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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.

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A measurement is a dynamical process that aims to estimate the true value of a measurand. The measurand is the input that excites a sensor, and, as a consequence, the sensor output is a transient response. The main approach to estimate the input is applying the sensor transient response to another dynamical system. This dynamical system is designed by deconvolution to invert the sensor dynamics and compensate the sensor response. Digital signal processors enable an alternative approach to estimate the unknown input. There exists a data-driven subspace-based signal processing method that estimates a measurand, assuming it is constant during the measurement. To estimate the parameters of a measurand that varies at a constant rate, we extended the data-driven input estimation method to make it adaptive to the affine input.In this paper, we describe the proposed subspace signal processing method for the measurement of an affine measurand and compare its performance to a maximum-likelihood input estimation method and to an existing time-varying compensation filter. The subspace method is recursive and allows real-time implementations since it directly estimates the input without identifying a sensor model. The maximum-likelihood method is model-based and requires very high computational effort.In this form, the maximum-likelihood method cannot be implemented in real-time, however, we used it as a reference to evaluate the subspace method and the time-varying compensation filter results. The effectiveness of the subspace method is validated in a simulation study with a time-varying sensor. The results show that the subspace method estimation has relative errors that are one order of magnitude smaller and converges two times faster than the compensation filter.

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