
1896:i
1893:i
1889:M
1855:i
1836:2
1831:i
1828:,
1825:x
1815:2
1807:+
1802:2
1797:i
1794:,
1791:y
1783:=
1778:i
1775:i
1771:M
1742:i
1738:x
1731:+
1725:=
1722:)
1716:,
1711:i
1707:x
1703:(
1700:f
1687:W
1683:M
1661:.
1652:y
1648:r
1639:f
1627:=
1622:y
1618:K
1611:,
1603:x
1599:r
1590:f
1578:=
1573:x
1569:K
1562:;
1557:T
1552:y
1548:K
1542:y
1538:M
1532:y
1528:K
1524:+
1519:T
1514:x
1510:K
1504:x
1500:M
1494:x
1490:K
1486:=
1483:M
1469:M
1455:,
1451:y
1446:1
1439:M
1433:T
1429:X
1425:=
1417:X
1412:1
1405:M
1399:T
1395:X
1370:,
1366:y
1358:1
1351:M
1345:T
1341:X
1337:=
1326:X
1321:1
1314:M
1308:T
1304:X
1285:m
1268:.
1264:0
1261:=
1250:X
1242:y
1238:r
1229:y
1225:r
1216:f
1202:x
1198:r
1189:x
1185:r
1176:f
1164:y
1158:=
1155:F
1141:m
1126:)
1118:,
1113:y
1109:r
1105:,
1100:x
1096:r
1092:(
1089:f
1078:y
1074:x
1058:y
1053:r
1029:x
1024:r
999:,
993:y
989:r
983:1
975:y
971:M
965:T
960:y
956:r
952:+
947:x
943:r
937:1
929:x
925:M
919:T
914:x
910:r
905:=
902:S
877:y
872:M
848:x
843:M
831:y
827:x
805:.
801:y
798:W
793:T
789:X
785:=
777:X
774:W
769:T
765:X
750:y
734:y
729:M
714:W
710:x
706:n
704:×
702:m
698:X
694:n
690:m
686:β
682:n
678:y
674:m
657:.
648:X
642:y
639:=
636:r
596:W
588:r
571:,
567:r
564:W
559:T
555:r
550:=
547:S
534:S
470:y
466:x
462:y
458:y
454:x
437:e
430:t
423:v
20:)
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