Stationary Vector Research Articles

A Nonlinear Autoregressive Distributed Lag (NARDL) Analysis of West Texas Intermediate Oil Prices and the DOW JONES Index

The paper features an examination of the link between the behaviour of oil prices and DowJones Index in a nonlinear autoregressive distributed lag NARDL framework. The attraction of NARDL is that it represents the simplest method available of modelling combined short- and long-run asymmetries. The bounds testing framework adopted means that it can be applied to stationary and non-stationary time series vectors, or combinations of both. The data comprise a monthly West Texas Intermediate (WTI) crude oil series from FRED, commencing in January 2000 and terminating in February 2019, and a corresponding monthly DOW JONES index adjusted-price series obtained from Yahoo Finance. Both series are adjusted for monthly USA CPI values to create real series. The results of the analysis suggest that movements in the lagged real levels of monthly WTI crude oil prices have very significant effects on the behaviour of the DOW JONES Index. They also suggest that negative movements have larger impacts than positive movements in WTI prices, and that long-term multiplier effects take about 9 to 12 months to take effect.

A Structured Condition Number for Kemeny's Constant

Kemeny's constant is an interesting and useful quantifier describing the global average behavior of a Markov chain. In this article, we examine the sensitivity of Kemeny's constant to perturbations in the transition probabilities. That is, we consider the problem of generating a condition number for Kemeny's constant to give an indication of the size of the change in its value relative to the size of the perturbation. We provide a structured condition number and determine some illuminating upper and lower bounds which connect the conditioning of Kemeny's constant to well-studied condition numbers for the stationary vector of the Markov chain. We also investigate the behavior of this structured condition number for several infinite families of Markov chains.

Distributed Hierarchical Control for State Estimation With Robotic Sensor Networks

This paper addresses active state estimation with a team of robotic sensors. The states to be estimated are represented by spatially distributed, uncorrelated, stationary vectors. Given a prior belief on the geographic locations of the states, we cluster the states in moderately sized groups and propose a new hierarchical dynamic programming framework to compute optimal sensing policies for each cluster that mitigates the computational cost of planning optimal policies in the combined belief space. Then, we develop a decentralized assignment algorithm that dynamically allocates clusters to robots based on the precomputed optimal policies at each cluster. The integrated distributed state estimation framework is optimal at the cluster level but also scales very well to large numbers of states and robot sensors. We demonstrate efficiency of the proposed method in both simulations and real-world experiments using stereoscopic vision sensors.

Ideal flow of Markov Chain

Ideal flow network is a strongly connected network with flow, where the flows are in steady state and conserved. The matrix of ideal flow is premagic, where vector, the sum of rows, is equal to the transposed vector containing the sum of columns. The premagic property guarantees the flow conservation in all nodes. The scaling factor as the sum of node probabilities of all nodes is equal to the total flow of an ideal flow network. The same scaling factor can also be applied to create the identical ideal flow network, which has from the same transition probability matrix. Perturbation analysis of the elements of the stationary node probability vector shows an insight that the limiting distribution or the stationary distribution is also the flow-equilibrium distribution. The process is reversible that the Markov probability matrix can be obtained from the invariant state distribution through linear algebra of ideal flow matrix. Finally, we show that recursive transformation [Formula: see text] to represent [Formula: see text]-vertices path-tracing also preserved the properties of ideal flow, which is irreducible and premagic.

A Reinforcement Learning Approach for Solving the Mean Variance Customer Portfolio in Partially Observable Models

In problems involving control of financial processes, it is usually complicated to quantify exactly the state variables. It could be expensive to acquire the exact value of a given state, even if it may be physically possible to do so. In such cases it may be interesting to support the decision-making process on inaccurate information pertaining to the system state. In addition, for modeling real-world application, it is necessary to compute the values of the parameters of the environment (transition probabilities and observation probabilities) and the reward functions, which are typically, hand-tuned by experts in the field until it has acquired a satisfactory value. This results in an undesired process. To address these shortcomings, this paper provides a new Reinforcement Learning (RL) framework for computing the mean-variance customer portfolio with transaction costs in controllable Partially Observable Markov Decision Processes (POMDPs). The solution is restricted to finite state, action, observation sets and average reward problems. For solving this problem, a controller/actor-critic architecture is proposed, which balance the difficult tasks of exploitation and exploration of the environment. The architecture consists of three modules: controller, fast-tracked portfolio learning and an actor-critic module. Each module involves the design of a convergent Temporal Difference (TD) learning algorithm. We employ three different learning rules to estimate the real values of: (a) the transition matrices [Formula: see text], (b) the rewards [Formula: see text] and (c) the resources destined for carrying out a promotion [Formula: see text]. We present a proof for the estimated transition matrix rule [Formula: see text] and showing that it converges when t → ∞. For solving the optimization programming problem we extend the c-variable method for partially observable Markov chains. The c-variable is conceptualized as joint strategy given by the product of the control policy, the observation kernel Q(y|s) and the stationary distribution vector. A major advantage of this procedure is that it can be implemented efficiently for real settings in controllable POMDP. A numerical example illustrates the results of the proposed method.

Detecting Co‐Movements in Non‐Causal Time Series

AbstractThis paper introduces the notion of common non‐causal features and proposes tools to detect them in multivariate time series models. We argue that the existence of co‐movements might not be detected using the conventional stationary vector autoregressive (VAR) model as the common dynamics are present in the non‐causal (i.e. forward‐looking) component of the series. We show that the presence of a reduced rank structure allows to identify purely causal and non‐causal VAR processes of order P>1 even in the Gaussian likelihood framework. Hence, usual test statistics and canonical correlation analysis can be applied, where either lags or leads are used as instruments to determine whether the common features are present in either the backward‐ or forward‐looking dynamics of the series. The proposed definitions of co‐movements are also valid for the mixed causal—non‐causal VAR, with the exception that a non‐Gaussian maximum likelihood estimator is necessary. This means however that one loses the benefits of the simple tools proposed. An empirical analysis on Brent and West Texas Intermediate oil prices illustrates the findings. No short run co‐movements are found in a conventional causal VAR, but they are detected when considering a purely non‐causal VAR.

Suppressing Chemotactic Blow-Up Through a Fast Splitting Scenario on the Plane

We revisit the question of global regularity for the Patlak-Keller-Segel (PKS) chemotaxis model. The classical 2D hyperbolic-elliptic model blows up for initial mass M>8\pi. We consider more realistic scenario which takes into account the flow of the ambient environment induced by harmonic potentials, and thus retain the identical elliptic structure as in the original PKS. Surprisingly, we find that already the simplest case of linear stationary vector field, $Ax^\top$, with large enough amplitude $A$, prevents chemotactic blow-up. Specifically, the presence of such an ambient fluid transport creates what we call a 'fast splitting scenario', which competes with the focusing effect of aggregation so that 'enough mass' is pushed away from concentration along the $x_1$-axis, thus avoiding a finite time blow-up, at least for M<16\pi. Thus, the enhanced ambient flow doubles the amount of allowable mass which evolve to global smooth solutions.

Consistent estimation of time-varying loadings in high-dimensional factor models

In this paper, we develop a two-step maximum likelihood estimator of time-varying loadings in high-dimensional factor models. We specify the loadings to evolve as stationary vector autoregressions (VAR) and show that consistent estimates of the loadings parameters can be obtained. In the first step, principal components are extracted from the data to form factor estimates. In the second step, the parameters of the loadings VARs are estimated as a set of linear regression models with time-varying coefficients. We document the finite-sample properties of the maximum likelihood estimator through an extensive simulation study and illustrate the empirical relevance of the time-varying loadings structure using a large quarterly dataset for the US economy.

APPLICATION OF MARKOV CHAINS TO THE ANALYSIS OF BLAST FURNACE OPERATION EFFICIENCY

The article presents the results of modeling in a dynamic format of one of the most important parameters of any research object – the efficiency of its work. As the object of investigation, a blast furnace with a volume of 2014 m3 was chosen. The main parameters of the efficiency of this object are traditionally used daily productivity and specific consumption of coke; these two parameters were generalized in this paper. In this case, various algebraic signs of the influence of these parameters were taken into account in the generalized efficiency index. Taking into account the variation of each of these parameters at 3 levels, the number of levels of the generalized efficiency index was determined as 32 = 9, therefore it was rational to take a 9-point scale with the measuring scale of profitability from the efficient operation of the blast furnace. The two-dimensional array of primary data of the volume N = 177 was transformed into a 9×9 transitional matrix for processing of random transitions of the efficiency index from one state to another by the Markov chain method with discrete states and time. The set of parameters of the random process is calculated: for the long-term forecast – the stationary vector of state probabilities, the average time of recurrence (reversal) for each efficiency state, the evaluation of the blast furnace efficiency in points; for a short-term forecast – the first time of transition from each state to any other state, the step number for a “burst” of probability for each reliable state at the initial moment of time, and the components of the efficiency index are obtained. It was established that the average level of the analyzed efficiency of the blast furnace (daily output 3702 tons and specific coke consumption 470 kg/ton) is achieved mainly due to short-term transitions of low-efficiency states to high-efficiency states and vice versa. The transfer of the system to more efficient and prolonged conditions is possible, and as practice has shown on the same blast furnace after repair works to eliminate the distortion of the furnace profile, the daily productivity has increased to 5048 tons with a specific coke consumption of 445 kg/t, but the structure of the transition matrix and the calculated indicators of the Markov chain have fundamentally changed in the direction of increasing the probabilities of stay and transitions of the system in more efficient states. The use of the Markov chain method with discrete states and time makes it possible to estimate the probable value of the change in the parameters of the operation of a blast furnace in a given time interval with constant levels of parameters characterizing the conditions of its operation.

Performance Measures of State Dependent MMPP/M/1 Queue

In this research work we are concerned with single unit server queue queue with Markov Modulated process in Poisson fashion and the service time follow exponential distribution. The system is framed as a state dependent with the arrival process as Markov Modulated input and service is rendered by a single server with variation in service rate based on the intensity of service state of the system. The rate matrix that is essential to compute the stationary probability vector is obtained and various performance measures are computed using matrix method.

Principal component analysis for second-order stationary vector time series

We extend the principal component analysis (PCA) to second-order stationary vector time series in the sense that we seek for a contemporaneous linear transformation for a $p$-variate time series such that the transformed series is segmented into several lower-dimensional subseries, and those subseries are uncorrelated with each other both contemporaneously and serially. Therefore, those lower-dimensional series can be analyzed separately as far as the linear dynamic structure is concerned. Technically, it boils down to an eigenanalysis for a positive definite matrix. When $p$ is large, an additional step is required to perform a permutation in terms of either maximum cross-correlations or FDR based on multiple tests. The asymptotic theory is established for both fixed $p$ and diverging $p$ when the sample size $n$ tends to infinity. Numerical experiments with both simulated and real data sets indicate that the proposed method is an effective initial step in analyzing multiple time series data, which leads to substantial dimension reduction in modelling and forecasting high-dimensional linear dynamical structures. Unlike PCA for independent data, there is no guarantee that the required linear transformation exists. When it does not, the proposed method provides an approximate segmentation which leads to the advantages in, for example, forecasting for future values. The method can also be adapted to segment multiple volatility processes.

Performance Analysis of Shock Models Governed by a Discrete-Time Markovian Arrival Process

This paper considers a system subject to discrete-time non-renewal process of shocks’ arrival, where only the impact of fatal shock is taken into account. The device survives any non fatal shock and instantaneously replaces the failure device by a new and identical one only after a fatal shock. Any shock to an operating device of the system having certain strength of damage leads to its failure only with some probability. We construct the transition probability matrix to compute the stationary probability vector of number of shocks. The system is analysed for both the restricted and unrestricted number of shocks. We provide several key performance measures such as the number of renewals, availability and occurrence of failure. Computational experiences with a variety of numerical results are discussed on system performance measures. This renders relevant information to the system analysts, researchers, and industry people who are interested to investigate congestion problems with the mechanism generating shocks.

Rate Distortion Function of Gaussian Asymptotically WSS Vector Processes.

In this paper, we obtain an integral formula for the rate distortion function (RDF) of any Gaussian asymptotically wide sense stationary (AWSS) vector process. Applying this result, we also obtain an integral formula for the RDF of Gaussian moving average (MA) vector processes and of Gaussian autoregressive MA (ARMA) AWSS vector processes.

QA-deformation of surface of negative Gaussian curvature

An infinitesimal deformation with the given law of changing the element of area of a surface in Euclidean three-space was considered in this article. Such deformation in the article was called the quasiareal infinitesimal deformation or, briefly, the QA-deformation. The problem of finding the QA-deformation, under which the unit normal vector to the surface is preserved, was reduced to the study of one nonhomogeneous partial differential equation of the second order with respect to one unknown function. The initial conditions, under which the only one QA-deformation with the stationary unit normal vector exists, were defined for the surfaces of a negative Gaussian curvature. In this case, for the above equation, the Cauchy and Goursat problems were applied. The initial conditions of these tasks were expressed through the deforming vector.

Random function representation of stationary stochastic vector processes for probability density evolution analysis of wind-induced structures

Abstract This paper develops a hybrid approach of spectral representation and random function for simulating stationary stochastic vector processes. In the proposed approach, the high-dimensional random variables, included in the original spectral representation (OSR) formula, could be effectively reduced to only two elementary random variables by introducing the random functions that serve as random constraints. Based on this, a satisfactory simulation accuracy can be guaranteed by selecting a small representative point set of the elementary random variables. The probability information of the stochastic excitations can be fully emerged through just several hundred of sample functions generated by the proposed approach. Therefore, combined with the probability density evolution method (PDEM), it could be able to implement dynamic response analysis and reliability assessment of engineering structures. For illustrative purposes, a stochastic turbulence wind velocity field acting on a frame-shear-wall structure is simulated by constructing three types of random functions to demonstrate the accuracy and efficiency of the proposed approach. Careful and in-depth studies concerning the probability density evolution analysis of the wind-induced structure have been conducted so as to better illustrate the application prospects of the proposed approach. Numerical examples also show that the proposed approach possesses a good robustness.

Detecting Co-Movements in Noncausal Time Series

This paper introduces the notion of common noncausal features and proposes tools to detect them in multivariate time series models. We argue that the existence of co-movements might not be detected using the conventional stationary vector autoregressive (VAR) model as the common dynamics are present in the noncausal (i.e. forward-looking) component of the series. In particular, we show that the presence of a reduced rank structure allows to identify purely causal and noncausal VAR processes of order two and higher even in the Gaussian likelihood framework. Hence, usual test statistics and canonical correlation analysis can still be applied, where both lags and leads are used as instruments to determine whether the common features are present in either the backward- or forward-looking dynamics of the series. The proposed de finitions of co-movements also valid for the mixed causal-noncausal VAR, with the exception that an approximate non-Gaussian maximum likelihood estimator is necessary for these cases. This means however that one loses the bene fits of the simple tools proposed in this paper. An empirical analysis on European Brent and U.S. West Texas Intermediate oil prices illustrates the main fi ndings. Whereas we fail to fi nd any short run co-movements in a conventional causal VAR, they are detected in the growth rates of the series when considering a purely noncausal VAR.

Approximating service-time distributions by phase-type distributions in single-server queues: a strong stability approach

Phase-type queueing systems are used to approximate queues with general service-time distributions. In this work, we provide by means of the strong stability method, the mathematical justification of the approximation method by phase-type distributions that is already used in several works. We consider the approximation of M/G/1 queueing system by a M/PH/1 system, where PH refers to a hyperexponential H2 or a hypoexponential HOE2 distribution depending on the value of the coefficient of variation of the original distribution. We prove the robustness of the underlying Markov chain in each case and estimate an upper bound of the deviation of the stationary vector, resulting from the perturbation of the service-time distribution. We provide numerical examples and compare the perturbation bounds obtained in this paper with the estimates of the real deviation of the stationary vector obtained by simulation.

The multivariate bullwhip effect

A multivariate bullwhip expression for m products with an order-up-to inventory policy is developed. The demand models under consideration are differenced stationary vector time series with a Wold representation for which general forecasting formulas are available, resulting in a large class of possible models (including nonstationary ones). Examples are provided for common demand models and implemented on sales data. It is found that the multivariate approach gives rise to mechanisms for understanding and reducing the bullwhip effect through horizontal information sharing, particularly for the nonstationary demand case. In the stationary setting, a more nuanced approach to bullwhip reduction can be achieved by managing the relationship between cross-correlations and lead-times. A method of determining whether a multivariate or univariate approach generates a lower bullwhip effect is proposed.

On Testing for Impropriety of Multivariate Complex-Valued Random Sequences

We consider the problem of testing whether a complex-valued vector random sequence is proper, i.e., if the sequence is uncorrelated with its complex conjugate. Previous nonparametric approaches to impropriety testing are limited to a sequence of independent random vectors, typically assumed to be Gaussian. In this paper, we extend the results to stationary vector sequences that can be non Gaussian. A binary hypothesis testing approach is formulated and a generalized likelihood ratio test (GLRT) is derived using the power spectral density estimator of an augmented sequence. An asymptotic analytical solution for calculating the test threshold is provided. The performance of the GLRT is analyzed by deriving an approximate asymptotic distribution of the GLRT statistic under the alternative hypothesis. The results are illustrated via simulations.

Gridless Line Spectrum Estimation and Low-Rank Toeplitz Matrix Compression Using Structured Samplers: A Regularization-Free Approach

This paper considers the problem of compressively sampling wide sense stationary random vectors with a low rank Toeplitz covariance matrix. Certain families of structured deterministic samplers are shown to efficiently compress a high-dimensional Toeplitz matrix of size $N\times N$ , producing a compressed sketch of size $O(\sqrt{r})\times O(\sqrt{r})$.The reconstruction problem can be cast as that of line spectrum estimation, whereby, in absence of noise, Toeplitz matrices of any size $N$ can be exactly recovered from compressive sketches of size $O(\sqrt{r})\times O(\sqrt{r})$, no matter how large $N$ is. In the presence of noise and finite data, the line spectrum estimation algorithm is combined with a novel denoising technique that only exploits a positive semidefinite (PSD) Toeplitz constraint to denoise the compressed sketch using a simple least-squares minimization framework. A major advantage of the algorithm is that it does not require any regularization parameter. It also enjoys lower computational complexity owing to its ability to predict the unobserved entries of the low-rank Toeplitz matrix. Explicit bounds on the reconstruction error are established and it is shown that the PSD constraint on the denoiser is sufficient to ensure stable reconstruction from a sketch of size $O(\sqrt{r})\times O(\sqrt{r})$. Extensive simulations demonstrate that the proposed algorithm provides better performance over random samplers and algorithms that use nuclear-norm-based regularizers.