A new technique is considered for parameter estimation in a linear measurement error model AX ≈ B, A = A 0 +Ã, B = B 0 +B, A 0 X 0 = B 0 with row-wise independent and non-identically distributed measurement errorsÃ,B. Here, A 0 and B 0 are the true values of the measurements A and B, and X 0 is the true value of the parameter X. The total least-squares method yields an inconsistent estimate of the parameter in this case. Modified total least-squares problem, called element-wise weighted total least-squares, is formulated so that it provides a consistent estimator, i.e., the estimateX converges to the true value X 0 as the number of measurements increases. The new estimator is a solution of an optimization problem with the parameter estimateX and the correction D = [ A B], applied to the measured data D = [A B], as decision variables. An equivalent unconstrained problem is derived by minimizing analytically over the correction D, and an iterative algorithm for its solution, based on the first order optimality condition, is proposed. The algorithm is locally convergent with linear convergence rate. For large sample size the convergence rate tends to quadratic.
The estimation of individual instabilities of N clocks, when only differences of clock readings are available, is examined without any a priori assumption about the lack of correlation between clocks. The instability of the N clocks is described by the N x N covariance matrix R but only the ( N -1)x ( N -1) covariance matrix S of the differences referred to the N-th clock may be estimated directly from measurements. The estimation of R from S is not unique but the solution domain is limited by the positive definiteness of R. The features of this domain are analysed and the conditions for the validity of the uncorrelation hypothesis are established. The modifications of the solution domain, when the number N of compared clocks increases, is examined showing that each added clock can reduce the domain and therefore the arbitrariness in estimating R. Examples with some experimental data illustrate the capabilities of this approach.
An entirely new algorithm to find all the equilibrium points Of piecewise-linear (PWL) circuits is presented. TO this aim, the new class of the so-called polyhedral circuits, associated to the PWL ones, are defined by replacing the PWL elements a genealogical tree, whose nodes represent specific polyhedral circuits. All the equilibrium points of the original PWL circuit can be captured by the analysis of these nodes. This analysis requires the solution of the Phase I of Linear Programming (LP) problems, one problem for each node. An example shows the capabilities of this algorithm. Complementarity problem [5], [6]. Unfortunately, the number of linear regions explodes because of the diode synthesis. first proposed by Chua and f ( Z ) = 0, where f : TIM + T I M and 2 groups the bmnch variables, in a canonical form based on the absolute value function. The linear regions whose images in f(z) contain the origin as an interior point are determined by of a sign test. Since the corresponding linear systems are solved only if a solution exists, this method is cheaper, even if the sign test A third class of with the polyhedral elements. The algorithm is structured as Ying 171, consists in writing the equations of a PWL circuit
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