On the Optimal Parameters for Reconstruction and Forecasting in Singular Spectrum Analysis

There are two main parameters in Singular Spectrum Analysis (SSA). The aim of this study is to determine whether the optimal values of these parameters are different for reconstruction and forecasting stages, and if those are worth the extra computational effort and time which they require. Here, we evaluate these issues using simulation study.     

The Density Functions of the Singular Quaternion Normal Matrix and the Singular Quaternion Wishart Matrix

In this article, we give the density functions of the singular quaternion normal matrix and the singular quaternion Wishart matrix. Furthermore, we also give the density functions of certain singular quaternion β-matrix and the singular quaternion F-matrix in terms of the density function of the singular quaternion Wishart matrix and hypergeometric functions of quaternion matrix argument.     

The Extended Skew-Exponential Power Distribution and Its Derivation

We consider an extended family of asymmetric univariate distributions generated using a symmetric density, f, and the cumulative distribution function, G, of a symmetric distribution, which depends on two real-valued parameters λ and β and is such that when β = 0 it includes the entire class of distributions with densities of the form g(z | λ) = 2 G(λ z) f(z). A key element in the construction of random variables distributed according to the family is that they can be represented stochastically as the product of two random variables. From this representation we can readily derive theoretical properties, easy-to-implement simulation schemes, as well as extensions to the multivariate case and an explicit procedure for obtaining the moments. We give special attention to the extended skew-exponential power distribution. We derive its information matrix in order to obtain the asymptotic covariance matrix of the maximum likelihood estimators. Finally, an application to a real data set is reported, which shows that the extended skew-exponential power model can provide a better fit than the skew-exponential power distribution.     

On hypothesis testing for Poisson processes. Singular cases

ABSTRACT

We consider the problem of hypothesis testing in the situation where the first hypothesis is simple and the second one is local one-sided composite. Wedescribe the choice of the thresholds and the power functions of different tests when the intensity function of the observed inhomogeneous Poisson process has two different types of singularity: cusp and discontinuity. The asymptotic results are illustrated by numerical simulations.     

奇异线性系统E_(n×n)=A_(n×n)X初值问题的通解公式

给出了奇异线性系统E_(nxn~x)=A_(nxn~x)的通解公式,并研究了初值问题解的结构.     

A comparison of stepwise common singular spectrum analysis and horizontal multi-channel singular spectrum analysis

The problem of forecasting a time series by using information provided by a second time series is considered. Two multivariate extensions of Singular Spectrum Analysis (SSA) are compared in terms of forecast error: Horizontal Multi-channel SSA and Stepwise Common SSA. Different signal structures, defined in terms of trend, period, amplitude and phase, are investigated. In broad terms we find that neither Horizontal Multichannel SSA nor Stepwise Common SSA is best in all cases. Horizontal MSSA is outperformed particularly in cases where different trends are considered.     

On linear regression designs which maximize information

Necessary and sufficient conditions are established when a continuous design contains maximal information for a prescribed s-dimensional parameter in a classical linear model. The development is based on a thorough study of a particular dual problem and its interplay with the optimal design problem, extending partial results and earlier approaches based on differential calculus, game theory, and other programming methods. The results apply in particular to a class of information functionals which covers c-, D-, A-, L-optimality, they include a complete account of the non-differentiable criterion of E-optimality, and provide a constructive treatment of those situations in which the information matrix is singular. Corollaries pertain to the case of s out of k parameters, simultaneous optimality with respect to several criteria, multiplicity of optimal designs, bounds on their weights, and optimality which is induced by admissibility.     

Modelling spatial intensity for replicated inhomogeneous point patterns in brain imaging

Summary.  Pharmacological experiments in brain microscopy study patterns of cellular activ- ation in response to psychotropic drugs for connected neuroanatomic regions. A typical ex- perimental design produces replicated point patterns having highly complex spatial variability. Modelling this variability hierarchically can enhance the inference for comparing treatments. We propose a semiparametric formulation that combines the robustness of a nonparametric kernel method with the efficiency of likelihood-based parameter estimation. In the convenient framework of a generalized linear mixed model, we decompose pattern variation by kriging the intensities of a hierarchically heterogeneous spatial point process. This approximation entails discretizing the inhomogeneous Poisson likelihood by Voronoi tiling of augmented point patterns. The resulting intensity-weighted log-linear model accommodates spatial smoothing through a reduced rank penalized linear spline. To correct for anatomic distortion between subjects, we interpolate point locations via an isomorphic mapping so that smoothing occurs relative to common neuroanatomical atlas co-ordinates. We propose a criterion for choosing the degree and spatial locale of smoothing based on truncating the ordered set of smoothing covariates to minimize residual extra-dispersion. Additional spatial covariates, experimental design factors, hierarchical random effects and intensity functions are readily accommodated in the linear predictor, enabling comprehensive analyses of the salient properties underlying replicated point patterns. We illustrate our method through application to data from a novel study of drug effects on neuronal activation patterns in the brain of rats.     

The dual-projection approach for two-mode networks

There have been two distinct approaches to two-mode data. The first approach is to project the data to one-mode and then analyze the projected network using standard single-mode techniques, also called the conversion method. The second approach has been to extend methods and concepts to the two-mode case and analyze the network directly with the two modes considered jointly. The direct approach in recent years has been the preferred method since it is assumed that the conversion method loses important structural information. Here we argue that this is not the case, provided both projections are used together in any analysis. We illustrate how this approach works using core/periphery, structural equivalence and centrality as examples.     

Singular Gaussian graphical models: Structure learning

ABSTRACT

The goal of this article is to introduce singular Gaussian graphical models and their conditional independence properties. In fact, we extend the concept of Gaussian Markov Random Field to the case of a multivariate normally distributed vector with a singular covariance matrix. We construct, then, the associated graph’s structure from the covariance matrix’s pseudo-inverse on the basis of a characterization of the pairwise conditional independence. The proposed approach can also be used when the covariance matrix is ill-conditioned, through projecting data on a smaller subspace. In this case, our method ensures numerical stability and consistency of the constructed graph and significantly reduces the inference problem’s complexity. These aspects are illustrated using numerical experiments.