Multivariate Brownian Motion Research Articles

Simultaneous Model Change Detection in Multivariate Linear Regression With Application to Indonesian Economic Growth Data

In this paper, we study asymptotic model change detection in multivariate linear regression by using the Kolmogorov–Smirnov function of the partial sum process of recursive residuals. We approximate the rejection region and also the power function of the test by establishing a functional central limit theorem for the sequence of the partial sum processes of the recursive residuals of the observations. When the assumed model is true, the limit process is given by the standard multivariate Brownian motion which does not depend on the regression functions. However, when the assumed model is not true (some models change), the limit process is represented by a vector of deterministic trend plus the standard multivariate Brownian motion. The finite sample size rejection region and the power of the test are investigated by means of Monte Carlo simulation. The simulation study shows evidence that the proposed test is consistent in the sense that it attains the power larger than the size of the test when the hypothesis is not true. We also demonstrate the application of the proposed test method to Indonesian economic growth data in which we test the adequacy of three‐variate low‐order polynomial model. The test result shows that the growth of the Indonesian economy is neither simultaneously constant nor linear. The test has successfully detect the appearance of a change in the model which is mainly caused by the COVID‐19 pandemic in 2020.

Estimation and asymptotics for vector autoregressive models with unit roots and Markov switching trends

We provide a formal definition of an M-state multivariate Markov switching trend, describe its asymptotic distribution, and consider vector autoregressive processes with trends which contain either unit roots or a stationary part. Then, we estimate the coefficients of such models via ordinary least squares , and determine the asymptotic distributions of estimators in terms of functionals on a multivariate Brownian motion.

Multivariate exponential power Lévy processes and random fields

Two multivariate exponential power Lévy processes are introduced in this paper via compound Poisson representations, which are so termed because their characteristic exponents are the linear combination of certain positive or negative power functions. They can be represented as the time changed multivariate Brownian motions using the time changes that are two exponential power subordinators, respectively. Two multivariate exponential power random fields are also proposed as elliptically contoured random fields, whose finite-dimensional characteristic functions are made up of the exponential power functions and enjoy the infinitely divisible property.

Multivariate Gaussian processes: definitions, examples and applications

Gaussian processes occupy one of the leading places in modern statistics and probability theory due to their importance and a wealth of strong results. The common use of Gaussian processes is in connection with problems related to estimation, detection, and many statistical or machine learning models. In this paper, we propose a precise definition of multivariate Gaussian processes based on Gaussian measures on vector-valued function spaces, and provide an existence proof. In addition, several fundamental properties of multivariate Gaussian processes, such as stationarity and independence, are introduced. We further derive two special cases of multivariate Gaussian processes, including multivariate Gaussian white noise and multivariate Brownian motion, and present a brief introduction to multivariate Gaussian process regression as a useful statistical learning method for multi-output prediction problems.

Regulated seasonal unit root process

Abstract Unfortunately, time series problems do not appear in data singly. We focus on the joint occurrence of nonstationarity, seasonality and bounded data. Seasonal unit root tests and bounded unit root tests already exist in the literature, yet when all these issues are combined their performance needs improvement. That is why we offer a testing procedure for bounded seasonal unit root processes. The combination of these tests is not straightforward as the nonlinearity coming from bounds causes the limiting distribution of the proposed test statistic to be multivariate Brownian motion while the others have univariate distributions. The simulation exercises reveal that the existing tests, which ignores the presence of bounds or seasonality, suffer significant size problems. Our statistic removes the size distortions and also maintain satisfactory power performance.

Gaussian process approximations for multicolor Pólya urn models

AbstractMotivated by mathematical tissue growth modelling, we consider the problem of approximating the dynamics of multicolor Pólya urn processes that start with large numbers of balls of different colors and run for a long time. Using strong approximation theorems for empirical and quantile processes, we establish Gaussian process approximations for the Pólya urn processes. The approximating processes are sums of a multivariate Brownian motion process and an independent linear drift with a random Gaussian coefficient. The dominating term between the two depends on the ratio of the number of time steps n to the initial number of balls N in the urn. We also establish an upper bound of the form $c(n^{-1/2}+N^{-1/2})$ for the maximum deviation over the class of convex Borel sets of the step-n urn composition distribution from the approximating normal law.

Applications of the Girsanov theorem for multivariate fractional Brownian motions

Applications of the Girsanov theorem for multivariate fractional Brownian motions

BROWNIAN MOTION MINUS THE INDEPENDENT INCREMENTS: REPRESENTATION AND QUEUING APPLICATION

This paper relaxes assumptions defining multivariate Brownian motion (BM) to construct processes with dependent increments as tractable models for problems in engineering and management science. We show that any Gaussian Markov process starting at zero and possessing stationary increments and a symmetric smooth kernel has a parametric kernel of a particular form, and we derive the unique unbiased, jointly sufficient, maximum-likelihood estimators of those parameters. As an application, we model a single-server queue driven by such a process and derive its transient distribution conditional on its history.

Gene expression of functionally-related genes coevolves across fungal species: detecting coevolution of gene expression using phylogenetic comparative methods

BackgroundResearchers often measure changes in gene expression across conditions to better understand the shared functional roles and regulatory mechanisms of different genes. Analogous to this is comparing gene expression across species, which can improve our understanding of the evolutionary processes shaping the evolution of both individual genes and functional pathways. One area of interest is determining genes showing signals of coevolution, which can also indicate potential functional similarity, analogous to co-expression analysis often performed across conditions for a single species. However, as with any trait, comparing gene expression across species can be confounded by the non-independence of species due to shared ancestry, making standard hypothesis testing inappropriate.ResultsWe compared RNA-Seq data across 18 fungal species using a multivariate Brownian Motion phylogenetic comparative method (PCM), which allowed us to quantify coevolution between protein pairs while directly accounting for the shared ancestry of the species. Our work indicates proteins which physically-interact show stronger signals of coevolution than randomly-generated pairs. Interactions with stronger empirical and computational evidence also showing stronger signals of coevolution. We examined the effects of number of protein interactions and gene expression levels on coevolution, finding both factors are overall poor predictors of the strength of coevolution between a protein pair. Simulations further demonstrate the potential issues of analyzing gene expression coevolution without accounting for shared ancestry in a standard hypothesis testing framework. Furthermore, our simulations indicate the use of a randomly-generated null distribution as a means of determining statistical significance for detecting coevolving genes with phylogenetically-uncorrected correlations, as has previously been done, is less accurate than PCMs, although is a significant improvement over standard hypothesis testing. These methods are further improved by using a phylogenetically-corrected correlation metric.ConclusionsOur work highlights potential benefits of using PCMs to detect gene expression coevolution from high-throughput omics scale data. This framework can be built upon to investigate other evolutionary hypotheses, such as changes in transcription regulatory mechanisms across species.

Decoupled jaws promote trophic diversity in cichlid fishes.

Functional decoupling of oral and pharyngeal jaws is widely considered to have expanded the ecological repertoire of cichlid fishes. But, the degree to which the evolution of these jaw systems is decoupled and whether decoupling has impacted trophic diversification remains unknown. Focusing on the large Neotropical radiation of cichlids, we ask whether oral and pharyngeal jaw evolution is correlated and how their evolutionary rates respond to feeding ecology. In support of decoupling, we find relaxed evolutionary integration between the two jaw systems, resulting in novel trait combinations that potentially facilitate feeding mode diversification. These outcomes are made possible by escaping the mechanical trade-off between force transmission and mobility, which characterizes a single jaw system that functions in isolation. In spite of the structural independence of the two jaw systems, results using a Bayesian, state-dependent, relaxed-clock model of multivariate Brownian motion indicate strongly aligned evolutionary responses to feeding ecology. So, although decoupling of prey capture and processing functions released constraints on jaw evolution and promoted trophic diversity in cichlids, the natural diversity of consumed prey has also induced a moderate degree of evolutionary integration between the jaw systems, reminiscent of the original mechanical trade-off between force and mobility.

Limit theorems in the context of multivariate long-range dependence

This article considers multivariate linear processes whose components are either short- or long-range dependent. The functional central limit theorems for the sample mean and the sample autocovariances for these processes are investigated, paying special attention to the mixed cases of short- and long-range dependent series. The resulting limit processes can involve multivariate Brownian motion marginals, operator fractional Brownian motions and matrix-valued versions of the so-called Rosenblatt process.

Self-decomposability of weak variance generalised gamma convolutions

Weak variance generalised gamma convolution processes are multivariate Brownian motions weakly subordinated by multivariate Thorin subordinators. Within this class, we extend a result from strong to weak subordination that a driftless Brownian motion gives rise to a self-decomposable process. Under moment conditions on the underlying Thorin measure, we show that this condition is also necessary. We apply our results to some prominent processes such as the weak variance alpha–gamma process, and illustrate the necessity of our moment conditions in some cases.

Asymptotic expansions of solutions of stochastic differential equations driven by multivariate fractional Brownian motions having Hurst indices greater than

In this paper, we consider n-dimensional stochastic differential equations driven by multivariate fractional Brownian motion with Hurst indices greater than . Using a Taylor type development we obtain an expansion of expectations for small t, where denotes the solution of the mentioned stochastic differential equation with initial value and is a sufficiently smooth function. Also we compute coefficients appearing in the expansion of in the case when the fractional Brownian motions have the Hurst indices greater than . Finally, we consider the case of commutative vector fields getting Kolmogorov’s backward partial differential equation for the function .

Estimating Correlated Rates of Trait Evolution with Uncertainty.

Correlated evolution among traits, which can happen due to genetic constraints, ontogeny, and selection, can have an important impact on the trajectory of phenotypic evolution. For example, shifts in the pattern of evolutionary integration may allow the exploration of novel regions of the morphospace by lineages. Here, we use phylogenetic trees to study the pace of evolution of several traits and their pattern of evolutionary correlation across clades and over time. We use regimes mapped to the branches of the phylogeny to test for shifts in evolutionary integration while incorporating the uncertainty related to trait evolution and ancestral regimes with joint estimation of all parameters of the model using Bayesian Markov chain Monte Carlo. We implemented the use of summary statistics to test for regime shifts based on a series of attributes of the model that can be directly relevant to biological hypotheses. In addition, we extend Felsenstein's pruning algorithm to the case of multivariate Brownian motion models with multiple rate regimes. We performed extensive simulations to explore the performance of the method under a series of scenarios. Finally, we provide two test cases; the evolution of a novel buccal morphology in fishes of the family Centrarchidae and a shift in the trajectory of evolution of traits during the radiation of anole lizards to and from the Caribbean islands. [Anolis; Centrarchidae; comparative methods; evolutionary integration; evolutionary rates; modularity; pruning algorithm.].

ASYMPTOTIC EXPANSIONS OF SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY MULTIVARIATE FRACTIONAL BROWNIAN MOTIONS

In this article, n-dimensional stochastic differential equations driven by multivariate fractional Brownian motions with the Hurst indices greater than 1/3 and a drift term are considered. We have obtained an expansion of expectationsPtg (x) = Eg (Xxt)for small t, whereXxtdenotes the solution of the mentioned equation with an initial valuex, andg: Rn→ Ris a sufficiently smooth function.

A Penalized Likelihood Framework for High-Dimensional Phylogenetic Comparative Methods and an Application to New-World Monkeys Brain Evolution.

Working with high-dimensional phylogenetic comparative data sets is challenging because likelihood-based multivariate methods suffer from low statistical performances as the number of traits $p $ approaches the number of species $n $ and because some computational complications occur when $p $ exceeds $n$. Alternative phylogenetic comparative methods have recently been proposed to deal with the large $p $ small $n $ scenario but their use and performances are limited. Herein, we develop a penalized likelihood (PL) framework to deal with high-dimensional comparative data sets. We propose various penalizations and methods for selecting the intensity of the penalties. We apply this general framework to the estimation of parameters (the evolutionary trait covariance matrix and parameters of the evolutionary model) and model comparison for the high-dimensional multivariate Brownian motion (BM), Early-burst (EB), Ornstein-Uhlenbeck (OU), and Pagel's lambda models. We show using simulations that our PL approach dramatically improves the estimation of evolutionary trait covariance matrices and model parameters when $p$ approaches $n$, and allows for their accurate estimation when $p$ equals or exceeds $n$. In addition, we show that PL models can be efficiently compared using generalized information criterion (GIC). We implement these methods, as well as the related estimation of ancestral states and the computation of phylogenetic principal component analysis in the R package RPANDA and mvMORPH. Finally, we illustrate the utility of the new proposed framework by evaluating evolutionary models fit, analyzing integration patterns, and reconstructing evolutionary trajectories for a high-dimensional 3D data set of brain shape in the New World monkeys. We find a clear support for an EB model suggesting an early diversification of brain morphology during the ecological radiation of the clade. PL offers an efficient way to deal with high-dimensional multivariate comparative data.

One Dimensional Random Motion on Segment with Reflecting Edges and Dependent Increments

In previous papers, the author considered the model of anomalous diffusion, defined by stable random process on an interval with reflecting edges. Estimates of the rate convergence of this process distribution to a uniform distribution are constructed. However, recent physical studies require consideration of models of diffusion, defined not only by stable random process with independent increments but multivariate fractional Brownian motion with dependent increments. This task requires the development of special mathematical techniques evaluation of the rate of convergence of the distribution of multivariate Brownian motion in a segment with reflecting boundaries to the limit. In the present work, this technology is developed and a power estimate of the rate of convergence to the limiting uniform distribution is built.

Estimation of Dynamic Mixed Hitting Time Model Using Characteristic Function Based Moments

We present a characteristic function-based method for the estimation of dynamic mixed hitting time model for duration between events and price changes. The model specifies duration between events as the first time a latent component of multivariate Brownian motion crosses a random boundary. Meanwhile, another (correlated) Brownian component generates the prices. The proposed estimation method facilitates computation and overcomes problems related to the discretization error in the moment conditions and the non-tractability in the joint probability density function. An empirical application using transaction level data on stocks of a large capitalization company traded on the Indonesia Stock Exchange is illustrated. Estimation results suggest that durations and return volatility have strong persistence and, further, there is a negative instantaneous relation between volatility and contemporaneous duration. The impact of considering the causality relation between volatilities and durations on instantaneous volatility estimate are also investigated.

Multivariate subordination using generalised Gamma convolutions with applications to Variance Gamma processes and option pricing

We unify and extend a number of approaches related to constructing multivariate Madan–Seneta Variance-Gamma models for option pricing. Complementing Grigelionis’ (2007) class, an overarching model is derived by subordinating multivariate Brownian motion to a subordinator from Thorin’s (1977) [58, 59] class of generalised Gamma convolutions. Multivariate classes developed by Pérez-Abreu and Stelzer (2014), Semeraro (2008) and Guillaume (2013) are submodels. The classes are shown to be invariant under Esscher transforms, and quite explicit expressions for canonical measures are obtained, which permit applications such as option pricing using PIDEs or tree based methodologies. We illustrate with best-of and worst-of European and American options on two assets.

Trends in the sand: Directional evolution in the shell shape of recessing scallops (Bivalvia: Pectinidae)

Directional evolution is one of the most compelling evolutionary patterns observed in macroevolution. Yet, despite its importance, detecting such trends in multivariate data remains a challenge. In this study, we evaluate multivariate evolution of shell shape in 93 bivalved scallop species, combining geometric morphometrics and phylogenetic comparative methods. Phylomorphospace visualization described the history of morphological diversification in the group; revealing that taxa with a recessing life habit were the most distinctive in shell shape, and appeared to display a directional trend. To evaluate this hypothesis empirically, we extended existing methods by characterizing the mean directional evolution in phylomorphospace for recessing scallops. We then compared this pattern to what was expected under several alternative evolutionary scenarios using phylogenetic simulations. The observed pattern did not fall within the distribution obtained under multivariate Brownian motion, enabling us to reject this evolutionary scenario. By contrast, the observed pattern was more similar to, and fell within, the distribution obtained from simulations using Brownian motion combined with a directional trend. Thus, the observed data are consistent with a pattern of directional evolution for this lineage of recessing scallops. We discuss this putative directional evolutionary trend in terms of its potential adaptive role in exploiting novel habitats.