From: stewart@cs.umd.edu (G. W. Stewart)
Newsgroups: sci.math.num-analysis
Subject: Re: REQUEST: lst sqrs withs errors in both X and Y
Date: 16 Aug 92 05:26:47 GMT
Organization: U of Maryland, Dept. of Computer Science, Coll. Pk., MD 20742

In article <1992Aug4.135256.11081@gov.on.ca> diamong@gov.on.ca (Gerald Diamond) writes:
>I am looking for a good reference (or references) for 
>a least-squares type algorithm which treats both X and Y as 
>having errors rather than the usual case of assuming all the
>errors are in one variable.

Since this problem occurs frequently, I am taking the liberty of
appending a rather long list of references from my personal data base.
It is the result of a brute-force search on keywords and certainly
contains some irrelevant articles.  With equal certainty it also
omits some relevant articles.

Pete Stewart

\begin{thebibliography}{10}

\bibitem{abde:74}
N.~N. Abdelmalek.
\newblock On the solution of least squares problems and pseudo-inverses.
\newblock {\em Computing}, 13:215--228, 1974.

\bibitem{ande:76}
T.~W. Anderson.
\newblock Estimation of linear functional relationships: Approximate
  distributions and connections with simultaneous equations in econometrics.
\newblock {\em Journal of the Royal Statistical Society}, 38:1--31, 1976.

\bibitem{ande:84}
T.~W. Anderson.
\newblock Estimating linear statistical relationships.
\newblock {\em Annals of Statistics}, 12:1--45, 1984.

\bibitem{ardr:89}
M.~Arioli, I.~S. Duff, and P.~P.~M. van Rijk.
\newblock On the augmented system approach to sparse least-squares problems.
\newblock {\em Numerische Mathematik}, 55:667--685, 1989.

\bibitem{berb:76}
A.~E. Beaton, D.~B. Rubin, and J.~L. Barone.
\newblock The acceptability of regression solutions: Another look at
  computational accuracy.
\newblock {\em Journal of the American Statistical Association}, 71:158--168,
  1976.

\bibitem{bels:82}
D.~A. Belsley.
\newblock Assessing the presence of harmful collinearity and other forms of
  weak data through a test for signal-to-noise.
\newblock {\em Journal of Econometrics}, 20:211--253, 1982.

\bibitem{bels:84}
D.~A. Belsley.
\newblock Demeaning conditioning diagnostics through centering.
\newblock {\em The American Statistician}, 38:73--77, 1984.

\bibitem{bekw:80}
D.~A. Belsley, A.~E. Kuh, and R.~E. Welsch.
\newblock {\em Regression Diagnostics: Identifying Influential Data and Sources
  of Collinearity}.
\newblock John Wiley and Sons, New York, 1980.

\bibitem{berk:50}
J.~Berkson.
\newblock Are there two regressions.
\newblock {\em Journal of the American Statistical Association}, 45:164--180,
  1950.

\bibitem{bjor:88a}
{\AA}.~Bj{\"o}rck.
\newblock Error analysis of least squares algorithms.
\newblock Technical Report LiTH-MAT-R-1988-22, Department of Mathematics,
  Link{\"o}ping University, 1988.

\bibitem{bjor:89}
{\AA}.~Bj{\"o}rck.
\newblock Componentwise backward errors and condition estimates for linear
  least square problems.
\newblock Technical Report LiTH-MATH-R-1989-13, Department of Mathematics,
  Link{\"o}ping University, 1989.

\bibitem{bjor:91a}
{\AA}.~Bj{\"o}rck.
\newblock Component-wise perturbation analysis and errors bounds for linear
  least square solutions.
\newblock {\em BIT}, 31:238--244, 1991.
\newblock Cited in {\AA ke Bj\"orck's} bibliography on least squares, which is
  available by anonymous ftp from {\tt math.liu.se} in {\tt pub/references}.

\bibitem{brkr:74}
G.~F. Brown, J.~B. Kadane, and J.~G. Ramage.
\newblock The asymptotic bias and mean-squared error of double {$K$}-class
  estimators when the distrubances are small.
\newblock {\em International Economic Review}, 15:667--679, 1974.

\bibitem{dast:88}
Nancy David and G.~W. Stewart.
\newblock Hypothesis testing with errors in the variables.
\newblock Technical Report TR-1735, University of Maryland Department of
  Computer Science, 1988.

\bibitem{dahu:75}
R.~B. Davies and B.~Hutton.
\newblock The effects of errors in the independent variables in linear
  regression.
\newblock {\em Biometrika}, 62:383--391, 1975.

\bibitem{demm:88}
J.~Demmel.
\newblock The probability that a numerical analysis problem is difficult.
\newblock {\em Mathematics of Computation}, 50:449--480, 1988.

\bibitem{elde:80}
L.~Eld\'en.
\newblock Perturbation theory for the least squares problem with linear
  equality constraints.
\newblock {\em SIAM Journal on Numerical Analysis}, 17:338--350, 1980.

\bibitem{feld:74}
M.~Feldstein.
\newblock Errors in variables: A consistent estimator with smaller {MSE} in
  finite samples.
\newblock {\em Journal of the American Statistical Association}, 69:990--996,
  1974.

\bibitem{full:87}
W.~A. Fuller.
\newblock {\em Measurement Error Models}.
\newblock John Wiley, New York, 1987.

\bibitem{gall:81}
P.~P. Gallo.
\newblock Consistency of regression estimates when some variables are subject
  to error.
\newblock Mimeo Series 1328, Institute of Statistics, University of North
  Carolina at Chapel Hill, 1981.

\bibitem{gera:76}
V.~J. Geraci.
\newblock Identification of simultaneous equation models with measurement
  error.
\newblock {\em Journal of Econometrics}, 4:263--283, 1976.

\bibitem{gera:77}
V.~J. Geraci.
\newblock Estimation of simultaneous equation models with measurement error.
\newblock {\em Econometrica}, 45:1243--1255, 1977.

\bibitem{gope:73}
G.~H. Golub and V.~Pereyra.
\newblock The differentiation of pseudoinverses and nonlinear least squares
  problems whose variables separate.
\newblock {\em SIAM Journal on Numerical Analysis}, 10:413--432, 1973.

\bibitem{gope:76}
G.~H. Golub and V.~Pereyra.
\newblock Differentiation of pseudoinverses, separable nonlinear least squares
  problems and other tales.
\newblock In M.~Z. Nashed, editor, {\em Generalized Inverses and Applications},
  pages 303--324, New York, 1976. Academic Press.

\bibitem{govl:80}
G.~H. Golub and C.~F. Van~Loan.
\newblock An analysis of the total least squares problem.
\newblock {\em SIAM Journal on Numerical Analysis}, 17:883--893, 1980.

\bibitem{gowi:66}
G.~H. Golub and J.~H. Wilkinson.
\newblock Note on the iterative refinement of least squares solution.
\newblock {\em Numerische Mathematik}, 9:139--148, 1966.

\bibitem{gril:74}
Z.~Griliches.
\newblock Errors in variables and other unobservables.
\newblock {\em Econometrica}, 42:971--998, 1974.

\bibitem{high:90}
N.~J. Higham.
\newblock Computing error bounds for regression problems.
\newblock In P.~J. Brown and W.~A. Fuller, editors, {\em Contemporary
  Mathematics 112: Statistical Analysis of Measurement Error Models and
  Applications}, pages 195--210, Providence, Rhode Island, 1990. American
  Mathematical Society.

\bibitem{homo:72}
S.~D. Hodges and P.~G. Moore.
\newblock Data uncertainties and least squares regression.
\newblock {\em Applied Statistics}, 21:185--195, 1972.

\bibitem{kada:71}
J.~B. Kadane.
\newblock Comparison of $k$-class estimators when the disturbances are small.
\newblock {\em Econometrica}, 39:723--737, 1971.

\bibitem{kasw:92}
R.~Karlsson, J.-G. Sun, and B.~Wald\`en.
\newblock Optimal backward perturbation bounds for the linear least squares
  problem.
\newblock Technical Report LiTH-MAT-R-1992-06, Department of Mathematics,
  Link{\"o}ping University, 1992.
\newblock Cited in {\AA ke Bj\"orck's} bibliography on least squares, which is
  available by anonymous ftp from {\tt math.liu.se} in {\tt pub/references}.

\bibitem{koni:81}
H.~S. Konijn.
\newblock Maximum likelihood estimators and confidence intervals for a simple
  errors in variables model.
\newblock {\em Communications in Statistics: Part A}, 10:983--996, 1981.

\bibitem{kumm:1879}
C.~H. Kummell.
\newblock Reduction of observation equations which contain more than one
  observed quantity.
\newblock {\em The Analyst}, 6:97--105, 1879.

\bibitem{laha:74}
C.~L. Lawson and R.~J. Hanson.
\newblock {\em Solving Least Squares Problems}.
\newblock Prentice Hall, Englewood Cliffs, New Jersey, 1974.

\bibitem{lind:47}
D.~V. Lindley.
\newblock Regression lines and the linear functional relationship.
\newblock {\em Supplement to the Journal of the Royal Statistical Society},
  9:218--244, 1947.

\bibitem{lots:83}
P.~L{\"o}tstedt.
\newblock Perturbation bounds for the linear least squares problem subject to
  linear inequality constraints.
\newblock {\em BIT}, 23:500--519, 1983.

\bibitem{mada:59}
A.~Madansky.
\newblock The fitting of straight lines when both variables are subject to
  error.
\newblock {\em Journal of the American Statistical Association}, 54:173--205,
  1959.
\newblock Cited in \cite{govl:89}.

\bibitem{marq:80}
D.~W. Marquardt.
\newblock Comment on {Smith} and {Campbell} (1980).
\newblock {\em Journal of the American Statistical Association}, 75:87--91,
  1980.

\bibitem{mora:71}
P.~A.~P. Moran.
\newblock Estimating structural and functional relationships.
\newblock {\em Journal of Multivariate Analysis}, 1:232--255, 1971.

\bibitem{paar:79}
M.~Paardekooper.
\newblock Inverse perturbations for approximations of least squares solutions.
\newblock Technical Report 79.083, Katholieke Hogeschool Tilburg, Faculteit der
  Economische Wetenschappen, 1979.

\bibitem{paig:79}
C.~C. Paige.
\newblock Computer solution and perturbation analysis of generalized linear
  least squares problems.
\newblock {\em Mathematics of Computation}, 33:171--184, 1979.

\bibitem{reie:50}
O.~Reiers{\o}l.
\newblock Identifiability of a linear relation between variables which are
  subject to error.
\newblock {\em Econometrica}, 18:375--389, 1950.

\bibitem{schm:79}
K.~C. Schmidt.
\newblock Selecting subsets of variables in the linear regression problem.
\newblock Master's thesis, Department of Mathematics, University of Maryland,
  College Park, 1979.

\bibitem{scti:85}
H.~Schwetlick and V.~Tiller.
\newblock Numerical methods for estimating parameters in nonlinear models with
  errors in the variables.
\newblock {\em Technometrics}, 27:17--24, 1985.
\newblock Cited in {\AA ke Bj\"orck's} bibliography on least squares, which is
  available by anonymous ftp from {\tt math.liu.se} in {\tt pub/references}.

\bibitem{spre:66}
P.~Sprent.
\newblock A generalized least-squares approach to linear functional
  relationships.
\newblock {\em Journal of The Royal Statistical Society, Series B},
  28:278--297, 1966.

\bibitem{sriv:72}
V.~K. Srivastava.
\newblock Disturbance-variance estimation in simultaneous equations when
  disturbances are small.
\newblock {\em Journal of the American Statistical Association}, 67:164--168,
  1972.

\bibitem{stew:69}
G.~W. Stewart.
\newblock On the continuity of the generalized inverse.
\newblock {\em SIAM Journal on Applied Mathematics}, 17:33--45, 1969.

\bibitem{stew:77a}
G.~W. Stewart.
\newblock On the perturbation of pseudo-inverses, projections, and linear least
  squares problems.
\newblock {\em SIAM Review}, 19:634--662, 1977.

\bibitem{stew:77b}
G.~W. Stewart.
\newblock Sensitivity coefficients for the effects of errors in the independent
  variables in a linear regression.
\newblock Technical Report TR-571, Computer Science Department, University of
  Maryland, 1977.

\bibitem{stew:83}
G.~W. Stewart.
\newblock A generalization of the {Eckart-Young} approximation theorem.
\newblock Technical Report TR-1325, Department of Computer Science, the
  University of Maryland, 1983.

\bibitem{stew:83a}
G.~W. Stewart.
\newblock A nonlinear version of {Gauss's} minimum variance theorem with
  applications to an errors-in-the-variables model.
\newblock Technical Report TR-1263, Department of Computer Science, University
  of Maryland, 1983.

\bibitem{stew:87d}
G.~W. Stewart.
\newblock Collinearity and least squares regression.
\newblock {\em Statistical Science}, 2:68--100, 1987.

\bibitem{stew:90}
G.~W. Stewart.
\newblock Perturbation theory and least squares with errors in the variables.
\newblock In P.~J. Brown and W.~A. Fuller, editors, {\em Contemporary
  Mathematics 112: Statistical Analysis of Measurement Error Models and
  Applications}, pages 171--181, Providence, Rhode Island, 1990. American
  Mathematical Society.

\bibitem{stew:90d}
G.~W. Stewart.
\newblock Stochastic perturbation theory.
\newblock {\em SIAM Review}, 32:576--610, 1990.

\bibitem{swbo:72}
B.~F. Swindel and D.~R. Bower.
\newblock Rounding errors in the independent variables in a general linear
  model.
\newblock {\em Technometrics}, 14:215--218, 1972.

\bibitem{vdsl:69}
A.~van~der Sluis.
\newblock Condition numbers and equilibration of matrices.
\newblock {\em Numerische Mathematik}, 14:14--23, 1969.

\bibitem{vdsl:75}
A.~van~der Sluis.
\newblock Stability of the solutions of linear least squares problems.
\newblock {\em Numerische Mathematik}, 23:241--254, 1975.

\bibitem{vhva:87a}
S.~Van~Huffel and J.~Vandewalle.
\newblock Subset selection using the total least squares approach in
  collinearity problems with errors in the variables.
\newblock {\em Linear Algebra and its Applications}, 88/89:695--714, 1987.
\newblock Cited in \cite{govl:89}.

\bibitem{waks:92}
B.~Wald\'en, R.~Karlson, and J.-G. Sun.
\newblock Optimal backward perturbation bounds for the linear least squares
  problem.
\newblock Technical Report LiTH-MAT-R-1992-06, Department of Mathematics,
  Link{\"o}ping University, 1992.

\bibitem{wedi:73a}
P.-{\AA}. Wedin.
\newblock On the almost rank deficient case of the least squares problem.
\newblock {\em BIT}, 13:344--354, 1973.

\bibitem{wedi:78}
P.-{\AA}. Wedin.
\newblock Perturbations of the pseudo-inverse and the linear least squares
  problem analysed with suitable matrix decompositions.
\newblock Technical Report UMINF--69.78, Institute of Information Processing,
  University of Ume\aa, 1978.

\bibitem{wedi:85}
P.-{\AA}. Wedin.
\newblock Perturbation theory and condition numbers for generalized and
  constrained linear least squares problems.
\newblock Technical Report UMINF--125.85, Institute of Information Processing,
  University of Ume\aa, 1985.

\bibitem{wedi:87}
P.-{\AA}. Wedin.
\newblock Perturbation theory and condition numbers for generalized and
  constrained least squares problems.
\newblock Technical Report S-901-87, Institute of Information Processing,
  University of Ume\aa, 1987.

\bibitem{wei:90}
M.~Wei.
\newblock Perturbation of the least squares problem.
\newblock {\em Linear Algebra and Its Applications}, 141:177--182, 1990.

\end{thebibliography}

Newsgroups: sci.math.num-analysis