Multivariate Stochastic Processes Research Articles

Multivariate Stochastic Volatility with Co-Heteroscedasticity

Abstract A new methodology that decomposes shocks into homoscedastic and heteroscedastic components is developed. This specification implies there exist linear combinations of heteroscedastic variables that eliminate heteroscedasticity; a property known as co-heteroscedasticity. The heteroscedastic part of the model uses a multivariate stochastic volatility inverse Wishart process. The resulting model is invariant to the ordering of the variables, which is shown to be important for volatility estimation. By incorporating testable co-heteroscedasticity restrictions, the specification allows estimation in moderately high-dimensions. The computational strategy uses a novel particle filter algorithm, a reparameterization that substantially improves algorithmic convergence and an alternating-order particle Gibbs that reduces the amount of particles needed for accurate estimation. An empirical application to a large Vector Autoregression (VAR) is provided, finding strong evidence for co-heteroscedasticity and that the new method outperforms some previously proposed methods in terms of forecasting at all horizons. It is also found that the structural monetary shock is 98.8 % homoscedastic, and that investment and the SP 500 index are nearly 100 % determined by fat tail heteroscedastic shocks. A Monte Carlo experiment illustrates that the new method estimates well the characteristics of approximate factor models with heteroscedastic errors.

Degradation and life prediction of mechanical equipment based on multivariate stochastic process

IntroductionAccurately predicting the remaining mechanical equipment is of great significance for ensuring the safe operation of the equipment and improving economic efficiency.MethodsTo accurately assess the mechanical equipment degradation, predict its remaining useful life, and ensure efficient, stable, and safe operation, a degradation and life prediction model for mechanical equipment based on multivariate stochastic processes is studied. The study innovatively predicts the remaining life of mechanical equipment using multivariate stochastic processes, and facilitates the correlation analysis between performance indicators based on the characteristics of Copula functions.Results and discussionThe results showed that the Root Mean Squared Error value of the prediction results based on the trivariate Wiener process was 2.58, which decreased by 46.91% and 35.82% compared with the univariate and bivariate Wiener processes, respectively. The prediction value based on the trivariate gamma process was 3.49, which decreased by 44.95% and 40.54% compared with the univariate and bivariate gamma processes, respectively. In conclusion, the degradation and life prediction model with multivariate stochastic processes can effectively assess the mechanical equipment degradation and predict its remaining useful life. This provides an important reference for the maintenance and management of mechanical equipment, improving equipment efficiency and extend its service life.

Simulation of multivariate ergodic stochastic processes using adaptive spectral sampling and non-uniform fast Fourier transform

The simulation of multivariate ergodic stochastic processes is critical for structural dynamic analysis and reliability evaluation. Although the traditional spectral representation method (SRM) has a wide application in many areas, it is highly inefficient in simulating stochastic processes with many simulation points or long durations due to the significant computational cost associated with matrix factorizations concerning frequency. To address the encountered challenge, this paper presents an efficient approach for simulating ergodic stochastic processes with limited frequencies. Central to this approach is a fusion of the adaptive spectral sampling and the non-uniform fast Fourier transform (NUFFT) techniques. The adaptive spectral sampling of the envelope spectrum enables the determination of limited non-equispaced frequencies, which are randomly sampled according to a uniform distribution. Thus, the Cholesky decomposition is only required at limited specific frequencies, which dramatically reduces the computational cost of matrix factorizations. Since the randomly sampled frequencies are not equispaced, utilizing FFT to accelerate the summation of trigonometric functions becomes impractical. Then, the NUFFT that adapts the non-equispaced sampling points is employed instead to expedite this process with the non-uniform increment approximated through reduced interpolation. By taking the wind field simulation of a long-span suspension bridge as an example, a parametric analysis is conducted to investigate the effect of random frequencies on the simulation error of the developed approach and the convergence of spectra. Finally, the developed approach is further validated by focusing on the spectra and probabilistic density functions of the simulated wind samples, and the simulation performance is compared with that of the traditional approach. The analytical results demonstrate the efficiency and accuracy of the developed approach in simulating ergodic stochastic processes.

Multivariate stochastic Vasicek diffusion process: computational estimation and application to the analysis of $$CO_2$$ and $$N_2O$$ concentrations

Multivariate stochastic Vasicek diffusion process: computational estimation and application to the analysis of $$CO_2$$ and $$N_2O$$ concentrations

Deep impulse control: application to interest rate intervention

We propose a deep learning framework for impulse control problems involving multivariate stochastic processes, which can be controllable or uncontrollable. We use this framework to estimate central bank interventions on the (controllable) interest rate to stabilize the (uncontrollable) inflation rate, where the two rates are correlated and cointegrated. This method is useful for small banks or insurance companies with high exposure to Treasury securities to predict and stress-test their potential losses from central bank interventions. We also study the mathematical properties of the proposed framework.

An efficient methodology for simulating multivariate non-Gaussian stochastic processes

Simulation of multivariate non-Gaussian stochastic processes is still a challenging task, where the efficiency and accuracy may not be balanced. In this paper, an efficient methodology is presented to address this challenge. This methodology makes use of a probability weighted moments-based Hermite polynomial model and a wavenumber–frequency spectrum density-based spectral representation approach. First, a probability weighted moments-based Hermite polynomial model is employed to establish a transformation relationship between Gaussian and non-Gaussian samples. Importantly, this model only requires the solution of a solvable system of equations. Second, the target non-Gaussian wavenumber–frequency spectrum density can be promptly transformed into the Gaussian one, where a fast root matching strategy and fast Fourier transform are utilized. Finally, the underlying Gaussian samples are generated by the spectral representation method, and the non-Gaussian samples are obtained via the proposed transformation model. The computational results from numerical examples demonstrate that the proposed methodology is considered to be an accurate, applicable, and efficient approach for simulating multivariate non-Gaussian stochastic processes.

Network dynamical stability analysis reveals key “mallostatic” natural variables that erode homeostasis and drive age-related decline of health

Using longitudinal study data, we dynamically model how aging affects homeostasis in both mice and humans. We operationalize homeostasis as a multivariate mean-reverting stochastic process. We hypothesize that biomarkers have stable equilibrium values, but that deviations from equilibrium of each biomarker affects other biomarkers through an interaction network—this precludes univariate analysis. We therefore looked for age-related changes to homeostasis using dynamic network stability analysis, which transforms observed biomarker data into independent “natural” variables and determines their associated recovery rates. Most natural variables remained near equilibrium and were essentially constant in time. A small number of natural variables were unable to equilibrate due to a gradual drift with age in their homeostatic equilibrium, i.e. allostasis. This drift caused them to accumulate over the lifespan course and makes them natural aging variables. Their rate of accumulation was correlated with risk of adverse outcomes: death or dementia onset. We call this tendency for aging organisms to drift towards an equilibrium position of ever-worsening health “mallostasis”. We demonstrate that the effects of mallostasis on observed biomarkers are spread out through the interaction network. This could provide a redundancy mechanism to preserve functioning until multi-system dysfunction emerges at advanced ages.

Relations between entropy rate, entropy production and information geometry in linear stochastic systems

In this work, we investigate the relation between the concept of ‘information rate’, an information geometric method for measuring the speed of the time evolution of the statistical states of a stochastic process, and stochastic thermodynamics quantities like entropy rate and entropy production. Then, we propose the application of entropy rate and entropy production to different practical applications such as abrupt event detection, correlation analysis, and control engineering. Specifically, by utilising the Fokker–Planck equation of multi-variable linear stochastic processes described by Langevin equations, we calculate the exact value for information rate, entropy rate, and entropy production and derive various inequalities among them. Inspired by classical correlation coefficients and control techniques, we create entropic-informed correlation coefficients as abrupt event detection methods and information geometric cost functions as optimal thermodynamic control policies, respectively. The methods are analysed via the numerical simulations of common prototypical systems.

Generalized autocovariance matrices for multivariate time series

The paper treats the modeling of stationary multivariate stochastic processes via frequency domain, and extends the notion of generalized autocovariance function, given by Proietti and Luati (2015) for univariate time series, to the multivariate setting. The generalized autocovariance matrices are defined for stationary multivariate stochastic processes as the Fourier transform of the power transformation of the spectral density matrix. Then we prove the consistency and derive the asymptotic distribution of frequency domain non-parametric estimators of the generalized autocovariance matrices, based on the power transformation of the periodogram matrix. Generalized autocovariance matrices are used to construct white noise hypothesis testing, to discriminate stochastic processes, and to introduce a generalized Yule–Walker estimator for the spectrum. A so-called λ–squared distance between two multivariate stochastic processes is also defined by using their generalized autocovariance matrices, and it serves for clustering time series and estimation by feature matching. Another use is in discriminant analysis.

A Continuous Multisite Multivariate Generator for Daily Temperature Conditioned by Precipitation Occurrence

Temperature is one of the most influential weather variables necessary for numerous studies, such as climate change, integrated water resources management, and water scarcity, among others. The temperature and precipitation are relevant in river basins because they may be particularly affected by modifications in the variability, for example, due to climate change. We developed a stochastic model for daily precipitation occurrences and their influence on maximum and minimum temperatures with a straightforward approach. The Markov model has been used to determine everyday occurrences of rainfall. Moreover, we developed a multisite multivariate autoregressive model to represent the short-term memory of daily temperature, called MASCV. The reduction of parameters is an essential factor addressed in this approach. For this reason, the normalization of the temperatures was performed through different nonparametric transformations. The case study is the Jucar River Basin in Spain. The multisite multivariate stochastic model of two states and a lag-one accurately represents both occurrences as well as maximum and minimum temperature. The simulation and generation of occurrences and temperature is considered a continuous multivariate stochastic process. Additionally, time series of multiple correlated climate variables are completed. Therefore, we simplify the complexity and reduce the computational time for the simulation.

Comparing climate time series – Part 3: Discriminant analysis

Abstract. In parts I and II of this paper series, rigorous tests for equality of stochastic processes were proposed. These tests provide objective criteria for deciding whether two processes differ, but they provide no information about the nature of those differences. This paper develops a systematic and optimal approach to diagnosing differences between multivariate stochastic processes. Like the tests, the diagnostics are framed in terms of vector autoregressive (VAR) models, which can be viewed as a dynamical system forced by random noise. The tests depend on two statistics, one that measures dissimilarity in dynamical operators and another that measures dissimilarity in noise covariances. Under suitable assumptions, these statistics are independent and can be tested separately for significance. If a term is significant, then the linear combination of variables that maximizes that term is obtained. The resulting indices contain all relevant information about differences between data sets. These techniques are applied to diagnose how the variability of annual-mean North Atlantic sea surface temperature differs between climate models and observations. For most models, differences in both noise processes and dynamics are important. Over 40 % of the differences in noise statistics can be explained by one or two discriminant components, though these components can be model dependent. Maximizing dissimilarity in dynamical operators identifies situations in which some climate models predict large-scale anomalies with the wrong sign.

Multivariate Threshold Regression Models with Cure Rates: Identification and Estimation in the Presence of the Esscher Property

The first hitting time of a boundary or threshold by the sample path of a stochastic process is the central concept of threshold regression models for survival data analysis. Regression functions for the process and threshold parameters in these models are multivariate combinations of explanatory variates. The stochastic process under investigation may be a univariate stochastic process or a multivariate stochastic process. The stochastic processes of interest to us in this report are those that possess stationary independent increments (i.e., Lévy processes) as well as the Esscher property. The Esscher transform is a transformation of probability density functions that has applications in actuarial science, financial engineering, and other fields. Lévy processes with this property are often encountered in practical applications. Frequently, these applications also involve a ‘cure rate’ fraction because some individuals are susceptible to failure and others not. Cure rates may arise endogenously from the model alone or exogenously from mixing of distinct statistical populations in the data set. We show, using both theoretical analysis and case demonstrations, that model estimates derived from typical survival data may not be able to distinguish between individuals in the cure rate fraction who are not susceptible to failure and those who may be susceptible to failure but escape the fate by chance. The ambiguity is aggravated by right censoring of survival times and by minor misspecifications of the model. Slightly incorrect specifications for regression functions or for the stochastic process can lead to problems with model identification and estimation. In this situation, additional guidance for estimating the fraction of non-susceptibles must come from subject matter expertise or from data types other than survival times, censored or otherwise. The identifiability issue is confronted directly in threshold regression but is also present when applying other kinds of models commonly used for survival data analysis. Other methods, however, usually do not provide a framework for recognizing or dealing with the issue and so the issue is often unintentionally ignored. The theoretical foundations of this work are set out, which presents new and somewhat surprising results for the first hitting time distributions of Lévy processes that have the Esscher property.

Multi-attribute Bayesian fault prediction for hidden-state systems under condition monitoring

Although Bayesian approaches have been utilized in engineering systems for health prognostics, very little work has been done using Bayesian methods for fault prediction of systems under multiple attributes. To address this issue, in this paper a novel multi-attribute Bayesian control chart is presented for predicting failures of hidden-state systems by jointly considering two performance measures of system operation. The system actual status is represented by a three-state multivariate hidden stochastic process with a normal state, an abnormal state, and a failure state. The working states are unobservable and failure state is observable. Based on the built hidden-state model, a fault prediction scheme integrating both system availability and cost objectives is constructed via a multi-attribute Bayesian control chart to monitor and predict impending risks of the operational systems. The Bayesian control chart alarms when the probability of impending risks reaches a certain control limit, which is optimized and determined by a computational algorithm developed in a semi-Markov decision process framework. The proposed fault prediction scheme provides an appearing feature to jointly consider multiple attributes for hidden-state systems. A real case study of mechanical generators is presented and a comparison with other Bayesian and non-Bayesian methods is also given, which demonstrates the effectiveness and superiority of the proposed approach.

Learning delay dynamics for multivariate stochastic processes, with application to the prediction of the growth rate of COVID-19 cases in the United States

Delay differential equations form the underpinning of many complex dynamical systems. The forward problem of solving random differential equations with delay has received increasing attention in recent years. Motivated by the challenge to predict the COVID-19 caseload trajectories for individual states in the U.S., we target here the inverse problem. Given a sample of observed random trajectories obeying an unknown random differential equation model with delay, we use a functional data analysis framework to learn the model parameters that govern the underlying dynamics from the data. We show the existence and uniqueness of the analytical solutions of the population delay random differential equation model when one has discrete time delays in the functional concurrent regression model and also for a second scenario where one has a delay continuum or distributed delay. The latter involves a functional linear regression model with history index. The derivative of the process of interest is modeled using the process itself as predictor and also other functional predictors with predictor-specific delayed impacts. This dynamics learning approach is shown to be well suited to model the growth rate of COVID-19 for the states that are part of the U.S., by pooling information from the individual states, using the case process and concurrently observed economic and mobility data as predictors.

Quantile cross-spectral density: A novel and effective tool for clustering multivariate time series

Clustering of multivariate time series is a central problem in data mining with applications in many fields. Frequently, the clustering target is to identify groups of series generated by the same multivariate stochastic process. Most of the approaches to address this problem include a prior step of dimensionality reduction which may result in a loss of information or consider dissimilarity measures based on correlations and cross-correlations but ignoring the serial dependence structure. We propose a novel approach to measure dissimilarity between multivariate time series aimed at jointly capturing both cross dependence and serial dependence. Specifically, each series is characterized by a set of matrices of estimated quantile cross-spectral densities, where each matrix corresponds to a pair of quantile levels. Then the dissimilarity between every couple of series is evaluated by comparing their estimated quantile cross-spectral densities, and the pairwise dissimilarity matrix is taken as starting point to develop a partitioning around medoids algorithm. Since the quantile-based cross-spectra capture dependence in quantiles of the joint distribution, the proposed metric has a high capability to discriminate between high-level dependence structures. An extensive simulation study shows that our clustering procedure outperforms a wide range of alternative methods and exhibits robustness to noise distribution besides being computationally efficient. A real data application involving bivariate financial time series illustrates the usefulness of the proposed approach. The procedure is also applied to cluster nonstationary series from the UEA multivariate time series classification archive.

A micro‐to‐macro approach to returns, volumes and waiting times

AbstractModelling stock prices has been a research topic for many decades and it is still an open question. Different approaches have been used in the literature, the majority of which can be classified within the so‐called econometric framework and sometimes also referred to as the macro‐to‐micro approach. Another strand of literature relies on the modelling of directly observable quantities, the so‐called micro‐to‐macro approach. Based on this second line of research, we propose a new multivariate stochastic process to model simultaneously price returns, trading volumes and the time interval between changes in trades, price and volume. The proposed model is based on a generalization of semi‐Markov chain models and copulas and is motivated by empirical evidence that the three mentioned variables are correlated and long‐range autocorrelated. Utilizing Monte Carlo simulations, we compared our model with real data from the Italian stock market and show that it can reproduce many empirical pieces of evidence. The proposed model can be used in the field of portfolio optimization, development of risk measure and volatility forecasting.

Correlation bounds, mixing and m-dependence under random time-varying network distances with an application to Cox-processes

We will consider multivariate stochastic processes indexed either by vertices or pairs of vertices of a dynamic network. Under a dynamic network, we understand a network with a fixed vertex set and an edge set which changes randomly over time. We will assume that the spatial dependence-structure of the processes conditional on the network behaves in the following way: Close vertices (or pairs of vertices) are dependent, while we assume that the dependence decreases conditionally on that the distance in the network increases. We make this intuition mathematically precise by considering three concepts based on correlation, β-mixing with time-varying β-coefficients and conditional independence. These concepts allow proving weak-dependence results, for example, an exponential inequality, which might be of independent interest. In order to demonstrate the use of these concepts in an application, we study the asymptotics (for growing networks) of a goodness of fit test in a dynamic interaction network model based on a Cox-type model for counting processes. This model is then applied to bike-sharing data.

Analyzing Commodity Futures Using Factor State-Space Models with Wishart Stochastic Volatility

A factor state-space approach with stochastic volatility is proposed for modeling and forecasting the maturity structure of future commodity contracts. The proposed approach builds upon the dynamic 3-factor Nelson-Siegel model and its 4-factor Svensson extension and assumes for the latent level, slope and curvature factors a Gaussian vector autoregression with a multivariate Wishart stochastic volatility process. A computationally fast and easy to implement MCMC algorithm for the Bayesian posterior analysis is developed, which exploits the conjugacy of the Wishart and the Gaussian distribution. An empirical application to daily prices for contracts on crude oil with stipulated delivery dates ranging from one to 24 months ahead show that the estimated 4-factor Svensson model with two curvature factors provides a good parsimonious representation of the serial correlation in the individual prices and their volatility. It also shows that this model has a good out-of-sample forecast performance.

Simulation of ergodic multivariate stochastic processes: An enhanced spectral representation method

The spectral representation method (SRM) has been extensively used in the simulation of multivariate stationary Gaussian stochastic processes. In this study, an enhanced SRM for the simulation of ergodic multivariate stochastic processes is developed. Through shifting a frequency increment, the enhanced SRM offers a faster convergence rate of probabilistic characteristics in comparison with conventional SRM. The computational efficiency is enhanced as the Cholesky decomposition is only required at single-indexed frequencies. Additionally, a two-dimensional fast Fourier transform (FFT) algorithm is proposed to expedite simultaneously estimation of the double summation of cosine functions in both frequency and space dimensions. Accordingly, compared to the traditional FFT-based SRM, in which FFT is only used for the summation in the frequency dimension, the simulation efficiency is further enhanced significantly. The numerical example concerning the simulation of wind velocity field demonstrates that the proposed method offers a faster convergence rate and a higher level of efficiency.

Simulation of fluctuating wind of high-speed vehicle–bridge united system

This study introduces a novel multivariate stochastic process simulation algorithm to generate fluctuating wind velocity time histories of the vehicle–bridge united system. The Kaimal spectrum and wind spectrum relative to a moving point were applied to the bridge and the vehicle (train) in operation, respectively. The wind time series of the vehicle–bridge united system were generated simultaneously by considering the correlation between the moving point fixed on a vehicle in operation and the static points on a bridge. This algorithm is of high computational efficiency, as it takes advantage of explicit decomposition for the coherence function matrix and frequency truncation method. Based on an error estimation, this simulation method is proven to be accurate. Numerical examples of a vehicle running on a simply supported continuous beam bridge with the speeds of 20, 40, 60 and 80 m/s were performed. The result shows that the auto-/cross-correlation functions and auto-spectrum of the simulated wind samples are in good agreement with the targets. The proposed method can facilitate the analysis of wind–vehicle–bridge aerodynamic response with high efficiency and accuracy, and avoids the discontinuity problem of wind time history of the moving point.