Stationary Vector Research Articles

Structured prior distributions for the covariance matrix in latent factor models

Factor models are widely used for dimension reduction in the analysis of multivariate data. This is achieved through decomposition of a p×p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p \ imes p$$\\end{document} covariance matrix into the sum of two components. Through a latent factor representation, they can be interpreted as a diagonal matrix of idiosyncratic variances and a shared variation matrix, that is, the product of a p×k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p \ imes k$$\\end{document} factor loadings matrix and its transpose. If k≪p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k \\ll p$$\\end{document}, this defines a parsimonious factorisation of the covariance matrix. Historically, little attention has been paid to incorporating prior information in Bayesian analyses using factor models where, at best, the prior for the factor loadings is order invariant. In this work, a class of structured priors is developed that can encode ideas of dependence structure about the shared variation matrix. The construction allows data-informed shrinkage towards sensible parametric structures while also facilitating inference over the number of factors. Using an unconstrained reparameterisation of stationary vector autoregressions, the methodology is extended to stationary dynamic factor models. For computational inference, parameter-expanded Markov chain Monte Carlo samplers are proposed, including an efficient adaptive Gibbs sampler. Two substantive applications showcase the scope of the methodology and its inferential benefits.

Information flow and catalytic dyad in SARS-CoV2 main protease Mpro enzyme using embedded discrete Markov chains and centrality measures

Abstract In this work, we use a network representation of the globular crystalline structure of a given protein as a graph structure with N nodes and E edges in order to analyze quantitatively the flow information and to identify key sites within the globular structure. Each node nj represents a $ C_{\alpha}^{i}$ carbon of the main backbone whereas the node’s degree ki is a measure of its physical interactions. In order to identify biologically relevant and active nodes, we compute local per residue closeness $ C_{c}(i)$, betweenness $ C_{b}(i)$ and eigenvector centralities $ C_{e}(i)$. Further analysis is done by embedding a stochastic dynamic discrete Markov chain in order to evaluate the dynamics of a set of normal random walkers (NRW’s) within the network. From this, we compute the mean first passage time matrix M and the stationary occupation probability vector ψi for each node. These two measures provide very useful information on the dynamical process embed within the $ C_{\alpha}$ network. We apply this to SARS-CoV2 Mpro main protease which is a key enzyme in the virus replication cycle. In particular, we focus our attention to the properties of the catalytic dyad integrated by His41-Cys145 in Mpro main protease since this active site has been under intense scrutiny as a pharmaceutical target. In addition, our results show the existence of additional relevant aminoacids that might play a fundamental role on signal propagation and allosteric pathways in SARS-CoV main proteases.

Stationarity and Fredholm Theory in Subextremal Kerr-de Sitter Spacetimes

In a recent paper, we proved that solutions to linear wave equations in a subextremal Kerr-de Sitter spacetime have asymptotic expansions in quasinormal modes up to a decay order given by the normally hyperbolic trapping, extending the results of Vasy (2013). One central ingredient in the argument was a new definition of quasinormal modes, where a non-standard choice of stationary Killing vector field had to be used in order for the Fredholm theory to be applicable. In this paper, we show that there is in fact a variety of allowed choices of stationary Killing vector fields. In particular, the horizon Killing vector fields work for the analysis, in which case one of the corresponding ergoregions is completely removed.

Simulation algorithms for stationary sequences with distributions in the form of a mixture of Gaussian distributions

Abstract In this paper, we present three algorithms for simulation of intervals of stationary vector and scalar sequences with partial distributions of their subsequences of fixed length in the form of a mixture of Gaussian distributions. The first algorithm is based on superposition of two Gaussian vector processes and the second and third ones use the method of conditional distributions and the method of superpositions to simulate the mixtures and to select realizations for approximate construction of conditional realizations. Some properties of these algorithms are presented.

An investigation into the abnormal dynamic connection mechanism of generalized anxiety disorders based on non-homogeneous Markov models

BackgroundThe dynamic and hierarchical nature of the functional brain network. The neural dynamical systems tend to converge to multiple attractors (stable fixed points or dynamical states) in long run. Little is known about how the changes in this brain dynamic “long-term” behavior of the connectivity flow of brain network in generalized anxiety disorder (GAD). MethodsThis study recruited 92 patients with GAD and 77 healthy controls (HC). We applied a reachable probability approach combining a Non-homogeneous Markov model with transition probability to quantify all possible connectivity flows and the hierarchical structure of brain functional systems at the dynamic level and the stationary probability vector (10-step transition probabilities) to describe the steady state of the system in the long run. A random forest algorithm was conducted to predict the severity of anxiety. ResultsThe dynamic functional patterns in distributed brain networks had larger possibility to converge in bilateral thalamus, posterior cingulate cortex (PCC), right superior occipital gyrus (SOG) and smaller possibility to converge in bilateral superior temporal gyrus (STG) and right parahippocampal gyrus (PHG) in patients with GAD compared to HC. The abnormal transition probability pattern could predict anxiety severity in patients with GAD. LimitationsSmall samples and subjects taking medications may have influenced our results. Future studies are expected to rule out the potential confounding effects. ConclusionOur results have revealed abnormal dynamic neural communication and integration in emotion regulation in patients with GAD, which give new insights to understand the dynamics of brain function of patients with GAD.

Consistency of averaged impulse response estimators in vector autoregressive models

We show root‐T consistency of the smoothed AIC and smoothed BIC model averaging estimators (sAIC, sBIC) of impulse response coefficients in stationary vector autoregressive models of finite lag order. We also show that there is not one unique way to define the sAIC and sBIC estimators, but that instead there is a whole class of each of these defined by a weight scaling factor that allows the averaging estimator to become more similar to either its model selection counterpart or the equal weights averaging estimator. We also show asymptotic validity of a bootstrap method for estimating the averaging estimators' distributions. Simulations illustrate the benefits of using sAIC instead of AIC estimators.

Position-specific evolution in transcription factor binding sites, and a fast likelihood calculation for the F81 model.

Transcription factor binding sites (TFBS), like other DNA sequence, evolve via mutation and selection relating to their function. Models of nucleotide evolution describe DNA evolution via single-nucleotide mutation. A stationary vector of such a model is the long-term distribution of nucleotides, unchanging under the model. Neutrally evolving sites may have uniform stationary vectors, but one expects that sites within a TFBS instead have stationary vectors reflective of the fitness of various nucleotides at those positions. We introduce 'position-specific stationary vectors' (PSSVs), the collection of stationary vectors at each site in a TFBS locus, analogous to the position weight matrix (PWM) commonly used to describe TFBS. We infer PSSVs for human TFs using two evolutionary models (Felsenstein 1981 and Hasegawa-Kishino-Yano 1985). We find that PSSVs reflect the nucleotide distribution from PWMs, but with reduced specificity. We infer ancestral nucleotide distributions at individual positions and calculate 'conditional PSSVs' conditioned on specific choices of majority ancestral nucleotide. We find that certain ancestral nucleotides exert a strong evolutionary pressure on neighbouring sequence while others have a negligible effect. Finally, we present a fast likelihood calculation for the F81 model on moderate-sized trees that makes this approach feasible for large-scale studies along these lines.

Cross-spectral purity of nonstationary vector optical fields: A similarity with stationary fields

This study establishes a reduction formula for nonstationary cross-spectrally pure vector (or electromagnetic) light fields with any spectral bandwidth. In general, for nonstationary cross-spectrally pure electromagnetic fields, the reduction formula of the two-time Stokes parameters normalized by the corresponding usual Stokes parameters is not similar to the one for stationary fields since the correlation properties for nonstationary fields depend on two-time instants t1 and t2, in addition to their time difference. Here, we derive a reduction formula by including time-integrated correlations that share a similar mathematical structure to that of reduction formulas used for stationary vector fields if the normalized spectra of the interfering fields are identical. Additionally, we examine the requirement of strict cross-spectral purity for nonstationary electromagnetic fields using a time-integrated coherence function, which exhibits a mathematical form similar to that of strict cross-spectral purity in stationary vector fields. This investigation sheds light on the cross-spectral purity of pulse-type fields, which holds potential applications in statistical optics.

Fixed points of doubly stochastic operators with application to Markov chains

In this work, we characterize the fixed points of the doubly stochastic operators defined on discrete spaces ℓ p ( I ) , where I is a nonempty set and p ∈ [ 1 , + ∞ ) . We use our results to characterize all Markov chains with infinite state space and doubly stochastic transition probability matrix that have at least one stationary distribution vector.

Oriented spanning trees and stationary distribution of digraphs

By using biclique partitions of digraphs, this paper gives reduction formulas for the number of oriented spanning trees, stationary distribution vector and Kemeny's constant of digraphs. As applications, we give a method for enumerating spanning trees of undirected graphs by vertex degrees and biclique partitions. The biclique partition formula also extends the results of Knuth and Levine from line digraphs to general digraphs.

A Strong Limit Theorem of the Largest Entries of a Sample Correlation Matrices under a Strong Mixing Assumption

We are interested in an n by p matrix Xn where the n rows are strictly stationary α-mixing random vectors and each of the p columns is an independent and identically distributed random vector; p=pn goes to infinity as n→∞, satisfiying 0<c1≤pn/nτ≤c2<∞, where τ>0, c2≥c1>0. We obtain a logarithmic law of Ln=max1≤i<j≤pn|ρij| using the Chen–Stein Poisson approximation method, where ρij denotes the sample correlation coefficient between the ith column and the jth column of Xn.

Real-time observation of stationary and pulsating noise-like vector pulses in a fiber laser.

Rapid progress in real-time measurement technology has uncovered varieties of transient pulse dynamics. Here, we report the vector nature of noise-like pulse (NLP) in a passive fiber laser based on the nonlinear optical loop mirror (NOLM) as the polarization independent saturable absorber. After achieving the basic operation regime of NLP, various types of vector pulses, namely, the polarization locked noise-like vector pulse (PLNLVP), the group velocity locked noise-like vector pulse (GVLNLVP), and the transitional state of combined characteristics of GVLNLVP and polarization rotation noise-like vector pulse (PRNLVP) are also obtained in the cavity. Besides, by utilizing the Dispersive Fourier transform (DFT) technique, the spectral evolution and the energy vibration of pulsating PLNLVP, GVLNLVP, and the transitional state of combined characteristics of GVLNLVP and PRNLVP are also analyzed in real time. Particularly, the coexisting pulsation vector state of NLP and soliton is also captured. All these findings will help to complement our understanding of noise-like vector pulses (NLVPs) in a fiber laser.

Multipole analysis on stationary massive vector and symmetric tensor fields with irreducible Cartesian tensors

The multipole expansions for massive vector and symmetric tensor fields in the region outside spatially compact stationary sources are obtained by using the symmetric and trace-free formalism in terms of the irreducible Cartesian tensors, and the closed-form expressions for the source multipole moments are provided. The expansions show a Yukawa-like dependence on the massive parameters of the fields, and the integrals of the stationary source multipole moments are all modulated by a common radial factor. For stationary massive vector field, there are two types of ‘magnetic’ multipole moments, among which one is the generalization of that of the magnetostatic field, and another, being an additional set of multipole moments of the stationary massive vector field, can not be transformed away. As to the stationary massive symmetric tensor field, its multipole expansion is presented when the trace of its spatial part is specified, where besides the counterparts of the mass and spin multipole moments of massless symmetric tensor field, three additional sets of multipole moments also appear. The multipole expansions of the tensor field under two typical cases are discussed, where it is shown that if the spatial part of the tensor field is trace-free, the monopole and dipole moments in the corresponding expansion will vanish.

Bayesian prior modeling in vector autoregressions via the Yule-Walker equations

In multivariate time series analysis, the Yule-Walker method refers to a system of equations relating the cross-covariances of a stationary vector autoregressive (VAR) model with the matrices of the autoregressive coefficients and the covariance matrix of the noise, both of which are unknown to be estimated. In Bayesian inference of VAR models, one of the key problems is the setting of the prior distributions on these unknown parameters. The Yule-Walker equations are used here to develop a novel prior specification that exploits the reparameterization of the unknowns in terms of the mean, the cross-covariances, and the covariance of the process. Further, the cross-covariance matrices are separated out in terms of the standard deviations and the correlations. All these new quantities are easier to handle because it is more common to have prior information on the mean and the correlation structure of a multiple time series rather than the underlying autoregressive coefficients and the white noise process. The proposed prior specification is suitable for both non informative and informative settings. Through the Yule-Walker based prior, parameter estimation and structure learning of the stationary VAR models are performed via Markov chain Monte Carlo methods. The methodology is illustrated via some synthetic data sets, a benchmark example, and an environmental time series.

APPLICATION OF POWER NUMERICAL METHOD FOR THE STATIONARY DISTRIBUTION OF MARKOV CHAIN

The evolution of this model is represented by transitions from one state to the next. Also, the physical or mathematical behavior of this system can also be illustrated by identifying all of the possible states and explaining how it transitions between them. The iterative solution approaches for the stationary distribution of Markov chains, which begin with an initial estimate of the solution vector and it becomes closer and closer to the true solution with each iteration are investigated. Our goal is to compute solutions of stationary distribution of Markov chain by utilizing the power iterative method which leaves the transition matrices unchanged and saves time by considering the discretization effect, and the convergency. Matrices operations such as multiplication with one or more vectors, lower, diagonal and upper concepts of matrix, with the help of several existing Markov chain laws, theorems, formulas, and the normalization principle are applied. For the illustrative examples, the stationary distribution vectors and table of convergence are obtained.

An Optimization of Home Delivery Services in a Stochastic Modeling with Self and Compulsory Vacation Interruption

This study presents and discusses the home delivery services in stochastic queuing-inventory modeling (SQIM). This system consists of two servers: one server manages the inventory sales processes, and the other server provides home delivery services at the doorstep of customers. Based on the Bernoulli schedule, a customer served by the first server may opt for a home delivery service. If any customer chooses the home delivery option, he hands over the purchased item for home delivery and leaves the system immediately. Otherwise, he carries the purchased item and leaves the system. When the delivery server returns to the system after the last home delivery service and finds that there are no items available for delivery, he goes on vacation. Such a vacation of a delivery server is to be interrupted compulsorily or voluntarily, according to the prefixed threshold level. The replenishment process is executed due to the (s,Q) reordering policy. The unique solution of the stationary probability vector to the finite generator matrix is found using recursive substitution and the normalizing condition. The necessary and sufficient system performance measures and the expected total cost of the system are computed. The optimal expected total cost is obtained numerically for all the parameters and shown graphically. The influence of parameters on the expected number of items that need to be delivered, the probability that the delivery server is busy, and the expected rate at which the delivery server’s self and compulsory vacation interruptions are also discussed.

Nonlocal two-color vector dark solitons

In this paper, vector dark solitons with two colors are demonstrated in nonlocal nonlinear Kerr medium. Using Lagrangian method and rectangular profile of nonlocal response of the media, solutions of the solitons are obtained analytically. The stability of the solitons is also studied numerically, which confirms the analytical results. Nonlocality can suppress instability induced by the difference of diffraction coefficients, leading to formation of stationary two-color vector dark solitons which are always unstable in local nonlinear media.

Application of Jacobi and Gauss–Seidel Numerical Iterative Solution Methods for the Stationary Distribution of Markov Chain

The Physical or Mathematical behaviour of this model may be represented by describing all the different states it may occupy and by indicating how it moves among these states. In this study, the stationary distribution of Markov chains was solved using iterative methods that begin with an initial estimate of the solution vector and then modified it in a way that brings it closer and closer to the real solution with each step or iteration. These methods also involved matrix operations like multiplication with one or more vectors, which preserves the transition matrices while speeding up the process. We computed the solutions using Jacobi iterative method and Gauss-Seidel iterative method in order to shed more light on the solutions of stationary distribution in Markov chain. This was done with the aid of several already-existing laws, theorems, and formulas of Markov chain and the application of normalization principle and matrix operations such as lower, upper, and diagonal matrices. The stationary distribution vector’s 𝜋𝑖,𝑖=1,2,…,4 are obtained for the illustrative example one as 𝜋(3) = (0.078125,0.109375,0.21875,0.59375) as well as the four eigenvalues of the matrix as 𝜆1=1.0, 𝜆2=−0.7718, 𝜆3,4=−0.1141±0.5576𝑖 using Jacobi iterative technique, and for illustrative example two using Gauss-Siedel method as 𝝅(𝟑) = (0.090909, 0.181818, 0.363636, 0.363636). The research shown that Gauss Siedel method converged faster than Jacobi method.

Strongly mixing smooth planar vector field without asymptotic directions

We use a Voronoi-type tessellation based on a compound Poisson point process to construct a polynomially mixing stationary random smooth planar vector field with bounded nonnegative components such that, with probability one, none of the associated integral curves possess an asymptotic direction.

Necessary and sufficient conditions for AR vector processes to be stationary: Applications in information theory and in statistical signal processing

As the correlation matrices of stationary vector processes are block Toeplitz, autoregressive (AR) vector processes are non-stationary. However, in the literature, an AR vector process of finite order is said to be “stationary” if it satisfies the so-called stationarity condition (i.e., if the spectral radius of the associated companion matrix is less than one). Since the term “stationary” is used for such an AR vector process, its correlation matrices should “somehow approach” the correlation matrices of a stationary vector process, but the meaning of “somehow approach” has not been mathematically established in the literature. In the present paper we give necessary and sufficient conditions for AR vector processes to be “stationary”. These conditions show in which sense the correlation matrices of an AR “stationary” vector process asymptotically behave like block Toeplitz matrices. Applications in information theory and in statistical signal processing of these necessary and sufficient conditions are also given.