Research On Extended Theories Of Structured Total Least-squares And Their Applications

The Gauss-Morkov(GM)model is the basis of the surveying adjustment theory,and least-squares(LS)estimation can obtain the optimal parameter estimates of the GM model.During the surveying data processing,the coefficient matrix may also contain random errors.A GM model with a random coefficient matrix is called the errors-in-variables(EIV)model.In the sense of mean square error(MSE),the parameter estimates obtained by the total least-squares(TLS)estimation are superior to LS ones.Since Gloub and Van Load(1980)proposed the singular value decomposition(SVD)algorithm for the TLS problem,the TLS estimation has attracted widespread attention in various fields,such as,signal processing,automatic control,system identification,econometrics,biomedicine,and geodetic surveying,etc.,and becomes a basic parameter estimation method.In recent years,the TLS estimation has gradually evolved into a complete theoretical system,however,the related theories and methods are still not mature and need to be further improved.In this paper,classification of weighted total least-squares(WTLS)solution methods,structured total least-squares(STLS)estimation,non-negative least-squares variance component estimation of the STLS problem,robust total least-squares(RTLS)estimation based on the total residual,RTLS estimation with a high BP have been systematically studied.The main conclusions and contributions of this paper involve the aspects as following:(1)Solution methods for the weighted total least-squares(WTLS)estimation are systematically summarized.This paper classifies the WTLS solution methods from three aspects,such as mathematical model,optimization method and iterative formula.Under different model assumptions,the Gauss-Newton method is used to derive the corresponding iterative formulae.All of these iterative formulae have the form of the least-squares(LS)solution.This part lays the methodological foundation for other extended research of TLS.(2)In order to solve the problem that the coefficient matrix and the observation vector have structural characteristics for many practical applications in the field of surveying and mapping,that is,the coefficient matrix and the observation vector contain fixed quantities(even fixed columns)and random quantities,and the random quantities at different positions are linearly correlated.Starting from the EIV function model,the variable projection method is introduced to extract the structural characteristics in the coefficient matrix and the observation vector,and then two structural total least squares(STLS)estimation algorithms are constructed based on the Gauss-Newton method,which have the same form as the LS estimation.The example analysisshows that the STLS estimation algorithm proposed in this paper can reduce the number of predicted residuals compared with the WTLS estimation and the number of iterations compared with the SLTS estimation algorithm proposed by Markavsky.The equivalence between the STLS estimation and the WTLS estimation is also illustrated.(3)The characteristics of predicted residuals obtained by the TLS estimation are studied through a multi-linear fitting problem.It shows that the TLS estimation is not suitable for the residual prediction compared with the LS estimation.In this paper,total residuals are recommended to predict equation errors for variance component estimation(VCE)and robust estimation.(4)On the basis of the above research results,if considering that the random quantities in the coefficient matrix and the observation vector may come from different observations or adjustments during different periods,they may have different variance components.If considering the correlation between random quantities,they may have different covariance components.Least-squares variance component estimation(LS-VCE)is applied to estimate variance and covariance components of the STLS problem based on total residuals,and conclusions of Xu and Liu(2014)on the estimability of variance and covariance components are further expanded.At the same time,the non-negative function is used to reparameterize variance components,and the nonnegative function constrained least squares variance component estimation(NFC-LS-VCE)is derived for the structured EIV(SEIV)model.The example analysis shows that it can effectively solve the negative variance problem caused by the improper initial value selection.(5)Due to the influence of external environment,measuring instrument and human factor,observations may also contaminated by outliers.In the field of surveying and mapping,data-snooping method and M-estimation are two basic methods to deal with the above problem,and both of them are implemented based on predicted residuals.Although the TLS estimation can give a predicted residual for each observation,it is not suitable for residual predication compared with the LS one.In order to solve this problem,the median method is applied to calculate the posteriori unit weight variance based on standardized total residuals,and then an M-estimation algorithm,RSTLS-LS-IGG?,is proposed combined with IGG? weight function.The example analysis shows that its performance is superior to the reweighting strategy based on each observation.(6)Although M-estimator has high efficiency,the breakdown point(BP)of M-estimator is still zero and its robustness highly depends on the initial value.The WTLS estimation and the STLS estimation are not robust,and thus three robust estimation methods with a high BP are proposed,namely,repeated median(RM)estimation based on enumeration,total least median ofsquares(TLMS)estimation based on resampling and total least trimmed squares(TLTS)estimation based on branch and bound(BAB).Regardless of the correlation of random quantities among different observation equations,all three have a maximal 50% gradual BP.The example analysis shows that efficiency of the above three algorithms is low.Meanwhile,there are obvious deviations for the parameter estimates obtained by them because less observation equations are used to search for the global optimal solution.For the above reasons,they are not suitable for the final parameter estimation.In practice,the robust estimation method with a high BP can be used to calculate the initial value of parameters,and then the posterior unit weight variance is caculated based on standardized total residuals.Finally,M-estimator is used to calculate the final parameter estimates.This strategy is called MM-estimation,which can hold high efficiency with a high BP.