Simultaneous Decompositions Research Articles

On the complementary nature of ANOVA simultaneous component analysis (ASCA+) and Tucker3 tensor decompositions on designed multi‐way datasets

AbstractThe complementary nature of analysis of variance (ANOVA) Simultaneous Component Analysis (ASCA+) and Tucker3 tensor decompositions is demonstrated on designed datasets. We show how ASCA+ can be used to (a) identify statistically sufficient Tucker3 models; (b) identify statistically important triads making their interpretation easier; and (c) eliminate non‐significant triads making visualization and interpretation simpler. For multivariate datasets with an experimental design of at least two factors, the data matrix can be folded into a multi‐way tensor. ASCA+ can be used on the unfolded matrix, and Tucker3 modeling can be used on the folded matrix (tensor). Two novel strategies are reported to determine the statistical significance of Tucker3 models using a previously published dataset. A statistically sufficient model was created by adding factors to the Tucker3 model in a stepwise manner until no ASCA+ detectable structure was observed in the residuals. Bootstrap analysis of the Tucker3 model residuals was used to determine confidence intervals for the loadings and the individual elements of the core matrix and showed that 21 out of 63 core values of the 3 × 7 × 3 model were not significant at the 95% confidence level. Exploiting the mutual orthogonality of the 63 triads of the Tucker3 model, these 21 factors (triads) were removed from the model. An ASCA+ backward elimination strategy is reported to further simplify the Tucker3 3 × 7 × 3 model to 36 core values and associated triads. ASCA+ was also used to identify individual factors (triads) with selective responses on experimental factors A, B, or interactions, A × B, for improved model visualization and interpretation.

Decomposing Random Permutations into Order-Isomorphic Subpermutations

Two permutations and are -similar if they can be decomposed into subpermutations and such that is order-isomorphic to for all . Recently, Dudek, Grytczuk, and Ruciński Variations on twins in permutations, Electron. J. Combin., 28 (2021), P3.19. posed the problem of determining the minimum for which two permutations chosen independently and uniformly at random are -similar. We show that two such permutations are -similar with high probability, which is tight up to a polylogarithmic factor. Our result also generalizes to simultaneous decompositions of multiple permutations.

Non-Linear Load Characterisation Using Orthogonal Apparent Power Decompositions

There is no generally accepted definition for apparent power components in circuits with non-sinusoidal conditions. The decomposition of apparent power has been the subject of many discussions and is still an open topic. Over the years, a number of apparent power decompositions have been proposed and analysed, usually in the context of compensation. These analyses were performed by means of simulations or by calculating quantities using measurements on a simple test circuit. Here, we present a unique system capable of simultaneous apparent power decompositions for non-sinusoidal conditions in real-time. The system can calculate power components related to various power decompositions. The system’s capabilities are demonstrated on non-linear load characterization.

Catalecticant intersections and confinement of decompositions of forms

We introduce the notion of confinement of decompositions for forms or vector of forms. The confinement, when it holds, lowers the number of parameters that one needs to consider, in order to find all the possible decompositions of a given set of data. With the technique of confinement, we obtain here two results. First, we give a new, shorter proof of a result by London (1890) that 3 general plane cubics have 2 simultaneous Waring decompositions of rank 6. Then we compute, with the software Bertini, that 4 general plane quartics have 18 different decompositions of rank 10 (a result which was not known before).

Structure, Properties and Applications of Some Simultaneous Decompositions for Quaternion Matrices Involving $$\phi $$ ϕ -Skew-Hermicity

Let $${\mathbb {H}}$$ be the real quaternion algebra and $${\mathbb {H}}^{m\times n}$$ denote the set of all $$m\times n$$ matrices over $${\mathbb {H}}$$ . For $$A\in {\mathbb {H}}^{m\times n},$$ we denote by $$A_{\phi }$$ the $$n\times m$$ matrix obtained by applying $$\phi $$ entrywise to the transposed matrix $$A^{t},$$ where $$\phi $$ is a nonstandard involution of $${\mathbb {H}}$$ . $$A\in {\mathbb {H}}^{n\times n}$$ is said to be $$\phi $$ -skew-Hermitian if $$A=-A_{\phi }$$ . In this paper, we investigate and analyze in detail the structure and properties of a simultaneous decomposition for six quaternion matrices involving $$\phi $$ -skew-Hermicity: where A and F are $$\phi $$ -skew-Hermitian. Using this simultaneous decomposition, we give some practical necessary and sufficient conditions for the existence of a $$\phi $$ -skew-Hermitian solution (X, Y, Z) to the system of quaternion matrix equations Apart from proving an expression for the general $$\phi $$ -skew-Hermitian solution to this system, we derive the $$\beta (\phi )$$ -signature bounds of the $$\phi $$ -skew-Hermitian solution in terms of the coefficient matrices. Moreover, we obtain necessary and sufficient conditions for the system to have $$\beta (\phi )$$ -positive definite, $$\beta (\phi )$$ -positive semidefinite, $$\beta (\phi )$$ -negative definite and $$\beta (\phi )$$ -negative semidefinite solutions. We also give some numerical examples to illustrate our results.

Specificity of mental transformations involved in understanding spatial structures

Spatial abilities are thought to play an important role in learning from visualizations. The specificity of spatial abilities for particular mental transformations involved in spatial learning from visual media (rotation, perspective taking, spatial integration) was investigated. For each of the transformations, a dedicated criterion task was constructed and a specific ability was measured. Furthermore, reasoning ability (g) was considered. Hierarchical multiple regression analyses and simultaneous decompositions of variance were utilized. Mental rotation was a specific predictor of the rotation transformation, perspective transformation was predicted both by perspective taking ability and visual-spatial working memory (VSWM) capacity, and mental integration was predicted by reasoning ability, but not by VSWM capacity, contrary to expectations. It is concluded that correspondences between criterion tasks and ability measures may differ although links between ability and task requirements appear conceivable.

Subspace Identification for Linear Periodically Time-varying Systems

In this paper, an extension of the output-only subspace identification, to the class of linear periodically time-varying (LPTV) systems, is proposed. The goal is to identify a useful information about the system's stability using the Floquet theory which gives a necessary and sufficient condition for stability analysis. This information is retrieved from a matrix called the monodromy matrix, which is extracted by some simultaneous singular value decompositions (SVD) and from a resolution of a least squares criterion. The method is, finally, illustrated by a simulation of a model of a helicopter with a hinged-blades rotor.

Increase of achievable film thickness by UV-lamp irradiation in a fluorine-free metal-organic deposition process of YBa 2Cu 3O 7

We have studied the effects of UV-lamp irradiation in a fluorine-free metal-organic deposition (FF-MOD) process for epitaxial YBa 2Cu 3O 7 (YBCO) films. We found that the irradiation dramatically improves the elemental homogeneity in a precursor film, and increases the thickness of epitaxial growth by twofold with respect to the same process without irradiation. We obtained a 1.05-μm-thick epitaxial YBCO film, which is thickest for FF-MOD processes. A possible reason for the improved element uniformity is simultaneous decompositions of three metal-organic ingredients by photons, which strongly reduce grain growths in a precursor film that have been known as an unavoidable feature in the conventional all-pyrolytic MOD.

A simultaneous decomposition of a matrix triplet with applications

Researches on ranks of matrix expressions have posed a number of challenging questions, one of which is concerned with simultaneous decompositions of several given matrices. In this paper, we construct a simultaneous decomposition to a matrix triplet (A, B, C), where A=±A*. Through the simultaneous matrix decomposition, we derive a canonical form for the matrix expressions A−BXB*−CYC* and then solve two conjectures on the maximal and minimal possible ranks of A−BXB*−CYC* with respect to X=±X* and Y=±Y*. As an application, we derive a sufficient and necessary condition for the matrix equation BXB* + CYC*=A to have a pair of Hermitian solutions, and then give the general Hermitian solutions to the matrix equation. Copyright © 2010 John Wiley & Sons, Ltd.

Computing the complete CS decomposition

An algorithm for computing the complete CS decomposition of a partitioned unitary matrix is developed. Although the existence of the CS decomposition (CSD) has been recognized since 1977, prior algorithms compute only a reduced version. This reduced version, which might be called a 2-by-1 CSD, is equivalent to two simultaneous singular value decompositions. The algorithm presented in this article computes the complete 2-by-2 CSD, which requires the simultaneous diagonalization of all four blocks of a unitary matrix partitioned into a 2-by-2 block structure. The algorithm appears to be the only fully specified algorithm available. The computation occurs in two phases. In the first phase, the unitary matrix is reduced to bidiagonal block form, as described by Sutton and Edelman. In the second phase, the blocks are simultaneously diagonalized using techniques from bidiagonal SVD algorithms of Golub, Kahan, Reinsch, and Demmel. The algorithm has a number of desirable numerical features.

Higher Multifractal Analysis for Simultaneous Actions of Fuchsian Groups

We consider groups Γ1, Γ2, …, Γ d acting on the hyperbolic plane H 2 and by the method of Series for coding hyperbolic geodesics we can do simultaneous multifractal decompositions from the action of these groups. We obtain a variational formula for a dimension of the multifractal sets which generalizes that presented in our previous article for d = 1.

The Waring Locus of Binary Forms#

For integers m, d, n, we study the locus of m-dimensional subspaces of degree d binary forms whose elements admit simultaneous decompositions as sums of powers of n linear forms. We show that this locus has rational singularities and it is arithmetically Cohen–Macaulay in its natural Plücker imbedding. We describe invariant theoretic formulations for the equations of these loci.

Computation of the Canonical Decomposition by Means of a Simultaneous Generalized Schur Decomposition

The canonical decomposition of higher-order tensors is a key tool in multilinear algebra. First we review the state of the art. Then we show that, under certain conditions, the problem can be rephrased as the simultaneous diagonalization, by equivalence or congruence, of a set of matrices. Necessary and sufficient conditions for the uniqueness of these simultaneous matrix decompositions are derived. In a next step, the problem can be translated into a simultaneous generalized Schur decomposition, with orthogonal unknowns [A.-J. van der Veen and A. Paulraj, IEEE Trans. Signal Process., 44 (1996), pp. 1136--1155]. A first-order perturbation analysis of the simultaneous generalized Schur decomposition is carried out. We discuss some computational techniques (including a new Jacobi algorithm) and illustrate their behavior by means of a number of numerical experiments.

Simultaneous decompositions of two states

Simultaneous decompositions of a pair of states into pure ones are examined. There are privileged decompositions which are distinguished from all the other ones.

Importance of Orthogonalization Algorithm in Modeling Conditional Distributions by Orthogonal Transformed Indicator Methods

The orthogonal transformed indicator approach to conditional cumulative distribution functions is based on the decomposition of the indicator variogram matrix as a matrix product. This paper explores the manner in which the decomposition algorithm affects the conditional cumulative distribution function as estimated by orthogonal transformed indicator kriging. Five decomposition algorithms are considered: spectral, Cholesky, symmetric, Cholesky-spectral, and simultaneous decompositions. Impact of the algorithms on spatial orthogonality and order relations problems is examined and their performances together with indicator kriging are compared using a real dataset.

Hedging American contingent claims with constrained portfolios

The valuation theory for American Contingent Claims, due to Bensous- san (1984) and Karatzas (1988), is extended to deal with constraints on portfolio choice, including incomplete markets and borrowing/short-selling constraints, or with different interest rates for borrowing and lending. In the unconstrained case, the classical theory provides a single arbitrage-free price u0; this is expressed as the supremum, over all stopping times, of the claim's expected discounted value under the equivalent martingale measure. In the presence of constraints, fu0g is replaced by an entire interval (hlow; hup) of arbitrage-free prices, with endpoints characterized as hlow = inf2D u; hup = sup 2D u . Here u is the analogue of u0, the arbitrage-free price with unconstrained portfolios, in an auxiliary market model M ; and the family fMg2D is suitably chosen, to contain the original model and to reflect the constraints on portfolios. For several such constraints, explicit computations of the endpoints are carried out in the case of the American call-option. The analysis involves novel results in martingale theory (including simultaneous Doob-Meyer decompositions), optimal stopping and stochastic con- trol problems, stochastic games, and uses tools from convex analysis.

Simultaneous decompositions of sequential machines

Large sequential machines are difficult to design, to optimize, to implement and to verify. Therefore, methods and CAD tools are needed that can decompose sequential machines. In this paper, we briefly describe the theoretical and practical results that were obtained in the field of simultaneous decompositions which divide the process described by a given sequential machine into a number of interacting parallel partial processes, each implemented by one partial machine.

Content-Driven Progressive Transmission of Grey-Scale Images

A lossless progressive transmission method for grey-scale images which concentrates early transmission efforts on areas of greater image information content is described. The receiver does not have a priori knowledge of which image areas are to receive preferential treatment, and the preferential level of resolution is the pixel. The method makes use of simultaneous geometric and information content decompositions. The method is computationally simple with a complexity which grows linearly with the number of pixels. Compression achieved approaches that obtained by nonprogressive lossless methods, and is approximately the same as for homogeneous progressive lossless methods. Extensions of the method for progressive transmission with limited distortion and greater compression are also discussed.

On some characterizations of the “star” partial ordering for matrices and rank subtractivity

The main result is that Drazin's “star” partial ordering A ⩽ ∗ B holds if and only if A ∠ B and B † −A † =(B−A) † , where A ⩽ ∗ B is defined by A ∗ A = A ∗ B and AA ∗ = BA ∗ , and where A ∠ B denotes rank subtractivity. Here A and B are m × n complex matrices and the superscript † denotes the Moore-Penrose inverse. Several other characterizations of A ⩽ ∗ B are given, with particular emphasis on what extra condition must be added in order that rank subtractivity becomes the stronger “star” order; a key tool is a new canonical form for rank subtractivity. Connections with simultaneous singular-value decompositions, Schur complements, and idempotent matrices are also mentioned.

Matrices and pairs of modules

It is proved that each matrix over a principal ideal ring is equivalent to some diagonal matrix. Partial results are obtained on the uniqueness of the diagonal form obtained. These results are obtained by specializing some general properties about simultaneous decompositions of a projective module and a homomorphic image of finite (composition) length over any ring. These general results are also specialized to obtain results about matrices and projective modules over hereditary prime rings.