Generalized Cramér–von Mises goodness-of-fit tests for multivariate distributions

A class of statistics for testing the goodness-of-fit for any multivariate continuous distribution is proposed. These statistics consider not only the goodness-of-fit of the joint distribution but also the goodness-of-fit of all marginal distributions, and can be regarded as generalizations of the multivariate Cramér–von Mises statistic. Simulation shows that these generalizations, using the Monte Carlo test procedure to approximate their finite-sample p-values, are more powerful than the multivariate Kolmogorov–Smirnov statistic.     

2‐DOF discrete‐time nonlinear ℋ︁∞‐filters

In this paper, a new theory of two‐degrees‐of‐freedom (2‐DOF)‐??_∞ and certainty‐equivalent filters is presented. Exact and approximate solutions to the nonlinear ??_∞ filtering problem using this class of filters are derived in terms of discrete‐time Hamilton–Jacobi–Isaacs equations. The expressions for the filter gains are determined as functions of the filter state and the system's output in contrast to earlier results. Hence, it is shown that coupled with the additional degree‐of‐freedom, these filters are a substantial improvement over the earlier 1‐DOF case. The theory presented is also generalized to n‐DOF filters, which bore strong connections to linear infinite‐impulse response filters and hence are generalizations of this class of filters to the nonlinear setting. Simulation results are also given to show the usefulness of the new approach. Copyright © 2009 John Wiley & Sons, Ltd.     

Computations with quasiseparable polynomials and matrices

In this paper, we survey several recent results that highlight an interplay between a relatively new class of quasiseparable matrices and univariate polynomials. Quasiseparable matrices generalize two classical matrix classes, Jacobi (tridiagonal) matrices and unitary Hessenberg matrices that are known to correspond to real orthogonal polynomials and Szegö polynomials, respectively. The latter two polynomial families arise in a wide variety of applications, and their short recurrence relations are the basis for a number of efficient algorithms. For historical reasons, algorithm development is more advanced for real orthogonal polynomials. Recent variations of these algorithms tend to be valid only for the Szegö polynomials; they are analogues and not generalizations of the original algorithms.     

On a coding theorem connected with entropy of order α and type β

We prove two generalizations of Shannon's inequality for the case of entropy of order α and type β in a simple way. Noiseless coding theorems are proved using the two generalizations.     

Generalized Bezoutian and Sylvester matrices in multivariable linear control

Generalized Bezoutian and Sylvester matrices are defined and discussed in this short paper. The relationship between these two forms of matrices is established. It is shown that the McMillan degree of a real rational function can be ascertained by checking the rank of either one of these generalized matrices formed using a polynomial matrix fraction decomposition of the prescribed transfer function matrix. Earlier established results by Rowe and Munro are obtained as a special case. Several theorems related to the rank testing and other properties of the generalized matrices are discussed and various research problems are listed in the conclusion.     

A class of marked invariant subspaces with an application to algebraic Riccati equations

Invariant subspaces of a matrix A are considered which are obtained by truncation of a Jordan basis of a generalized eigenspace of A. We characterize those subspaces which are independent of the choice of the Jordan basis. An application to Hamilton matrices and algebraic Riccati equations is given.     

Generalized band matrices and their inverses

The idea of defining the generalized band matrices is based on the recognition that several pattern matrices and their inverses have low rank submatrices in certain domains. Theoretical considerations concerning the generalized band matrices enable us to give uniform treatment for several well known classes of matrices like band matrices, block band matrices, band matrices with low rank corrections, sparse matrices and their inverses. Making use of the new notions of information content and of compact representation of matrices, the concept of proper matrices is extended for generalized band matrices. Some reduction algorithms are presented which help to discover certain hidden structural properties of the generalized band matrices. The theoretical results are enlightened by a large number of figures illustrating numerical examples. Work supported by the Progetto Finalizzato Calcolo Parallelo e Sistemi Informatici of CNR. Visiting Professor at the University of Pisa under the support of GNIM-CNR.     

Well-posedness and attainability of indefinite stochastic linear quadratic control in infinite time horizon

This paper is concerned with a stochastic linear-quadratic (LQ) problem in an infinite time horizon with multiplicative noises both in the state and the control. A distinctive feature of the problem under consideration is that the cost weighting matrices for the state and the control are allowed to be indefinite. A new type of algebraic Riccati equation – called a generalized algebraic Riccati equation (GARE) – is introduced which involves a matrix pseudo-inverse and two additional algebraic equality/inequality constraints. It is then shown that the well-posedness of the indefinite LQ problem is equivalent to a linear matrix inequality (LMI) condition, whereas the attainability of the LQ problem is equivalent to the existence of a “stabilizing solution” to the GARE. Moreover, all possible optimal controls are identified via the solution to the GARE. Finally, it is proved that the solution to the GARE can be obtained via solving a convex optimization problem called semidefinite programming.     

No free lunch theorems for optimization

A framework is developed to explore the connection between effective optimization algorithms and the problems they are solving. A number of “no free lunch” (NFL) theorems are presented which establish that for any algorithm, any elevated performance over one class of problems is offset by performance over another class. These theorems result in a geometric interpretation of what it means for an algorithm to be well suited to an optimization problem. Applications of the NFL theorems to information-theoretic aspects of optimization and benchmark measures of performance are also presented. Other issues addressed include time-varying optimization problems and a priori “head-to-head” minimax distinctions between optimization algorithms, distinctions that result despite the NFL theorems' enforcing of a type of uniformity over all algorithms     

A note on the recursions of multichannel complex subset autoregressions

The recursive algorithm to select the optimum multivariate real subset autoregressive model (AR) [1] is generalized to apply to multichannel complex subset AR's. It is initiated by fitting all "forward" and "backward" one-lag AR's. The method then allows one to develop successively all complex subset AR's of sizek(the number of lags with nonzero coefficient matrices) from 1 toK. Finally, the best subsets of each size with the minimum generalized residual power for that size are compared to any one of three model selection criteria to find the optimum multichannel complex subset AR.     

集值决策信息系统的知识约简与规则提取

本文用集值信息系统描述不完备信息系统。在集值信息系统中定义了两种不同的关系：相容关系和优势关系,给出了两种不同关系下集合的上下近似概念及其性质。研究了集值决策信息系统在两种不同关系下的广义决策约简,得到了广义决策约简的判定定理和辨识矩阵,从而得到了约简的具体操作方法。最后,基于两种关系从集值决策信息系统提取了最优广义决策规则,并进行了讨论。     

A comparison of generalized linear discriminant analysis algorithms

Linear discriminant analysis (LDA) is a dimension reduction method which finds an optimal linear transformation that maximizes the class separability. However, in undersampled problems where the number of data samples is smaller than the dimension of data space, it is difficult to apply LDA due to the singularity of scatter matrices caused by high dimensionality. In order to make LDA applicable, several generalizations of LDA have been proposed recently. In this paper, we present theoretical and algorithmic relationships among several generalized LDA algorithms and compare their computational complexities and performances in text classification and face recognition. Towards a practical dimension reduction method for high dimensional data, an efficient algorithm is proposed, which reduces the computational complexity greatly while achieving competitive prediction accuracies. We also present nonlinear extensions of these LDA algorithms based on kernel methods. It is shown that a generalized eigenvalue problem can be formulated in the kernel-based feature space, and generalized LDA algorithms are applied to solve the generalized eigenvalue problem, resulting in nonlinear discriminant analysis. Performances of these linear and nonlinear discriminant analysis algorithms are compared extensively.     

Splines over regular triangulations in numerical simulation

We investigate the use of smooth spline spaces over regular triangulations as a tool in (isogeometric) Galerkin methods. In particular, we focus on box splines over three-directional meshes. Box splines are multivariate generalizations of univariate cardinal B-splines sharing the same properties. Tensor-product B-splines with uniform knots are a special case of box splines. The use of box splines over three-directional meshes has several advantages compared with tensor-product B-splines, including enhanced flexibility in the treatment of the geometry and stiffness matrices with stronger sparsity. Boundary conditions are imposed in a weak form to avoid the construction of special boundary functions. We illustrate the effectiveness of the approach by means of a selection of numerical examples.     

The bivariate generalized linear failure rate distribution and its multivariate extension

The two-parameter linear failure rate distribution has been used quite successfully to analyze lifetime data. Recently, a new three-parameter distribution, known as the generalized linear failure rate distribution, has been introduced by exponentiating the linear failure rate distribution. The generalized linear failure rate distribution is a very flexible lifetime distribution, and the probability density function of the generalized linear failure rate distribution can take different shapes. Its hazard function also can be increasing, decreasing and bathtub shaped. The main aim of this paper is to introduce a bivariate generalized linear failure rate distribution, whose marginals are generalized linear failure rate distributions. It is obtained using the same approach as was adopted to obtain the Marshall-Olkin bivariate exponential distribution. Different properties of this new distribution are established. The bivariate generalized linear failure rate distribution has five parameters and the maximum likelihood estimators are obtained using the EM algorithm. A data set is analyzed for illustrative purposes. Finally, some generalizations to the multivariate case are proposed.     

Generalized -KKM type theorems in topological spaces with an application

In this paper, we first introduce a new concept of generalized L-KKM mapping and establish some new generalized L-KKM type theorems without any convexity structure in topological spaces. As an application, an existence theorem of equilibrium points for an abstract generalized vector equilibrium problem is proved in topological spaces. The results presented in this paper unify and generalize some known results in recent literature.     

A characterization theorem of multivariate splines in blossoming form

It is known that polynomials in m variables of total degree n are equivalent to symmetric polynomials in n variables that are linear in each single variable. This principle, called the blossoming principle, has applied to the study of multivariate splines in this paper. For any spline on a simplicial partition Δ, a smoothness condition on its polynomial pieces on any two simplices of Δ which may not be adjacent is given. This smoothness condition presented in blossoming form generalizes the well-known smoothness conditions in B-form.     

Fast computation of the Smith form of a sparse integer matrix

We present a new probabilistic algorithm to compute the Smith normal form of a sparse integer matrix . The algorithm treats A as a “black box”—A is only used to compute matrix-vector products and we do not access individual entries in A directly. The algorithm requires about black box evaluations for word-sized primes p and , plus additional bit operations. For sparse matrices this represents a substantial improvement over previously known algorithms. The new algorithm suffers from no “fill-in” or intermediate value explosion, and uses very little additional space. We also present an asymptotically fast algorithm for dense matrices which requires about bit operations, where O(MM(m)) operations are sufficient to multiply two matrices over a field. Both algorithms are probabilistic of the Monte Carlo type — on any input they return the correct answer with a controllable, exponentially small probability of error. Received: March 9, 2000.     

Generalizations of the Continuous Logic

An order logic is studied, i.e., a generalization of the continuous logic to the case in an arbitrary argument of order r is determined instead of the operations of determination of maximum (disjunction) and minimum (conjunction). The new operation is expressed as a superposition of disjunctions and conjunctions of the continuous logic. Different classes of logical determinants—numeric characteristics of matrices that can be expressed through the operations of the continuous logic—are studied. Order determinants that are the generalizations of order logic operations to arguments in matrix form are studied. Determinants with different types of constraints on matrix subsets defining the matrix characteristic are described. For logical determinants, the properties that partly resemble the properties of algebraic determinants and computation formulas based on the operations of continuous logic are described. A predicate decision algebra generalizing the continuous logic to modeling of discontinuous functions is elaborated. A hybrid logic algebra is generalized to hybrid (continuous and discrete) variables. A logical arithmetic algebra, which includes continuous logical operations along with the four arithmetical operations, is described. A complex logic algebra in which the carrier set C is a field of complex numbers is developed. For all these logical algebras, main laws are formulated and their similarity to and distinction from the laws of the continuous logic are described. Generalizations of continuous logic operations as operations over matrices and random and interval variables are investigated. Their fields of application are described.     

New necessary and sufficient conditions for local controllability and local observability of 2-D separable denominator systems

New necessary and sufficient conditions for local controllability and observability of 2-D separable denominator systems (SDS's) are presented on the basis of the reduced-dimensional decomposition of 2-D SDS's. It is proved that local controllability of a 2-D SDS is equivalent to controllability of two 1-D systems which are a special decomposition pair of the 2-D SDS. Furthermore, local observability of a 2-D SDS is equivalent to a full rank condition of the coefficient matrices of the 2-D SDS in addition to observability of two 1-D systems which are another special decomposition pair of the 2-D SDS. Thus, these new conditions which use only 1-D controllability matrices, 1-D observability matrices, and coefficient matrices of the 2-D SDS are much simpler than any previous conditions which use complex 2-D controllability and observability matrices.