Academic literature on the topic 'Multivariate stationary process'

A criterion is given for the existence of a stationary and causal multivariate integer-valued autoregressive process, MGINAR(p). The autocovariance function of this process being identical to the autocovariance function of a standard Gaussian MAR(p), we deduce that the MGINAR(p) process is nothing but a MAR(p) process. Consequently, the spectral density is directly found and gives good insight into the stochastic structure of a MGINAR(p). The estimation of parameters of the model, as well as the forecasting of the series, is discussed.

A criterion is given for the existence of a stationary and causal multivariate integer-valued autoregressive process, MGINAR(p). The autocovariance function of this process being identical to the autocovariance function of a standard Gaussian MAR(p), we deduce that the MGINAR(p) process is nothing but a MAR(p) process. Consequently, the spectral density is directly found and gives good insight into the stochastic structure of a MGINAR(p). The estimation of parameters of the model, as well as the forecasting of the series, is discussed.

LetX= {Xt:t≥ 0} be a stationary piecewise continuousRd-valued process that moves between jumps along the integral curves of a given continuous vector field, and letS⊂Rdbe a smooth surface. The aim of this paper is to derive a multivariate version of Rice's formula, relating the intensity of the point process of (localized) continuous crossings ofSbyXto the distribution ofX0. Our result is illustrated by examples relating to queueing networks and stress release network models.

In the characterization of multivariate extremal indices of multivariate stationary processes, multivariate maxima of moving maxima processes, or M4 processes for short, have been introduced by Smith and Weissman. Central to the introduction of M4 processes is that the extreme observations of multivariate stationary processes may be characterized in terms of a limiting max-stable process under quite general conditions, and that a max-stable process can be arbitrarily closely approximated by an M4 process. In this paper, we derive some additional basic probabilistic properties for a finite class of M4 processes, each of which contains finite-range clustered moving patterns, called signature patterns, when extreme events occur. We use these properties to construct statistical estimation schemes for model parameters.

In the characterization of multivariate extremal indices of multivariate stationary processes, multivariate maxima of moving maxima processes, or M4 processes for short, have been introduced by Smith and Weissman. Central to the introduction of M4 processes is that the extreme observations of multivariate stationary processes may be characterized in terms of a limiting max-stable process under quite general conditions, and that a max-stable process can be arbitrarily closely approximated by an M4 process. In this paper, we derive some additional basic probabilistic properties for a finite class of M4 processes, each of which contains finite-range clustered moving patterns, called signature patterns, when extreme events occur. We use these properties to construct statistical estimation schemes for model parameters.

Let X = {Xt: t ≥ 0} be a stationary piecewise continuous Rd-valued process that moves between jumps along the integral curves of a given continuous vector field, and let S ⊂ Rd be a smooth surface. The aim of this paper is to derive a multivariate version of Rice's formula, relating the intensity of the point process of (localized) continuous crossings of S by X to the distribution of X0. Our result is illustrated by examples relating to queueing networks and stress release network models.

We derive multivariate moment generating functions for the conditional and stationary distributions of a discrete sample path of n observations of a square-root diffusion (CIR) process, X(t). For any fixed vector of observation times t1,…,tn, we find the conditional joint distribution of (X(t1),…,X(tn)) is a multivariate noncentral chi-squared distribution and the stationary joint distribution is a Krishnamoorthy-Parthasarathy multivariate gamma distribution. Multivariate cumulants of the stationary distribution have a simple and computationally tractable expression. We also obtain the moment generating function for the increment X(t + δ) - X(t), and show that the increment is equivalent in distribution to a scaled difference of two independent draws from a gamma distribution.

Dans un grand éventail d'applications allant des sciences du climat à la finance, des événements extrêmes avec une probabilité loin d'être négligeable peuvent se produire, entraînant des conséquences désastreuses. Les extrêmes d'évènements climatiques tels que le vent, la température et les précipitations peuvent profondément affecter les êtres humains et les écosystèmes, entraînant des événements tels que des inondations, des glissements de terrain ou des vagues de chaleur. Lorsque l'emphase est mise sur l'étude de variables mesurées dans le temps sur un grand nombre de stations ayant une localisation spécifique, comme les variables mentionnées précédemment, le partitionnement de variables devient essentiel pour résumer et visualiser des tendances spatiales, ce qui est crucial dans l'étude des événements extrêmes. Cette thèse explore plusieurs modèles et méthodes pour partitionner les variables d'un processus stationnaire multivarié, en se concentrant sur les dépendances extrémales.Le chapitre 1 présente les concepts de modélisation de la dépendance via les copules, fondamentales pour la dépendance extrême. La notion de variation régulière est introduite, essentielle pour l'étude des extrêmes, et les processus faiblement dépendants sont abordés. Le partitionnement est discuté à travers les paradigmes de séparation-proximité et de partitionnement basé sur un modèle. Nous abordons aussi l'analyse non-asymptotique pour évaluer nos méthodes dans des dimensions fixes.Le chapitre 2 est à propos de la dépendance entre valeurs maximales est cruciale pour l'analyse des risques. Utilisant la fonction de copule de valeur extrême et le madogramme, ce chapitre se concentre sur l'estimation non paramétrique avec des données manquantes. Un théorème central limite fonctionnel est établi, démontrant la convergence du madogramme vers un processus Gaussien tendu. Des formules pour la variance asymptotique sont présentées, illustrées par une étude numérique.Le chapitre 3 propose les modèles asymptotiquement indépendants par blocs (AI-blocs) pour le partitionnement de variables, définissant des clusters basés sur l'indépendance des maxima. Un algorithme est introduit pour récupérer les clusters sans spécifier leur nombre à l'avance. L'efficacité théorique de l'algorithme est démontrée, et une méthode de sélection de paramètre basée sur les données est proposée. La méthode est appliquée à des données de neurosciences et environnementales, démontrant son potentiel.Le chapitre 4 adapte des techniques de partitionnement pour analyser des événements extrêmes composites sur des données climatiques européennes. Les sous-régions présentant une dépendance des extrêmes de précipitations et de vitesse du vent sont identifiées en utilisant des données ERA5 de 1979 à 2022. Les clusters obtenus sont spatialement concentrés, offrant une compréhension approfondie de la distribution régionale des extrêmes. Les méthodes proposées réduisent efficacement la taille des données tout en extrayant des informations cruciales sur les événements extrêmes.Le chapitre 5 propose une nouvelle méthode d'estimation pour les matrices dans un modèle linéaire à facteurs latents, où chaque composante d'un vecteur aléatoire est exprimée par une équation linéaire avec des facteurs et du bruit. Contrairement aux approches classiques basées sur la normalité conjointe, nous supposons que les facteurs sont distribués selon des distributions de Fréchet standards, ce qui permet une meilleure description de la dépendance extrémale. Une méthode d'estimation est proposée garantissant une solution unique sous certaines conditions. Une borne supérieure adaptative pour l'estimateur est fournie, adaptable à la dimension et au nombre de facteurs
In a wide range of applications, from climate science to finance, extreme events with a non-negligible probability can occur, leading to disastrous consequences. Extremes in climatic events such as wind, temperature, and precipitation can profoundly impact humans and ecosystems, resulting in events like floods, landslides, or heatwaves. When the focus is on studying variables measured over time at numerous specific locations, such as the previously mentioned variables, partitioning these variables becomes essential to summarize and visualize spatial trends, which is crucial in the study of extreme events. This thesis explores several models and methods for partitioning the variables of a multivariate stationary process, focusing on extreme dependencies.Chapter 1 introduces the concepts of modeling dependence through copulas, which are fundamental for extreme dependence. The notion of regular variation, essential for studying extremes, is introduced, and weakly dependent processes are discussed. Partitioning is examined through the paradigms of separation-proximity and model-based clustering. Non-asymptotic analysis is also addressed to evaluate our methods in fixed dimensions.Chapter 2 study the dependence between maximum values is crucial for risk analysis. Using the extreme value copula function and the madogram, this chapter focuses on non-parametric estimation with missing data. A functional central limit theorem is established, demonstrating the convergence of the madogram to a tight Gaussian process. Formulas for asymptotic variance are presented, illustrated by a numerical study.Chapter 3 proposes asymptotically independent block (AI-block) models for partitioning variables, defining clusters based on the independence of maxima. An algorithm is introduced to recover clusters without specifying their number in advance. Theoretical efficiency of the algorithm is demonstrated, and a data-driven parameter selection method is proposed. The method is applied to neuroscience and environmental data, showcasing its potential.Chapter 4 adapts partitioning techniques to analyze composite extreme events in European climate data. Sub-regions with dependencies in extreme precipitation and wind speed are identified using ERA5 data from 1979 to 2022. The obtained clusters are spatially concentrated, offering a deep understanding of the regional distribution of extremes. The proposed methods efficiently reduce data size while extracting critical information on extreme events.Chapter 5 proposes a new estimation method for matrices in a latent factor linear model, where each component of a random vector is expressed by a linear equation with factors and noise. Unlike classical approaches based on joint normality, we assume factors are distributed according to standard Fréchet distributions, allowing a better description of extreme dependence. An estimation method is proposed, ensuring a unique solution under certain conditions. An adaptive upper bound for the estimator is provided, adaptable to dimension and the number of factors

AbstractGeodetic measurements rely on high-resolution sensors, but produce data sets with many observations which may contain outliers and correlated deviations. This paper proposes a powerful solution using Bayesian inference. The observed data is modeled as a multivariate time series with a stationary autoregressive (VAR) process and multivariate t-distribution for white noise. Bayes’ theorem integrates prior knowledge. Parameters, including functional, VAR coefficients, scaling, and degree of freedom of the t-distribution, are estimated with Markov Chain Monte Carlo using a Metropolis-within-Gibbs algorithm.

Here we introduce the notion of asymptotic weakly associated dependence conditions, the practical applications of which will be discussed in the next chapter. The theoretical importance of this class of random variables is that it leads to the functional CLT without the need to estimate rates of convergence of mixing coefficients. More precisely, because of the maximal moment inequalities established in the previous chapter, we are able to prove tightness for a stochastic process constructed from a negatively dependent sequence. Furthermore, we establish the convergence of the partial sums process, either to a Gaussian process with independent increments or to a diffusion process with deterministic time-varying volatility. We also provide the multivariate form of these functional limit theorems. The results are presented in the non-stationary setting, by imposing Lindeberg’s condition. Finally, we give the stationary form of our results for both asymptotic positively and negatively associated sequences of random variables.

Abstract The theoretical evolution of the multivariate trait mean has been most explored in the case of a hill-shaped adaptive landscape that has a consistent position, shape, and orientation. Under particular but general conditions, the mean tends to evolve uphill on this stationary landscape, towards the adaptive peak. When two or more traits are genetically correlated, the evolutionary path towards the peak is generally curved rather than straight. The big challenge is to develop generalizations about how multivariate evolution will proceed on different kinds of adaptive peaks. The effect of peak configuration on multivariate trait radiations has been explored by assuming that the peak is multivariate Gaussian. Theory is especially well developed for systems in which a single male trait (ornament) evolves in response to sexual selection exerted by a second, female trait (sexual preference). Under these conditions, the equilibrium of the bivariate mean may be a stable point, a line or an elliptical cycle. Unstable dynamics are also possible, but probably unlikely. Finite population size adds an element of uncertainty to the outcome, changing stable points into clouds, for example. Predictions from these models are consistent with important features of actual sexual radiations, but only a few discriminating, quantitative tests have been conducted.

Abstract This chapter deals with empirical periodic models for multivariate time series with one or more stochastic trends. In the previous two chapters we have seen that for univariate variables periodic time series models are most easily analysed in the vector of quarters (VQ) form. This amounts to investigating the properties of a quarterly observed time series y1via an analysis of the properties of the (4 x 1) vector process YT ‘ which contains the observations in each of the quarters. As noted in Section 7.5, the analysis of the stationarity property of a periodic VAR (PVAR) model can be quite involved, even for the simple PVAR(l) model (see equation (7.51) ). In this chapter I propose two reasonably simple alternative approaches to analysing a multivariate time series using a periodic model.

In the present work, a nonstationary stochastic model, which is suitable for the analysis and simulation of multivariate time series of wind and wave data, is being presented and validated. This model belongs to the class of periodically correlated stochastic processes with yearly periodic mean value and standard deviation (periodically correlated or cyclostationary stochastic process). First, the time series is appropriately transformed to become Gaussian using the Box-Cox transformation. Then, the series is decomposed, using an appropriate seasonal standardization procedure, to a periodic (deterministic) mean value and a (stochastic) residual time series multiplied by a periodic (deterministic) standard deviation. The periodic components are estimated using appropriate time series of monthly data. The residual stochastic part, which is proved to be stationary, is modelled as a VARMA process. This way the initial process can be given the structure of a multivariate periodically correlated process. The present methodology permits a reliable reproduction of available information about wind and wave conditions, which is required for a number of applications.

Abstract Early detection and characterization of anomalous events during drilling operations are critical to avoid costly downtime and prevent hazardous events, such as a stuck pipe or a well control event. A key aspect of real-time drilling data analysis is the capability to make precise predictions of specific drilling parameters based on past time series information. The ideal models should be able to deal with multivariate time series and perform multi-step predictions. The recurrent neural network with a long short-term memory (LSTM) architecture is capable of the task, however, given that drilling is a long process with high data sampling frequency, LSTMs may face challenges with ultra-long-term memory. The transformer-based deep learning model has demonstrated its superior ability in natural language processing and time series analysis. The self-attention mechanism enables it to capture extremely long-term memory. In this paper, transformer-based deep learning models have been developed and applied to real-time drilling data prediction. It comprises an encoder and decoder module, along with a multi-head attention module. The model takes in multivariate real-time drilling data as input and predicts a univariate parameter in advance for multiple time steps. The proposed model is applied to the Volve field data to predict real-time drilling parameters such as mud pit volume, surface torque, and standpipe pressure. The predicted results are observed and evaluated. The predictions of the proposed models are in good agreement with the ground truth data. Four Transformer-based predictive models demonstrate their applicability to forecast real-time drilling data of different lengths. Transformer models utilizing non-stationary attention exhibit superior prediction accuracy in the context of drilling data prediction. This study provides guidance on how to implement and apply transformer-based deep learning models applied to drilling data analysis tasks, with a specific focus on anomaly detection. When trained on dysfunction-free datasets, the proposed model can predict real-time drilling data with high precision, whereas when a downhole anomaly starts to build, the significant error in the prediction can be used as an alarm indicator. The model can consider extremely long-term memory and serve as the alternative algorithm to LSTM. Furthermore, this model can be extended to a wide range of sequence data prediction problems in the petroleum engineering discipline.

Virtual reality (VR) has many applications. However, not all users can enjoy them equally due to cybersickness, a form of visually induced motion sickness in VR. To increase the accessibility of VR, countermeasures against cybersickness are needed. The requirements for a good countermeasure are a reasonable effect size, especially since susceptibility varies between individuals, while reducing immersion as little as possible. One idea that seems to meet these requirements, the virtual nose, has been tested with small samples – from which large effect sizes can be derived – and allows universal applicability. The mode of action of the virtual nose derives from the rest frame hypothesis: Certain objects that are perceived as stationary serve as a rest frame, facilitating the self-calibration of the body. In addition, the rest frame may not only act as a postural corrector, which should be observable by a reduction of postural sway, but also as a fixation cross, which should be observable by longer and more frequent fixations. This study tested whether a virtual nose (treatment group) significantly reduced cybersickness compared to a group without a virtual nose (control group) and whether physiological process indicators, namely head and eye tracking, differed between the groups with a larger sample size than previous studies. Participants were matched into the treatment and control group according to their gender and previous VR experience, as these aspects are discussed to influence cybersickness susceptibility. Experience groups were divided into three: none, less than 30 min of VR experience, and more than 30 min. A total of 124 participants were recruited, of which 110 were eligible for the analyses (multivariate repeated measures analysis and Holm-corrected univariate post-hoc tests). During the VR exposure, the participants’ task was to explore a virtual city and collect checkpoints. The questionnaires used were the Virtual Reality Sickness Questionnaire (VRSQ) for a pre-post-comparison and the Misery Scale (MISC) applied every 2 min during the VR exposure. The continuously sampled process indicators were cut into these fixed 2-minute intervals for the analyses. The results show no mitigating effect of the treatment. Nevertheless, the reported cybersickness was significantly lower in the more experienced group and significantly higher in the inexperienced group compared to the low-experienced group. The process indicators head and eye tracking mostly confirm the mitigating effect of previous VR experience on cybersickness susceptibility but do not differ between the treatment and control group. It can be argued that the artificiality of a virtual nose that is added to a scene nullifies the mitigating effect by reducing immersion. It may also be that the stimulus needs to be more salient to be effective. In summary, prior experience with VR was the mitigating factor. As the process indicators and the controller input differ, one explanation could be a behavioral adaptation with increasing VR experience. Alternative explanations, such as a gender- or experience-specific pre-selection effect for VR studies, are discussed.

Multivariate time series (MTS) forecasting has been extensively applied across diverse domains, such as weather prediction and energy consumption. However, current studies still rely on the vanilla point-wise self-attention mechanism to capture cross-variable dependencies, which is inadequate in extracting the intricate cross-correlation implied between variables. To fill this gap, we propose Variable Correlation Transformer (VCformer), which utilizes Variable Correlation Attention (VCA) module to mine the correlations among variables. Specifically, based on the stochastic process theory, VCA calculates and integrates the cross-correlation scores corresponding to different lags between queries and keys, thereby enhancing its ability to uncover multivariate relationships. Additionally, inspired by Koopman dynamics theory, we also develop Koopman Temporal Detector (KTD) to better address non-stationarity in time series. The two key components enable VCformer to extract both multivariate correlations and temporal dependencies. Our extensive experiments on eight real-world datasets demonstrate the effectiveness of VCformer, achieving top-tier performance compared to other state-of-the-art baseline models. Code is available at this repository: https://github.com/CSyyn/VCformer.

Abstract Maps of organic matter distributions are an essential part of the basin modeling process for evaluation of conventional and unconventional hydrocarbon resources. This approach has become widespread for both assessment of the initial oil and gas generation potential and final assessment of the resource base. An important step in this method is pyrolysis of the core material, which is carried out in order to assess the geochemical parameters of source rocks. The data can be used to create maps of the distributions of geochemical parameters, such as the organic carbon content, the hydrogen index, the oxygen index, S1,S2,S3, TMax. The accepted assumption regarding the distribution of these parameters determines the entire modeling result and the final generation volumes. It is difficult to overestimate the impact of this process on the result of assessing the prospects for both traditional and shale hydrocarbons. However, a lot of uncertainty is associated with interpolation of parameters analyzed in wells into the interwell space because different interpolation methods give fundamentally different results. Deterministic interpolation methods are limited because of their nature and thus cannot be used for probabilistic estimation, so they are not considered in this work. There are two interpolation methods that can perform this process in a multivariate form: the geostatistical approach (SGS, GRFS) and the new Amazonas method, which integrates elements of artificial intellect. The stochastic approach to modeling has the undeniable advantages of estimating probabilities, and it includes uncertainties, which demonstrates the general idea more clearly. In most geology-related fields, the deterministic approach is gradually fading into the background, giving way to stochastic methods. This article considers a comparison of the SGS and Amazonas methods, their prerequisites and limitations, advantages and disadvantages using the example of a database of geochemical measurements of the Bazhenov source rock data. The results of modeling by both methods are shown in this work along with the issues of statistical stability of the results. The geostatistical approach in interpolation has been used for decades. It has a simple and understandable operating principle, but there are several fundamental limitations. These limitations are associated with the assumption of stationarity of the random variable under study. The proposed Amazonas method also offers a simple and understandable way to calculate the quantity being analyzed at each point in space. However, the latter does not have the limitations of the geostatistical approach. Besides, this method does not require the condition of stationarity for the quantity being analyzed and is also capable, depending on the settings, of reproducing both smooth and abrupt transitions.