Square Root Of Matrix Research Articles

A New Adaptive Square-Root Unscented Kalman Filter for Nonlinear Systems

This paper describes a new adaptive filtering approach for nonlinear systems with additive noise. Based on Square-Root Unscented Kalman Filter (SRUKF), the traditional Maybeck’s estimator is modified and extended to the nonlinear systems, the estimation of square root of the process noise covariance matrix Q or measurement noise covariance matrix R is obtained straightforwardly. Then the positive semi-definiteness of Q or R is guaranteed, some shortcomings of traditional Maybeck’s algorithm are overcome, so the stability and accuracy of the filter is improved greatly.

A Note on the Square Roots of a Class of Circulant Matrices

We prove that anyk-circulant matrix and any even order skewk-circulant matrix are diagonalizable for anyk∈ℂ. Then, we propose two algorithms for computing the square roots of thek-circulant matrix and the skewk-circulant matrix, respectively. In particular, we show that the square roots of thek-circulant matrix are stillk-circulant matrices. Both the theoretical analysis and the numerical experiments show that our algorithms are faster than the standard Schur method.

The Square Root of a Matrix

AbstractThis note derives an explicit formula for the numerical calculation of the square root of a matrix, when this function exists. An example is given as an illustration of the formula. The condition for the existence of the square root is also illustrated.

Adaptive linear discriminant analysis for online feature extraction

In this paper, we present new adaptive algorithms for the computation of the square root of the inverse covariance matrix. In contrast to the current similar methods, these new algorithms are obtained from an explicit cost function that is introduced for the first time. The new adaptive algorithms are used in a cascade form with a well-known adaptive principal component analysis to construct linear discriminant features. The adaptive nature and fast convergence rate of the new adaptive linear discriminant analysis algorithms make them appropriate for online pattern recognition applications. All adaptive algorithms discussed in this paper are trained simultaneously using a sequence of random data. Experimental results using the synthetic and real multiclass, multidimensional input data demonstrate the effectiveness of the new adaptive algorithms to extract the optimal features for the purpose of classification.

Efficient methods for large‐scale linear inversion using a geostatistical approach

In geophysical inverse problems, such as estimating the unknown parameter field from noisy observations of dependent quantities, e.g., hydraulic conductivity from head observations, stochastic Bayesian and geostatistical approaches are frequently used. To obtain best estimates and conditional realizations it is required to perform several matrix‐matrix computations involving the covariance matrix of the discretized field of the parameters. In realistic three‐dimensional fields that are finely discretized, these operations as performed in conventional algorithms become extremely expensive and even prohibitive in terms of memory and computational requirements. Using Hierarchical Matrices, we show how to reduce the complexity of forming approximate matrix‐vector products involving the Covariance matrices in log linear complexity for an arbitrary distribution of points and a wide variety of generalized covariance functions. The resulting system of equations is solved iteratively using a matrix‐free Krylov subspace approach. Furthermore, we show how to generate unconditional realizations using an approximation to the square root of the covariance matrix using Chebyshev matrix polynomials and use the above to generate conditional realizations. We demonstrate the efficiency of our method on a few standard test problems, such as interpolation from noisy observations and contaminant source identification.

Combining the Ensemble Kalman Filter With Markov Chain Monte Carlo for Improved History Matching and Uncertainty Characterization

SummaryIt is well known that when applied to reservoir history-matching problems, the ensemble Kalman filter (EnKF) can lead to a large underestimation of uncertainty in the posterior probability-density function (PDF) for reservoir-model parameters. Here, we demonstrate that, regardless of whether covariance localization is used, EnKF also can lead to an overestimation of the uncertainty in future predictions of reservoir performance. This overestimation occurs because, even though the data matches obtained with EnKF tend to appear reasonable, these matches are significantly worse than those that can be obtained when history matching dynamic data with a gradient-based method. The relatively poor data match obtained with EnKF means that the ensemble of states generated by assimilating data with EnKF are of relatively low probability (low value of posterior PDF). Under reasonable assumptions, a Markov Chain Monte Carlo (MCMC) algorithm will theoretically generate an accurate sampling of the posterior PDF conditional to dynamic data. However, standard implementations of MCMC are not computationally feasible because each proposed state in the chain requires a run of the forward model (reservoir simulator). In this work, the EnKF and MCMC methodologies are combined to obtain a relatively efficient algorithm for sampling the posterior PDF for reservoir-model parameters. In this algorithm, a symmetric square root of the state vector's posterior covariance matrix is calculated from an ensemble of state vectors obtained from EnKF. It is shown that this square root can be used both to propose new states in the Markov chain and to evaluate the probability of accepting the state without running the simulator, which results in an efficient algorithm. In order to improve sampling, we also propose to generate multiple ensembles with EnKF and multiple Markov chains, which are then combined and resampled based on the normalized objective function. We tested the proposed EnKF-MCMC algorithm on a small 3D two-phase-flow reservoir example. The problem is sufficiently small that by using MCMC only, we are able to generate a long chain based on 2 million proposed states. The posterior PDF calculated from this long chain is assumed to be accurate. The posterior PDFs for predicted cumulative oil and water production generated with the new EnKF-MCMC algorithm provide results that are in reasonable agreement with the posterior PDFs for the cumulative oil and water production obtained from the long Markov chain, whereas the corresponding PDFs obtained using EnKF with and without covariance localization exhibit a significantly higher variance (i.e., overestimate uncertainty in the predicted cumulative reservoir production).

The Square Root of a Matrix

This note derives an explicit formula for the numerical calculation of the square root of a matrix, when this function exists. An example is given as an illustration of the formula. The condition for the existence of the square root is also given.

Structure-preserving Schur methods for computing square roots of real skew-Hamiltonian matrices

The contribution in this paper is two-folded. First, a complete characterization is given of the square roots of a real nonsingular skew-Hamiltonian matrix W. Using the known fact that every real skew-Hamiltonian matrix has infinitely many real Hamiltonian square roots, such square roots are described. Second, a structure-exploiting method is proposed for computing square roots of W, skew-Hamiltonian and Hamiltonian square roots. Compared to the standard real Schur method, which ignores the structure, this method requires significantly less arithmetic.

Superior performance of using hyperbolic sine activation functions in ZNN illustrated via time-varying matrix square roots finding

A special class of recurrent neural network (RNN), termed Zhang neural network (ZNN) depicted in the implicit dynamics, has recently been proposed for online solution of time-varying matrix square roots. Such a ZNN model can be constructed by using monotonically-increasing odd activation functions to obtain the theoretical time-varying matrix square roots in an error-free manner. Different choices of activation function arrays may lead to different performance of the ZNN model. Generally speaking, ZNN model using hyperbolic sine activation functions may achieve better performance, as compared with those using other activation functions. In this paper, to pursue the superior convergence and robustness properties, hyperbolic sine activation functions are applied to the ZNN model for online solution of time-varying matrix square roots. Theoretical analysis and computer-simulation results further demonstrate the superior performance of the ZNN model using hyperbolic sine activation functions in the context of large model-implementation errors, in comparison with that using linear activation functions.

Unconstrained Synthesis of Covariance Matrix for MIMO Radar Transmit Beampattern

Multiple-input multiple-output (MIMO) radars have many advantages over their phased-array counterparts: improved spatial resolution; better parametric identifiably and greater flexibility to design the transmit beampattern. The design of the transmit beampatterns generally requires the waveforms to have arbitrary auto- and cross-correlation properties. The correlation/covariance matrix, <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</b> , of the waveforms must be positive semidefinite, therefore synthesis of a desired beampattern is usually a constrained optimization problem. In this paper, to simplify the constrained optimization problem, two algorithms are proposed to synthesize the waveform covariance matrix for the desired beampattern. In the first proposed algorithm the elements of a square-root matrix of the covariance matrix <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</b> are parameterized using the coordinates of a hypersphere that implicitly fulfil the constraints. This yields an iterative algorithm, whose convergence speed can be increased significantly by providing good initial values. In the second algorithm the constraints and redundant information in the covariance matrix <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</b> are exploited to find a closed-form solution. The drawback of the second algorithm is that it may yield a pseudocovariance matrix (pseudo-CM) that is not guaranteed to be positive semidefinite. However, the pseudo-CM can be easily converted into a covariance matrix using eigenvalue decomposition and/or shrinkage methods. Moreover, a pseudo-CM can also be used to provide good initial values for the first algorithm to enable faster convergence.

Euclidean Nature of Phylogenetic Distance Matrices

Phylogenies are fundamental to comparative biology as they help to identify independent events on which statistical tests rely. Two groups of phylogenetic comparative methods (PCMs) can be distinguished: those that take phylogenies into account by introducing explicit models of evolution and those that only consider phylogenies as a statistical constraint and aim at partitioning trait values into a phylogenetic component (phylogenetic inertia) and one or multiple specific components related to adaptive evolution. The way phylogenetic information is incorporated into the PCMs depends on the method used. For the first group of methods, phylogenies are converted into variance-covariance matrices of traits following a given model of evolution such as Brownian motion (BM). For the second group of methods, phylogenies are converted into distance matrices that are subsequently transformed into Euclidean distances to perform principal coordinate analyses. Here, we show that simply taking the elementwise square root of a distance matrix extracted from a phylogenetic tree ensures having a Euclidean distance matrix. This is true for any type of distances between species (patristic or nodal) and also for trees harboring multifurcating nodes. Moreover, we illustrate that this simple transformation using the square root imposes less geometric distortion than more complex transformations classically used in the literature such as the Cailliez method. Given the Euclidean nature of the elementwise square root of phylogenetic distance matrices, the positive semidefinitiveness of the phylogenetic variance-covariance matrix of a trait following a BM model, or related models of trait evolution, can be established. In that way, we build a bridge between the two groups of statistical methods widely used in comparative analysis. These results should be of great interest for ecologists and evolutionary biologists performing statistical analyses incorporating phylogenies.

Lattice-reduction-aided MMSE equalization and the successive estimation of correlated data

Over the last years, novel low-complexity approaches to the equalization of MIMO channels have gained much attention. Thereby, methods based on lattice basis reduction are of special interest, as they achieve the optimum diversity order. In this paper, a tutorial overview on LRA equalization optimized according to the MMSE criterion is given. It is proven that applying the zero-forcing BLAST algorithm to a suitably augmented channel matrix – the inverse of the square root of the correlation matrix of the data symbols times the noise variance forms its lower part – results in the optimum solution. This fact is already widely used but lacks a formal proof. It turns out that it is more important to take the correlations of the data correctly into account than what type of lattice reduction actually is used.

Variable Tap-Length Mixed-Tone RLS-based Per-Tone Equalisation with Adaptive Implementation

In this paper, a methodology of mixed-tone recursive least squares algorithm development based on an orthogonal projection approach and a new variable tap-length mechanism is presented for per-tone equalisation (PTEQ) in discrete multitone systems. A mixed-tone cost function described as the sum of weight estimated errors is minimised to achieve the solutions for different per-tone equalisers simultaneously. We describe about the inverse square-root recursive least squares algorithm based upon the QRdecomposition which preserves the Hermitian symmetry of the inverse autocorrelation matrix by means of the product of square-root matrix and its Hermitian transpose. Such symmetrical property lends the benefit to the parallel implementation. In order to reduce the computational complexity, a new variable tap-length algorithm based on the sense of mean square mixted-tone errors is introduced to search a proper choice of tap-length of PTEQ. Simulation results show the improvement of achievable bit rate and signal to noise ratio performance as compared to the PTEQ exploiting conventional recursive least squares algorithm.

Why the phylogenetic regression appears robust to tree misspecification.

The phylogenetic comparative method uses estimates of evolutionary relationships to explicitly model the covariance structure of interspecific data. By accounting for common ancestry, the coevolution between 2 or more traits, as a response to one another or to environmental variables, can be studied without confounding similarities due to identity by descent. Because the true phylogeny is unknowable, an estimate must be used, introducing a source of error into phylogenetic comparative analysis that can be difficult to quantify. This manuscript aims to elucidate how tree misspecification is propagated through a comparative analysis. I focus on the phylogenetic regression under a Brownian motion model of evolution and consider the effect of local phylogenetic perturbations on the regression fit. Motivated by Felsenstein's method of independent contrasts, I derive a matrix square root of the phylogenetic covariance matrix that has an obvious phylogenetic interpretation. I use this result to transform the perturbed phylogenetic regression model into an ordinary linear regression in which one interpretable point has been affected. The simplicity of this formulation allows the contributions of data and phylogeny to be disentangled when studying the effect of tree misspecification. Consequentially, I find that branch length misspecification can be easily explained in terms of the reweighting of contrast scores between subtrees. An analytical consideration of this and other perturbations helps to explain why the phylogenetic regression appears generally to be robust to tree misspecification, and I am able to identify conditions under which the regression may not yield robust results. I discuss why soft polytomies do not meet these problematic conditions, leading to the conclusion that unresolved bifurcations should have only modest effects on the regression fit.

Efficient Local Error Parameterizations for Square Root or Ensemble Kalman Filters: Application to a Basin-Scale Ocean Turbulent Flow

Abstract In large-sized atmospheric or oceanic applications of square root or ensemble Kalman filters, it is often necessary to introduce the prior assumption that long-range correlations are negligible and force them to zero using a local parameterization, supplementing the ensemble or reduced-rank representation of the covariance. One classic algorithm to perform this operation consists of taking the Schur product of the ensemble covariance matrix by a local support correlation matrix. However, with this parameterization, the square root of the forecast error covariance matrix is no more directly available, so that any observational update algorithm requiring this square root must include an additional step to compute local square roots from the Schur product. This computation generates an additional numerical cost or produces high-rank square roots, which may deprive the observational update from its original efficiency. In this paper, it is shown how efficient local square root parameterizations can be obtained, for use with a specific square root formulation (i.e., eigenbasis algorithm) of the observational update. Comparisons with the classic algorithm are provided, mainly in terms of consistency, accuracy, and computational complexity. As an application, the resulting parameterization is used to estimate maps of dynamic topography characterizing a basin-scale ocean turbulent flow. Even with this moderate-sized system (a 2200-km-wide square basin with 100-km-wide mesoscale eddies), it is observed that more than 1000 ensemble members are necessary to faithfully represent the global correlation patterns, and that a local parameterization is needed to produce correct covariances with moderate-sized ensembles. Comparisons with the exact solution show that the use of local square roots is able to improve the accuracy of the updated ensemble mean and the consistency of the updated ensemble variance. With the eigenbasis algorithm, optimal adaptive estimates of scaling factors for the forecast and observation error covariance matrix can also be obtained locally at negligible additional numerical cost. Finally, a comparison of the overall computational cost illustrates the decisive advantage that efficient local square root parameterizations may have to deal simultaneously with a larger number of observations and avoid data thinning as much as possible.

Schrödinger equations for the square root density of an eigenmixture and the square root of an eigendensity spin matrix

We generalize a "one eigenstate" theorem of Levy, Perdew and Sahni (LPS) to the case of densities coming from eigenmixture density operators. The generalization is of a special interest for the radial density functional theory (RDFT) for nuclei, a consequence of the rotational invariance of the nuclear Hamiltonian; when nuclear ground states (GSs) have a finite spin, the RDFT uses eigenmixture density operators to simplify predictions of GS energies into one-dimensional, radial calculations. We also study Schr\"odinger equations governing spin eigendensity matrices.

Stochastic Integration for the Heterogeneous Correlation Modeling Using a Diffusion Equation

Abstract In this note, a stochastic integration scheme is proposed as an alternative to a deterministic integration scheme, usually employed for the diffusion operator in data assimilation. The stochastic integration scheme is no more than a simple interpolation of the initial condition in lieu of the deterministic integration. Furthermore, this also presents a potential in high performance computing. For the classic preconditioned minimizing problem, the stochastic integration is employed to implement the square root of the background error covariance matrix, while its adjoint is obtained from the adjoint code of the square root code. In a first part, the stochastic integration method and its weak convergence are detailed. Then the practical use of this approach in data assimilation is described. It is illustrated in a 1D test bed, where it is shown to run smoothly for background error covariance modeling, with nearest-neighbor interpolations, and O(100) particles.

Adaptive algorithms and networks for optimal feature extraction from Gaussian data

In this paper, new adaptive learning algorithms and correspondent networks are presented in order to extract optimal features from a sequence of multidimensional Gaussian data. For this purpose, novel adaptive algorithms for the estimation of the square root of the inverse covariance matrix Σ - 1 / 2 are introduced and applied for Gaussian optimal feature extraction. New adaptive algorithms are drawn by optimization of an appropriate cost function that is presented for the first time. Based on the proposed adaptive algorithms, related networks are implemented in order to extract optimal features from a sequence of multidimensional Gaussian data. Adaptive nature of the new feature extraction methods makes them appropriate for on-line pattern recognition applications. Experimental results using multidimensional Gaussian data demonstrated the effectiveness of the new adaptive feature extraction methods.

Efficient Adaptive Error Parameterizations for Square Root or Ensemble Kalman Filters: Application to the Control of Ocean Mesoscale Signals

Abstract In Kalman filter applications, an adaptive parameterization of the error statistics is often necessary to avoid filter divergence, and prevent error estimates from becoming grossly inconsistent with the real error. With the classic formulation of the Kalman filter observational update, optimal estimates of general adaptive parameters can only be obtained at a numerical cost that is several times larger than the cost of the state observational update. In this paper, it is shown that there exists a few types of important parameters for which optimal estimates can be computed at a negligible numerical cost, as soon as the computation is performed using a transformed algorithm that works in the reduced control space defined by the square root or ensemble representation of the forecast error covariance matrix. The set of parameters that can be efficiently controlled includes scaling factors for the forecast error covariance matrix, scaling factors for the observation error covariance matrix, or even a scaling factor for the observation error correlation length scale. As an application, the resulting adaptive filter is used to estimate the time evolution of ocean mesoscale signals using observations of the ocean dynamic topography. To check the behavior of the adaptive mechanism, this is done in the context of idealized experiments, in which model error and observation error statistics are known. This ideal framework is particularly appropriate to explore the ill-conditioned situations (inadequate prior assumptions or uncontrollability of the parameters) in which adaptivity can be misleading. Overall, the experiments show that, if used correctly, the efficient optimal adaptive algorithm proposed in this paper introduces useful supplementary degrees of freedom in the estimation problem, and that the direct control of these statistical parameters by the observations increases the robustness of the error estimates and thus the optimality of the resulting Kalman filter.

Verified Computation of Square Roots of a Matrix

We present methods to compute verified square roots of a square matrix $A$. Given an approximation $X$ to the square root, obtained by a classical floating point algorithm, we use interval arithmetic to find an interval matrix which is guaranteed to contain the error of $X$. Our approach is based on the Krawczyk method, which we modify in two different ways in such a manner that the computational complexity for an $n\times n$ matrix is reduced to $n^3$. The methods are based on the spectral decomposition or, in the case that the eigenvector matrix is ill conditioned, on a similarity transformation to block diagonal form. Numerical experiments prove that our methods are computationally efficient and that they yield narrow enclosures provided $X$ is a good approximation. This is particularly true for symmetric matrices, since their eigenvector matrix is perfectly conditioned.