Bias and covariance of the least squares estimate in a structured errors-in-variables problem

Abstract

A structured errors-in-variables (EIV) problem arising in metrology is studied. The observations of a sensor response are subject to perturbation. The input estimation from the transient response leads to a structured EIV problem. Total least squares (TLS) is a typical estimation method to solve EIV problems. The TLS estimator of an EIV problem is consistent, and can be computed efficiently when the perturbations have zero mean, and are independently and identically distributed (i.i.d). If the perturbation is additionally Gaussian, the TLS solution coincides with maximum-likelihood (ML). However, the computational complexity of structured TLS and total ML prevents their real-time implementation. The least-squares (LS) estimator offers a suboptimal but simple recursive solution to structured EIV problems with correlation, but the statistical properties of the LS estimator are unknown. To know the LS estimate uncertainty in EIV problems, either structured or not, to provide confidence bounds for the estimation uncertainty, and to find the difference from the optimal solutions, the bias and variance of the LS estimates should be quantified.Expressions to predict the bias and variance of LS estimators applied to unstructured and structured EIV problems are derived. The predicted bias and variance quantify the statistical properties of the LS estimate and give an approximation of the uncertainty and the mean squared error for comparison to the Cramér-Rao lower bound of the structured EIV problem.

Input parameters estimation from time-varying measurements

“…The results of the expected values B 1 , B 2 , B 3 , B 4 , C 1 , and C 2 , were described by the authors of this paper in [15].…”

“…The uncertainty of the subspace method is assessed using a Taylor expansion of the estimate and Monte Carlo random sampling approach [15]. The Monte Carlo approach requires a large set of generated random samples, and for simple systems it is the recommended method.…”

Experimental Validation of a Data-Driven Step Input Estimation Method for Dynamic Measurements

“…The expected values B 1 , B 2 , B 3 , B 4 , C 1 , and C 2 , are described in [19]. The bias and covariance were obtained to extend previous analysis conducted on EIV estimation problems without an imposed structure [20], [21].…”

“…The impact that the signal processing data-driven dynamic error correction has on the uncertainty is investigated in [18]. A statistical analysis of the data-driven step input estimation method [11] was investigated in [19] and the method uncertainty was obtained with a Monte Carlo simulation study.…”