Vector-valued Gaussian Processes Research Articles

Extremes of locally-homogenous vector-valued Gaussian processes

AbstractIn this paper, we study the asymptotical behaviour of high exceedence probabilities for centered continuous $$\mathbb {R}^n$$ R n -valued Gaussian random field $$\varvec{X}$$ X with covariance matrix satisfying $$\Sigma - R ( t + s, t ) \sim \sum _{l = 1}^n B_l ( t ) \, | s_l |^{\alpha _l}$$ Σ - R ( t + s , t ) ∼ ∑ l = 1 n B l ( t ) | s l | α l as $$s \downarrow 0$$ s ↓ 0 . Such processes occur naturally as time transformations of homogenous random fields, and we present two asymptotic results of this nature as applications of our findings. The technical novelty of our proof consists in showing that the Slepian-Gordon inequality technique, essential in the univariate case, can also be successfully applied in the multivariate setup. This is noteworthy because this technique was previously believed to be inaccessible in this particular context.

Multi-fidelity modeling using Gaussian processes for the frequency dependent Helmholtz equation

In scientific and engineering problems, accurate model predictions are generally accompanied with high resource and time expenses. In most applications, the data generation process can only be performed for a limited number of forward evaluations leading to insufficient model results. The concept of multi-fidelity modeling is to holistically evaluate data from different sources with varying complexity and costs. Ideally, the predictive quality of models with a high-fidelity can be combined with the efficiency of low-fidelity models. Usually, physical experiments and numerical simulations at large-scale are attributed to high-fidelity models, while analytical solutions or small-scale numerical models and experiments are considered as low-fidelity models. This contribution aims to develop a multi-fidelity model for an interior sound propagation problem. More specifically, a vehicle interior noise problem is investigated by using two boundary element models of different fidelity levels. The finer mesh corresponds to the high-fidelity model, whereas the coarser mesh is adopted as the low-fidelity model. As such, the multi-fidelity model is realized as a vector-valued Gaussian process.

Renewal theory for iterated perturbed random walks on a general branching process tree: Early generations

AbstractLet $(\xi_k,\eta_k)_{k\in\mathbb{N}}$ be independent identically distributed random vectors with arbitrarily dependent positive components. We call a (globally) perturbed random walk a random sequence $T\,{:\!=}\, (T_k)_{k\in\mathbb{N}}$ defined by $T_k\,{:\!=}\, \xi_1+\cdots+\xi_{k-1}+\eta_k$ for $k\in\mathbb{N}$ . Consider a general branching process generated by T and let $N_j(t)$ denote the number of the jth generation individuals with birth times $\leq t$ . We treat early generations, that is, fixed generations j which do not depend on t. In this setting we prove counterparts for $\mathbb{E}N_j$ of the Blackwell theorem and the key renewal theorem, prove a strong law of large numbers for $N_j$ , and find the first-order asymptotics for the variance of $N_j$ . Also, we prove a functional limit theorem for the vector-valued process $(N_1(ut),\ldots, N_j(ut))_{u\geq0}$ , properly normalized and centered, as $t\to\infty$ . The limit is a vector-valued Gaussian process whose components are integrated Brownian motions.

A Bayesian Optimization Approach to the Simultaneous Extraction of Intrinsic Physical Parameters fromT1 andT2 Relaxation Responses

SummaryNuclear magnetic resonance (NMR) relaxation responses in porous media provide a sensitive probe of the microstructure and yet are influenced by a number of factors which are not easily detangled. Low-field T2 transverse relaxation measurements can be carried out quickly and are frequently used as pore size distributions, while adding T1 longitudinal relaxation measurements provides additional insights into surface properties and fluid content. Here we present an inverse solution workflow extracting related intrinsic physical parameters of the system by fitting experiment and numerical simulation(s). An efficient NMR forward solver for the simultaneous calculation of T1 and T2 responses is introduced, which honors existing inequality relationships between T1 and T2 parameters. We propose a Bayesian optimization approach that jointly identifies T1- and T2-related properties satisfying physical constraints by simultaneously fitting T1 and T2 experiments to simulations. This dual-task inverse solution workflow (DT-ISW) identifies the solution by minimizing the sum of the L2 norm of the fitting residuals of both T1 and T2 distributions into a single objective and jointly models the two highly correlated objectives with high efficiency using the vector-valued Gaussian process (GP) kernel for transfer learning. A multimodal search strategy is used to identify nonunique solution sets of the problem. The workflow is demonstrated on Bentheimer sandstone, identifying five intrinsic physical parameters. The performance of the joint DT-ISW (DT-ISW-J) is compared to a sequential DT-ISW (DT-ISW-S) approach as well as an independent single-task ISW (ST-ISW) of the T1 and T2 responses. Both dual-task versions converge more than two times faster than the single-task version. DT-ISW-J equally minimizes the L2 norm of T1 and T2 fitting residuals whereas DT-ISW-S only preferentially minimizes the objective assigned higher importance. A Pareto optimal solution (POS) is provided to allow operators to subjectively balance the preference of T1 and T2 data fits for the slightly conflicting objectives. The ability to extract five intrinsic physical parameters simultaneously provides new techniques for tracking wettability alteration and assessing the influence of clay amount and distribution on petrophysical property estimates.

Limit theorem on the pointwise maxima of minimum of vector-valued Gaussian processes

In this paper, the limit properties of properly time-scaled and normalized maxima of minimum of vector-valued Gaussian processes are studied. It is shown that the maxima of dependent samples of those processes converge weakly on the space of continuous functions to a stochastic process with explicit finite-dimensional distributions.

LQG control and linear policies for noisy communication links with synchronized side information at the decoder

We consider a multidimensional closed-loop control problem where a stochastic linear time-invariant plant is connected via a communication system that includes an encoder, vector additive Gaussian noise (AGN) channel, decoder to a controller. For this setup, we first examine the impact of the communication link when causal side information (CSI) that is correlated to the plant is (possibly) allowed at the decoder and the performance criterion is the classical linear quadratic (LQ) cost. The CSI is modeled as a time-invariant vector-valued Gaussian process and when available at the decoder, it happens concurrently to the signal obtained from the vector AGN channel. Under the assumption that our encoder is linear, we leverage the separation principle of control and estimation, to obtain the following new results. (i) For multivariate Gaussian processes, we completely characterize a lower bound on the communication cost term of the optimal LQ Gaussian (LQG) cost in the form of a convex optimization problem and identify the best linear policies that achieve this bound. (ii) For scalar-valued Gaussian processes, we derive an analytical solution to the lower bound of the optimal communication cost term and show that for linear policies, CSI at the encoder does not improve the specific bound. (iii) For scalar-valued Gaussian processes, we also derive a linear time-invariant low delay joint source-channel coding (JSCC) scheme without bandwidth constraints, when CSI is available at the encoder, which demonstrates the case at which the specific lower bound communication cost expression is achievable. Further, we give a suboptimal linear time-invariant JSCC scheme when CSI is available only at the decoder. We supplement our results with various simulation studies.

Extremes of vector-valued Gaussian processes

The seminal papers of Pickands (Pickands, 1967; Pickands, 1969) paved the way for a systematic study of high exceedance probabilities of both stationary and non-stationary Gaussian processes. Yet, in the vector-valued setting, due to the lack of key tools including Slepian’s Lemma, there has not been any methodological development in the literature for the study of extremes of vector-valued Gaussian processes. In this contribution we develop the uniform double-sum method for the vector-valued setting, obtaining the exact asymptotics of the high exceedance probabilities for both stationary and n on-stationary Gaussian processes. We apply our findings to the operator fractional Brownian motion and Ornstein–Uhlenbeck process.

Extremes of vector-valued Gaussian processes with Trend

Let X(t)=(X1(t),…,Xn(t)),t∈T⊂R be a centered vector-valued Gaussian process with independent components and continuous trajectories, and h(t)=(h1(t),…,hn(t)),t∈T be a vector-valued continuous function. We investigate the asymptotics ofP{supt∈T⁡min1≤i≤n⁡(Xi(t)+hi(t))>u} as u→∞. As an illustration to the derived results we analyze two important classes of X(t): with locally-stationary structure and with varying variances of the coordinates, and calculate exact asymptotics of simultaneous ruin probability and ruin time in a Gaussian risk model.

A functional limit theorem for the profile of random recursive trees

Let $X_n(k)$ be the number of vertices at level $k$ in a random recursive tree with $n+1$ vertices. We prove a functional limit theorem for the vector-valued process $(X_{[n^t]}(1),\ldots , X_{[n^t]}(k))_{t\geq 0}$, for each $k\in \mathbb N$. We show that after proper centering and normalization, this process converges weakly to a vector-valued Gaussian process whose components are integrated Brownian motions. This result is deduced from a functional limit theorem for Crump-Mode-Jagers branching processes generated by increasing random walks with increments that have finite second moment. Let $Y_k(t)$ be the number of the $k$th generation individuals born at times $\leq t$ in this process. Then, it is shown that the appropriately centered and normalized vector-valued process $(Y_{1}(st),\ldots , Y_k(st))_{t\geq 0}$ converges weakly, as $s\to \infty $, to the same limiting Gaussian process as above.

Cardiorespiratory Model-Based Data-Driven Approach for Sleep Apnea Detection.

Obstructive sleep apnea (OSA) is a chronic sleep disorder affecting millions of people worldwide. Individuals with OSA are rarely aware of the condition and are often left untreated, which can lead to some serious health problems. Nowadays, several low-cost wearable health sensors are available that can be used to conveniently and noninvasively collect a wide range of physiological signals. In this paper, we propose a new framework for OSA detection in which we combine the wearable sensor measurement signals with the mathematical models of the cardiorespiratory system. Vector-valued Gaussian processes (GPs) are adopted to model the physiological variations among different individuals. The GP covariance is constructed using the sum of separable kernel functions, and the GP hyperparameters are estimated by maximizing the marginal likelihood function. A likelihood ratio test is proposed to detect OSA using the widely available heart rate and peripheral oxygen saturation (SpO ) measurement signals. We conduct experiments on both synthetic and real data to show the effectiveness of the proposed OSA detection framework compared to purely data-driven approaches.

Extremes of vector-valued Gaussian processes: Exact asymptotics

Let {Xi(t),t≥0},1≤i≤n be mutually independent centered Gaussian processes with almost surely continuous sample paths. We derive the exact asymptotics of P(∃t∈[0,T]∀i=1,…,nXi(t)>u) as u→∞, for both locally stationary Xi’s and Xi’s with a non-constant generalized variance function. Additionally, we analyze properties of multidimensional counterparts of the Pickands and Piterbarg constants that appear in the derived asymptotics. Important by-products of this contribution are the vector-process extensions of the Piterbarg inequality, the Borell–TIS inequality, the Slepian lemma and the Pickands–Piterbarg lemma which are the main pillars of the extremal theory of vector-valued Gaussian processes.

Estimating extremes of combined two Gaussian and non-Gaussian response processes

Assessment of structural performance under stochastic dynamic loadings requires estimation of the extremes of stochastic response components and the resultant responses as their linear and nonlinear combinations. This paper addresses the evaluations and combination rules for the extremes of scalar and vectorial resultant responses from two response components that may show non-Gaussian characteristics. The non-Gaussian response process is modeled as a translation process from an underlying Gaussian process. The mean crossing rates and extreme value distributions of resultant responses are calculated following the theory for vector-valued Gaussian processes. An extensive parameter study is conducted concerning the influence of statistical moments of non-Gaussian response components on the extremes of resultant responses. It is revealed that the existing combination rules developed for Gaussian processes are not applicable to the case of non-Gaussian process. New combination rules are suggested that permit predictions of the extremes of resultant responses directly from the extremes of response components.

Correlation Inequalities and Applications to Vector-Valued Gaussian Random Variables and Fractional Brownian Motion

In this paper we extend certain correlation inequalities for vector-valued Gaussian random variables due to Kolmogorov and Rozanov. The inequalities are applied to sequences of Gaussian random variables and Gaussian processes. For sequences of Gaussian random variables satisfying a correlation assumption, we prove a Borel-Cantelli lemma, maximal inequalities and several laws of large numbers. This extends results of Beśka and Ciesielski and of Hytonen and the author. In the second part of the paper we consider a certain class of vector-valued Gaussian processes which are α-Holder continuous in p-th moment. For these processes we obtain Besov regularity of the paths of order α. We also obtain estimates for the moments in the Besov norm. In particular, the results are applied to vector-valued fractional Brownian motions. These results extend earlier work of Ciesielski, Kerkyacharian and Roynette and of Hytonen and the author.

Distribution of wave crests in a non-Gaussian sea

The sea elevation at a fixed point is modelled as a quadratic form of a vector valued Gaussian process with arbitrary mean. With this model, saddlepoint methods are used to approximate the mean upcrossing intensity with which the sea level crosses upwards at a certain height. This estimated intensity is further used to determine the probability distribution of wave crests. The use of saddlepoint technique is particularly important here because it can approximate the crest distribution without the need to perform simulations or use fitted distributions. Several numerical examples are given, including two with measured data. In the cases of real data, the results obtained with the saddlepoint technique are also compared with the results obtained with well known methods commonly used in the industry.

Evaluation of peak resultant response for wind-excited tall buildings

The traditional peak factor approach and combination rules developed for Gaussian scalar combination effects are not applicable for vectorial combination effects which feature non-Gaussian characteristics. This paper addresses the evaluation of the peak value of a vectorial resultant response of a wind-excited tall building resulting from correlated three-dimensional responses with a special focus on building acceleration. The peak value of the resultant response over a given duration is analyzed based on the upcrossing theory for non-Gaussian processes and for vector-valued Gaussian processes. A parametric study including the correlation of response components and their derivative processes is conducted to identify the controlling parameters in estimating the upcrossing rate and peak resultant response. The accuracy of the current empirical combination approaches is evaluated. A practical combination scheme is proposed and demonstrated using a wind-excited tall building with three-dimensional mode shapes. The results show that the existence of eccentricity leads to coupled alongwind, crosswind and torsional accelerations, and considerably amplifies the peak resultant acceleration.

The expectation of solution of random continuity equation with Gaussian velocity field

The continuity equation in d-dimensional space with ran- dom velocity eld dened by means of a vector-valued Gaussian process is studied. The expectation of corresponding evolution family of operators is explicitly derived, generalizing in a sense the evolution family correspond- ing to the conventional diusion equation.

Wave Statistics in Non-Linear Random Sea

The sea elevation at a fixed point is modeled as a sum of a Gaussian process plus a quadratic random correction term. It is shown that the process can also be written as a quadratic form of a vector valued Gaussian process with arbitrary mean. The saddlepoint method is used to approximate the intensity μ (u), say, the sea level crosses the level u. The accuracy of the proposed method is studied. In examples the computed intensity is used to bound the wave crest distribution. The bounds are compared with empirical distributions derived from simulations.

Large Deviation Probabilities for Square-Gaussian Stochastic Processes

We investigate properties of square-Gaussian stochastic processes. These processes are formed by quadratic forms of Gaussian processes or by limits in the mean square of quadratic forms of Gaussian processes. Special classes of these processes are determined and investigated. For processes from these classes estimates of large deviation probability are obtained. These estimates we use to estimate the probability that Gaussian vector-valued process leave some region on some interval of time. We construct asymptotic confidence regions for estimates of covariance functions of vector-valued Gaussian processes. Criterion of hypothesis testing on covariance functions of these processes is constructed.