Unbiased Least-squares Research Articles

Unbiased Least-Squares Modelling

In this paper we analyze the bias in a general linear least-squares parameter estimation problem, when it is caused by deterministic variables that have not been included in the model. We propose a method to substantially reduce this bias, under the hypothesis that some a-priori information on the magnitude of the modelled and unmodelled components of the model is known. We call this method Unbiased Least-Squares (ULS) parameter estimation and present here its essential properties and some numerical results on an applied example.

Fast Graph Sampling Set Selection Using Gershgorin Disc Alignment

Graph sampling set selection, where a subset of nodes are chosen to collect samples to reconstruct a smooth graph signal, is a fundamental problem in graph signal processing (GSP). Previous works employ an unbiased least-squares (LS) signal reconstruction scheme and select samples via expensive extreme eigenvector computation. Instead, we assume a biased graph Laplacian regularization (GLR) based scheme that solves a system of linear equations for reconstruction. We then choose samples to minimize the condition number of the coefficient matrix---specifically, maximize the smallest eigenvalue $\lambda_{\min}$. Circumventing explicit eigenvalue computation, we maximize instead the lower bound of $\lambda_{\min}$, designated by the smallest left-end of all Gershgorin discs of the matrix. To achieve this efficiently, we first convert the optimization to a dual problem, where we minimize the number of samples needed to align all Gershgorin disc left-ends at a chosen lower-bound target $T$. Algebraically, the dual problem amounts to optimizing two disc operations: i) shifting of disc centers due to sampling, and ii) scaling of disc radii due to a similarity transformation of the matrix. We further reinterpret the dual as an intuitive disc coverage problem bearing strong resemblance to the famous NP-hard set cover (SC) problem. The reinterpretation enables us to derive a fast approximation scheme from a known SC error-bounded approximation algorithm. We find an appropriate target $T$ efficiently via binary search. Extensive simulation experiments show that our disc-based sampling algorithm runs substantially faster than existing sampling schemes and outperforms other eigen-decomposition-free sampling schemes in reconstruction error.

Least-squares algorithm for phase-stepping interferometry with an unknown relative step

A pointwise least-squares phase-stepping algorithm with an unknown relative phase step is proposed. In phase-stepping interferometry the recorded temporal intensity sequence is a discrete sinusoidal signal biased by a direct-current component. Its value at a certain time can be predicted from its three past samples by use of a recursive formula. Based on this linear prediction property, an unbiased least-squares estimator is deduced to determine the relative phase step from a sequence of intensity values, and the result is used to evaluate the phase value. The validity and performance of this algorithm are verified by computer simulations.

Orbital Configuration of the Valence Electrons, Ligand Field Symmetry, and Manganese Oxidation States of the Photosynthetic Water Oxidizing Complex: Analysis of the S(2) State Multiline EPR Signals.

A theoretical framework is presented for analysis of all three "multiline" EPR spectra (MLS) arising from the tetramanganese (Mn(4)) cluster in the S(2) oxidation state of the photosynthetic water oxidizing complex (WOC). Accurate simulations are presented which include anisotropy of the g and (four) (55)Mn hyperfine tensors, chosen according to a database of (55)Mn(III) and (55)Mn(IV) hyperfine tensors obtained previously using unbiased least-squares spectral fitting routines. In view of the large (30%) anisotropy common to Mn(III) hyperfine tensors in all complexes, previous MLS simulations which have assumed isotropic hyperfine constants have required physically unrealistic parameters. A simple model is found which offers good simulations of both the native "19-21-line" MLS and the "26-line" NH(3)-bound form of the MLS. Both a dimer-of-dimers and distorted-trigonal magnetic models are examined to describe the symmetry of the Heisenberg exchange interactions within the Mn(4) cluster and thus define the initial electronic basis states of the cluster. The effect of rhombic symmetry distortions is explicitly considered. Both magnetic models correspond to one of several possible structural models for the Mn(4) cluster proposed independently from Mn EXAFS studies. Simulated MLS were constructed for each of the eight (or seven) doublet states of the Mn(4) cluster in the WOC for the two viable oxidation models (3Mn(III)-1Mn(IV) or 3Mn(IV)-1Mn(III)), and using a wide range of axial Mn hyperfine tensors, with either coaxial or orthogonal tensor alignments. We find accurate simulations using the 3Mn(III)-1Mn(IV) oxidation model. In the dimer-of-dimers coupling model, the spin state conversion between two doublet states |S(12),S(34),S(T)|(7)/(2),4,(1)/(2)> and |(7)/(2),3,(1)/(2)> is found to explain the large (25%) contraction in the hyperfine splitting observed upon conversion from the native MLS to the NH(3)-bound MLS. Stabilization of this excited state as the new ground state is caused by change in the intermanganese exchange coupling, without appreciable change in the intrinsic hyperfine tensors. The lack of good simulations of the Ca(2+)-depleted MLS suggests that Ca(2+)-depletion changes both Mn ligation and intermanganese exchange coupling. The 3Mn(IV)-1Mn(III) oxidation model is disfavored because only approximate simulations could be found for the native MLS and no agreement with the NH(3)-bound MLS was obtained. The scalar part of the hyperfine tensors for both Mn(III) and Mn(IV) ions were found to approximate (+/-5%) the values for the dimanganese(III,IV) catalase enzyme, suggesting similar overall ligand types. However, the large (30%) anisotropic part of the Mn(III) hyperfine interaction is opposite in sign to that found in all tetragonally extended six-coordinate Mn(III) ions (i.e., the usual Jahn-Teller splitting). The distribution of spin density from the high-spin d(4) electron configuration of each Mn(III) ion corresponds to a flattened (oblate) ellipsoid. This electronic distribution is favored in five-coordinate ligand fields having trigonally compressed bipyramidal geometry, but it could also arise, in principle, in strained six-coordinate ligand fields having tetragonally compressed geometry, i.e. [Mn(2)(&mgr;-O)](4+) (reverse Jahn-Teller distortion). The resulting valence electronic configurations are described as e'(2)e"(2) and (d(pi))(3)(d(x)()()2(-)(y)()()2)(1), respectively, in contrast to the (d(pi))(3)(d(z)()()2)(1) configuration common to unstrained six-coordinate tetragonally-extended Mn(III) ions, such as found in the [Mn(2)(&mgr;-O)(2)](3+) core in several synthetic dimers and catalase. Both of the former geometries predict strongly oxidizing Mn(III) ions, thereby suggesting a structural basis for the oxidative reactivity of the Mn(4) cluster in the WOC. The magnetic model needed to explain the MLS is not readily reconciled with the simplest structural and electronic models deduced from EXAFS studies of the WOC.

Anatomy of Best Estimate

Every engineering computation is attended by some degree of uncertainty due to factors not taken into account and limitations of data. As a result, computed values differ from true values in nature, and the size and frequency of these differences determine if any particular estimate is a best estimate. A best estimate of any variable may be defined as one that produces an unbiased least-squares estimate in the objective variable, defined as the variable that is critical in application. It is not necessarily a least-squares estimate of that particular working variable. In project design, there is always some probability of overdesign and some probability of underdesign (due to uncertainties), and the consequences can be vastly different. If the objective variable is net expected cost where underdesign and overdesign are each appropriately penalized, minimizing net expected cost will automatically provide an appropriate factor of safety and consequently a best design.

On the use of term values in the least-squares fitting of spectra

The unbiased least-squares method of Curl and Dane ( J. Mol. Spectrosc. 128, 406–412 (1988)) for fitting spectra with perturbed upper states, which involves an initial determination of lowerstate constants and upper-state term values followed by a determination of upper-state constants from the term values, requires the inversion of the large variance-covariance matrix of the upperstate term values in the second step. It is shown here that this inverse can be calculated by a method which involves the numerical inversion of only smaller matrices, so that the numerical difficulties in inverting large matrices can be avoided.

Unbiased least-squares fitting of lower states

An unbiased least-squares method to determine lower state rotational, spin-rotational, and centrifugal distortion constants for vibrational or electronic spectra with perturbed upper states is described. Continuation to fitting procedures for the determination of upper state constants is discussed.

On-line system identification in real time using a minicomputer

The parameters of the dynamic model of a linear system are estimated from the measurements of the input and output data by using the recursive algorithm for matrix pseudoinverse. For measurements contaminated with noise, the stochastic approximation algorithm is employed in conjunction with the matrix pseudoinverse algorithm for obtaining unbiased least-squares estimates. Both the methods are tested on-line in real time using the PDP 11/45 minicomputer while the system is simulated on a TR-20 analogue computer. The results confirm the feasibility of using the algorithms to identify the parameters of a class of industrial processes on-line using a minicomputer. This has special application to the adaptive control of such processes, based on rapid identification of slowly-varying parameters.