Multivariate Stochastic Processes Research Articles

Long-term temperature field of steel-box girder of a long-span bridge: Measurement and simulation

The temperature field of a long-span bridge is variant with an underlying statistical property due to the periodically time-varying solar radiation and the unceasing structural heat exchange with surrounding environment. A comprehensive understanding of the long-term temperature field of a steel-box girder is essential for the design and maintenance of cable-supported bridges. However, it is challenging to obtain the long-term temperature variations because of the complex geometric configuration of the main girder and the complicated heat transfer process. In this context, this paper presents a strategy to overcome this problem by simulating the long-term temperature field of a steel-box girder based on limited monitored data. In the simulation, the long-term temperature field is treated as a stochastic process with prescribed spectral and probabilistic properties. By analyzing the monitored temperature data on Sutong Bridge, the models of power spectral density (PSD), coherence function, and high-order moments are developed. A noteworthy feature captured in measurements is that the temperature is non-Gaussian with considerable kurtosis and skewness. Hence, the temperature field simulation is converted to the simulation of multivariate non-Gaussian stochastic processes. Based on the spectral representation method accompanied with Hermite transformation, the temperature field of the steel-box girder of Sutong Bridge is simulated with a two-year duration considering the non-Gaussianity. Verifications of the simulated field are made through comparisons with the developed spectral and probabilistic models. It is shown that the simulations match well with the measured temperature characteristics, specifically including the time histories and their PSD, PDF, and coherence functions. The good agreement indicates the effectiveness of the simulated long-term temperature field.

3rd-order Spectral Representation Method: Simulation of multi-dimensional random fields and ergodic multi-variate random processes with fast Fourier transform implementation

This paper introduces a generalized 3rd-order Spectral Representation Method for the simulation of multi-dimensional random fields and ergodic multi-variate stochastic processes with asymmetric non-linearities. The formula for the simulation of general d-dimensional random fields is presented and the method is applied to simulate 2D and 3D random fields. The differences between samples generated by the proposed methodology and the existing classical Spectral Representation Method are analysed. The formula for the simulation of multi-variate random processes is subsequently developed. An important feature of the methodologies is that they can be implemented efficiently with the Fast Fourier Transform (FFT), details of which are presented. Computational savings are shown to grow exponentially with dimensionality as a testament of the scalability of the simulation methodology. Examples highlighting the salient features of these methodologies are also presented.

Stochastic modeling of assets and liabilities with mortality risk

ABSTRACT This paper describes a general approach for the stochastic modeling of assets returns and liability cash-flows of a typical pensions insurer. On the asset side, we model the investment returns on equities and various classes of fixed-income instruments including short- and long-maturity fixed-rate bonds as well as index-linked and corporate bonds. On the liability side, the risks are driven by future mortality developments as well as price and wage inflation. All the risk factors are modeled as a multivariate stochastic process that captures the dynamics and the dependencies across different risk factors. The model is easy to interpret and to calibrate to both historical data and to forecasts or expert views concerning the future. This feature is particularly useful in unprecedented circumstances like pandemics when historical data alone does not give a reasonable description of the future. The simple structure of the model allows for efficient computations. The construction of a million scenarios takes only a few minutes on a personal computer. The approach is illustrated with an asset-liability analysis of a defined benefit pension fund, pre- and post-COVID-19.

Dimension-reduction representation of stochastic ground motion fields based on wavenumber-frequency spectrum for engineering purposes

It is of great necessity to achieve the efficient and accurate representation of ground motion fields considering the spatial variation of which possess paramount importance to the seismic safety and reliability evaluation of lifeline systems. To circumvent the difficulties caused by the decomposition of cross power spectral density (PSD) matrix and the interpolation between discretized spatial points in the spectral representation method (SRM) for uni-dimensional multi-variate (1D-nV) stochastic vector process (regarded as discrete stochastic field), a wavenumber-frequency SRM (WSRM) for simulating multi-dimensional uni-variate (mD-1V) spatial-temporal stochastic field (regarded as continuous stochastic field) has been developed for years. In this paper, an updated WSRM (UWSRM), which can reduce the number of elementary random variables, is proposed for simulating the 2D-1V stochastic ground motion fields. Compared with the conventional WSRM (CWSRM) which belongs to the Monte Carlo simulation schemes, merely two elementary random variables are required to realize the precise modeling of ground motion fields. For the sake of facilitating the calculation, the UWSRM embedded in the Fast Fourier Transform (FFT) technique is applied to simulate the stochastic ground motion fields. Numerical results including the comparison of the accuracy and efficiency with the CWSRM verify the effectiveness of the UWSRM. Moreover, to illustrate the application prospect of the UWSRM, the refined random seismic response analysis of a buried pipeline network is implemented by combining with the probability density evolution method (PDEM). The analysis results preliminarily demonstrate the applicability of the UWSRM in lifeline systems.

Dimension Reduction of Polynomial Regression Models for the Estimation of Granger Causality in High-Dimensional Time Series

The causality analysis of multivariate time series and formation of complex networks relies on the estimation of the direct cause-effect from one observed variable to another accounting for the presence of other observed variables. Such effects are quantified by the conditional Granger causality index (CGCI), derived by linear vector autoregressive models (VAR). Dimension reduction approaches have been developed to restrict VAR models, such as the modified backward-in-time selection method (mBTS) giving the restricted CGCI (rCGCI). The rCGCI is limited to linear systems. In this study, we extend the mBTS dimension reduction scheme to polynomial VAR, so as to model nonlinear dynamics and cause-effect relationships and derive the Granger causality measure termed restricted polynomial CGCI (rpCGCI). The complications in the adaptation of mBTS to the restricted polynomial VAR and the construction of the Granger causality index are addressed, involving also a randomization significance test in the steps of mBTS. The rpCGCI has the advantage against other nonlinear Granger causality measures that it can be applied to short time series of high dimension (many observed variables). The simulation study on different types of multivariate stochastic processes and different lengths of generated time series showed the superiority of the proposed rpCGCI as compared to CGCI, rCGCI, pCGCI (using polynomial VAR) and other nonlinear causality measures. Further, rpCGCI was compared favorably to the other measures on signals of heart rate variability, respiration, and oxygen concentration in the blood, as well as multi-channel scalp electroencephalogram (EEG) recordings of epileptic patients containing epileptiform discharges.

Optimal Transport Between Gaussian Stationary Processes

We consider the optimal transport problem between multivariate Gaussian stationary stochastic processes. The transportation effort is the variance of the filtered discrepancy process. The main contribution of this article is to show that the corresponding solution leads to a weighted Hellinger distance between multivariate power spectral densities. Then, we propose a spectral estimation approach in the case of indirect measurements, which is based on this distance.

Maintenance optimisation for systems with multi-dimensional degradation and imperfect inspections

In this paper, we develop a maintenance model for systems subjected to multiple correlated degradation processes, where a multivariate stochastic process is used to model the degradation processes, and the covariance matrix is employed to describe the interactions among the processes. The system is considered failed when any of its degradation features hits the pre-specified threshold. Due to the dormancy of degradation-based failures, inspection is implemented to detect the hidden failures. The failed systems are replaced upon inspection. We assume an imperfect inspection, in such a way that a failure can only be detected with a specific probability. Based on the degradation processes, system reliability is evaluated to serve as the foundation, followed by a maintenance model to reduce the economic losses. We provide theoretical boundaries of the cost-optimal inspection intervals, which are then integrated into the optimisation algorithm to relieve the computational burden. Finally, a fatigue crack propagation process is employed as an example to illustrate the effectiveness and robustness of the developed maintenance policy. Numerical results imply that the inspection inaccuracy contributes significantly to the operating cost and it is suggested that more effort should be paid to improve the inspection accuracy.

Generalised liouville processes and their properties

AbstractWe define a new family of multivariate stochastic processes over a finite time horizon that we call generalised Liouville processes (GLPs). GLPs are Markov processes constructed by splitting Lévy random bridges into non-overlapping subprocesses via time changes. We show that the terminal values and the increments of GLPs have generalised multivariate Liouville distributions, justifying their name. We provide various other properties of GLPs and some examples.

Signature cumulants, ordered partitions, and independence of stochastic processes

The sequence of so-called signature moments describes the laws of many stochastic processes in analogy with how the sequence of moments describes the laws of vector-valued random variables. However, even for vector-valued random variables, the sequence of cumulants is much better suited for many tasks than the sequence of moments. This motivates us to study so-called signature cumulants. To do so, we develop an elementary combinatorial approach and show that in the same way that cumulants relate to the lattice of partitions, signature cumulants relate to the lattice of so-called "ordered partitions". We use this to give a new characterisation of independence of multivariate stochastic processes; finally we construct a family of unbiased minimum-variance estimators of signature cumulants.

Optimal consumption and portfolio decision with stochastic covariance in incomplete markets

Abstract This paper investigates finite and infinite horizon optimal consumption and portfolio choice problems in incomplete markets with stochastic covariance among stock returns. The stock price dynamics are assumed to follow a multivariate stochastic volatility process. The Hamilton-Jacobi-Bellman (HJB) equation is derived by using dynamic programming method and the closed-form expressions for portfolio–consumption strategies are derived in the case of logarithmic utility. In the case of infinite time horizon, we prove the existence of a classical solution and provide a verification theorem. The closed form approximate solutions to the optimal consumption and portfolio polices for a power utility investor are obtained by solving the same utility maximization problem in two fictitious complete markets. Moreover, we verify the existence and optimality of the solution of the original incomplete market and also provide the upper bound of the utility loss of approximate rules. Finally, a numerical application to an incomplete market with stochastic factors show that the utility loss of the approximation rules is small and the performance of the approximate rules is close to optimal.

On the two-step estimation of the cross-power spectrum for dynamical linear inverse problems

We consider the problem of reconstructing the cross-power spectrum of an unobservable multivariate stochastic process from indirect measurements of a second multivariate stochastic process, related to the first one through a linear operator. In the two-step approach, one would first compute a regularized reconstruction of the unobservable signal, and then compute an estimate of its cross-power spectrum from the regularized solution. We investigate whether the optimal regularization parameter for reconstruction of the signal also gives the best estimate of the cross-power spectrum. We show that the answer depends on the regularization method, and specifically we prove that, under a white Gaussian assumption: (i) when regularizing with truncated SVD the optimal parameter is the same; (ii) when regularizing with the Tikhonov method, the optimal parameter for the cross-power spectrum is lower than half the optimal parameter for the signal. We also provide evidence that a one-step approach would likely have better mathematical properties than the two-step approach. Our results apply particularly to the brain connectivity estimation from magneto/electro-encephalographic recordings and provide a formal interpretation of recent empirical results.

Markov equivalence of marginalized local independence graphs

Symmetric independence relations are often studied using graphical representations. Ancestral graphs or acyclic directed mixed graphs with $m$-separation provide classes of symmetric graphical independence models that are closed under marginalization. Asymmetric independence relations appear naturally for multivariate stochastic processes, for instance in terms of local independence. However, no class of graphs representing such asymmetric independence relations, which is also closed under marginalization, has been developed. We develop the theory of directed mixed graphs with $\mu$-separation and show that this provides a graphical independence model class which is closed under marginalization and which generalizes previously considered graphical representations of local independence. For statistical applications, it is pivotal to characterize graphs that induce the same independence relations as such a Markov equivalence class of graphs is the object that is ultimately identifiable from observational data. Our main result is that for directed mixed graphs with $\mu$-separation each Markov equivalence class contains a maximal element which can be constructed from the independence relations alone. Moreover, we introduce the directed mixed equivalence graph as the maximal graph with edge markings. This graph encodes all the information about the edges that is identifiable from the independence relations, and furthermore it can be computed efficiently from the maximal graph.

A general property for time aggregation

We classify all functions of multivariate stochastic processes having time-series estimates that are independent of data frequency. Such an estimator applied to high-frequency data may be used to infer properties of estimates relating to low-frequency data. Our property encompasses two previously-proposed time-aggregation properties (with limited solutions) as different special cases. Our general time-aggregating functions satisfy a pair of coupled second-order partial differential equations. We derive analytic solutions for arbitrary-dimensional martingales and log-martingales. The time-aggregation property of a time-series model is similar – indeed time-aggregating functions always correspond to point estimators based on expected values – but we do not propose a specific new forecasting model. However, we do derive time-aggregating unbiased and efficient estimators for nth-order moments of log returns, applying these results to problems facing portfolio managers who re-optimise portfolios or hedge their risks at lower frequencies than the frequency at which their risk premia are monitored.

Generalised cepstral models for the spectrum of vector time series

The paper treats the modeling of stationary multivariate stochastic processes via a frequency domain model expressed in terms of cepstrum theory. The proposed model nests the vector exponential model of [20] as a special case, and extends the generalised cepstral model of [36] to the multivariate setting, answering a question raised by the last authors in their paper. Contemporarily, we extend the notion of generalised autocovariance function of [35] to vector time series. Then we derive explicit matrix formulas connecting generalised cepstral and autocovariance matrices of the process, and prove the consistency and asymptotic properties of the Whittle likelihood estimators of model parameters. Asymptotic theory for the special case of the vector exponential model is a significant addition to the paper of [20]. We also provide a mathematical machinery, based on matrix differentiation, and computational methods to derive our results, which differ significantly from those employed in the univariate case. The utility of the proposed model is illustrated through Monte Carlo simulation from a bivariate process characterized by a high dynamic range, and an empirical application on time varying minimum variance hedge ratios through the second moments of future and spot prices in the corn commodity market.

Multivariate transformed Gaussian processes

We set up a general framework for modeling non-Gaussian multivariate stochastic processes by transforming underlying multivariate Gaussian processes. This general framework includes multivariate spatial random fields, multivariate time series, and multivariate spatio-temporal processes, whereas the respective univariate processes can also be seen as special cases. We advocate joint modeling of the transformation and the cross-/auto-correlation structure of the latent multivariate Gaussian process, for better estimation and prediction performance. We provide two useful models, the Tukey g-and-h transformed vector autoregressive model and the sinh-arcsinh-transformed multivariate Matérn random field. We evaluate them with a simulation study. Finally, we apply the two models to a wind data set for modeling the two perpendicular components of wind speed vectors. Both the simulation study and data analysis show the advantages of the joint modeling approach.

On OLS Estimation of Stochastic Linear Regression Model

This paper mainly discusses the formulation of stochastic linear statistical model and its assumptions and finally explores an important aspect namely the Ordinary Least Squares (OLS) estimation of stochastic linear regression model. In addition to these inference in stochastic linear regression model is also presented here. Nimitozbay et.al [1], in their paper proposed the weighted mixed regression estimation of the coefficient vector in a linear regression model with stochastic linear restrictions binding the regression coefficients. In 1980, P.A.V.B. Swamy et.al proposed a linear regression model where the coefficient vector is a weekly stationary multivariate stochastic process and that model provides a convenient representation of a general class of non-stationary processes. They proposed prediction and estimation methods which are linear and easy to compute. Daojiang et.al [2] in 2014, in their paper depicted an innovative estimation technique to the multicollinearity in statistical model which is linear in the case of existence of stochastic linear constraints on the parameters and a very different estimation technique was presented by mixing the OME and PCR estimator also known as SRPC regression estimator. In 2014, Shuling Wang et.al [3] in their paper proposed some diagnostic methods in restricted stochastic statistical models which are linear. Gil Gonjalez et.al [4], in 2007, in their paper, derived the LSEs for the simple linear statistical model and examined them from a theoretical perspective.

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

We propose a factor state-space approach with stochastic volatility to model and forecast the term structure of future contracts on commodities. Our 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. Exploiting the conjugacy of the Wishart and the Gaussian distribution, we develop a computationally fast and easy to implement MCMC algorithm for the Bayesian posterior analysis. 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.

Recursive Calculation Model for a Special Multivariate Normal Probability of First-Order Stationary Sequence

The consecutive hit probability of antiaircraft artillery corresponds to the multivariate normal probability distribution. The computational complexity depends on the length of the firing error sequence (i.e., the integral dimension may exceed 100). The traditional numerical integration and the Monte Carlo method are too slow for this calculation. This paper established the state equation of the firing error sequence, which was the bridge between the multivariate normal probability distribution and stochastic process theory. The recursive calculation model was given after the rigorous derivation process. The accuracy and computational complexity of the model were quantified by theoretical analysis and expressed intuitively by examples. The model shows the upper bound for the absolute error, and the computational efficiency is significantly improved.

Multivariate stochastic volatility modeling of neural data.

Because multivariate autoregressive models have failed to adequately account for the complexity of neural signals, researchers have predominantly relied on non-parametric methods when studying the relations between brain and behavior. Using medial temporal lobe (MTL) recordings from 96 neurosurgical patients, we show that time series models with volatility described by a multivariate stochastic latent-variable process and lagged interactions between signals in different brain regions provide new insights into the dynamics of brain function. The implied volatility inferred from our process positively correlates with high-frequency spectral activity, a signal that correlates with neuronal activity. We show that volatility features derived from our model can reliably decode memory states, and that this classifier performs as well as those using spectral features. Using the directional connections between brain regions during complex cognitive process provided by the model, we uncovered perirhinal-hippocampal desynchronization in the MTL regions that is associated with successful memory encoding.

Multivariate Stochastic Volatility Modeling of Neural Data

Because multivariate autoregressive models have failed to adequately account for the complexity of neural signals, researchers have predom- inantly relied on non-parametric methods when studying the relations between brain and behavior. Using medial temporal lobe (MTL) recordings from 96 neurosurgical patients, we show that time series models with volatility described by a multivariate stochastic latent-variable process and lagged interactions between signals in different brain regions provide new insights into the dynamics of brain function. The implied volatility inferred from our process positively correlates with high-frequency spectral activity, a signal that correlates with neuronal activity. We show that volatility features derived from our model can reliably decode memory states, and that this classifier performs as well as those using spectral features. Using the directional connections between brain regions during complex cognitive process provided by the model, we uncovered perirhinal-hippocampal desynchronization in the MTL regions that is associated with successful memory encoding.