Inverse Problems With Errors in the Independent Variables Errors-in-Variables, Total Least Squares, and Bayesian Inference

2008 ◽  
Author(s):  
A. F. Emery
Keyword(s):  
Bayesian Inference ◽  
Maximum Likelihood ◽  
Inverse Problems ◽  
Least Squares ◽  
Total Least Squares ◽  
Errors In Variables ◽  
Independent Variables ◽  
Error In Variables ◽  
Best Estimates ◽  
Normally Distributed

Most practioners of inverse problems use least squares or maximum likelihood (MLE) to estimate parameters with the assumption that the errors are normally distributed. When there are errors both in the measured responses and in the independent variables, or in the model itself, more information is needed and these approaches may not lead to the best estimates. A review of the error-in-variables (EIV) models shows that other approaches are necessary and in some cases Bayesian inference is to be preferred.

scholarly journals On the equivalence between total least squares and maximum likelihood PCA

2005 ◽  
Vol 544 (1-2) ◽  
pp. 254-267 ◽  
Author(s):  
M. Schuermans ◽  
I. Markovsky ◽  
Peter D. Wentzell ◽  
S. Van Huffel
Keyword(s):  
Maximum Likelihood ◽  
Least Squares ◽  
Total Least Squares

scholarly journals Modifying Cadzow's algorithm to generate the optimal TLS-solution for the structured EIV-Model of a similarity transformation

2012 ◽  
Vol 2 (2) ◽  
pp. 98-106 ◽  
Author(s):  
B. Schaffrin ◽  
F. Neitzel ◽  
S. Uzun ◽  
V. Mahboub
Keyword(s):  
Least Squares ◽  
Similarity Transformation ◽  
Optimal Solution ◽  
Total Least Squares ◽  
Errors In Variables ◽  
Parameter Adjustment ◽  
Structured Errors

Modifying Cadzow's algorithm to generate the optimal TLS-solution for the structured EIV-Model of a similarity transformationIn 2005, Felus and Schaffrin discussed the problem of a Structured Errors-in-Variables (EIV) Model in the context of a parameter adjustment for a classical similarity transformation. Their proposal, however, to perform a Total Least-Squares (TLS) adjustment, followed by a Cadzow step to imprint the proper structure, would not always guarantee the identity of this solution with the optimal Structured TLS solution, particularly in view of the residuals. Here, an attempt will be made to modify the Cadzow step in order to generate the optimal solution with the desired structure as it would, for instance, also result from a traditional LS-adjustment within an iteratively linearized Gauss-Helmert Model (GHM). Incidentally, this solution coincides with the (properly) Weighted TLS solution which does not need a Cadzow step.

Comparison of Maximum Likelihood, Generalized Least Squares, Ordinary Least Squares, and Asymptotically Distribution Free Parameter Estimates in Drug Abuse Latent Variable Causal Models

1983 ◽  
Vol 13 (4) ◽  
pp. 387-404 ◽  
Author(s):  
G. J. Huba ◽  
L. L. Harlow
Keyword(s):  
Drug Abuse ◽  
Maximum Likelihood ◽  
Least Squares ◽  
Latent Variable ◽  
Generalized Least Squares ◽  
Causal Models ◽  
Multivariate Normal ◽  
Distribution Free ◽  
Asymptotically Distribution Free ◽  
Normally Distributed

Latent variable causal modeling techniques are sometimes criticized when applied to drug abuse data because the commonly-employed maximum likelihood parameter estimation method requires that the data be normally distributed for the statistical tests to be accurate. In this article, four estimators for the parameters in two large latent variable causal models are compared in real drug abuse datasets. One estimator does not require that the data be multivariate normal and does, in fact, correct for data non-normality. Specifically, maximum likelihood and generalized least squares estimators for normally-distributed variables are compared with Browne's asymptotically distribution free techniques for continuous non-normally distributed data. Additionally, ordinary (unweighted) least squares estimates are used. Descriptions of the techniques are given and actual results in two “real” datasets are provided. It is concluded that the distribution free technique provides results which are generally comparable to those obtained with maximum likelihood estimation for datasets which depart in typical ways from the ideal of the multivariate normal distribution.

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