Multivariate Ornstein-Uhlenbeck Process Research Articles

Fast mvSLOUCH: Multivariate Ornstein–Uhlenbeck‐based models of trait evolution on large phylogenies

Abstract The PCMBase R package is a powerful computational tool that enables efficient calculations of likelihoods for a wide range of phylogenetic Gaussian models. Taking advantage of it, we redesigned the R package mvSLOUCH. Here, we demonstrate how the new version of the package can be used to thoroughly examine the evolution and adaptation of traits in a large dataset of 1252 vascular plants through the use of multivariate Ornstein–Uhlenbeck processes. The results of our analysis demonstrate the ability of the modelling framework to distinguish between various alternative hypotheses regarding the evolution of functional traits in angiosperms.

Fast mvSLOUCH: Multivariate Ornstein–Uhlenbeck‐based models of trait evolution on large phylogenies

Abstract The PCMBase R package is a powerful computational tool that enables efficient calculations of likelihoods for a wide range of phylogenetic Gaussian models. Taking advantage of it, we redesigned the R package mvSLOUCH. Here, we demonstrate how the new version of the package can be used to thoroughly examine the evolution and adaptation of traits in a large dataset of 1252 vascular plants through the use of multivariate Ornstein–Uhlenbeck processes. The results of our analysis demonstrate the ability of the modelling framework to distinguish between various alternative hypotheses regarding the evolution of functional traits in angiosperms.

Entropy production of multivariate Ornstein-Uhlenbeck processes correlates with consciousness levels in the human brain.

Consciousness is supported by complex patterns of brain activity which are indicative of irreversible nonequilibrium dynamics. While the framework of stochastic thermodynamics has facilitated the understanding of physical systems of this kind, its application to infer the level of consciousness from empirical data remains elusive. We faced this challenge by calculating entropy production in a multivariate Ornstein-Uhlenbeck process fitted to Functional magnetic resonance imaging brain activity recordings. To test this approach, we focused on the transition from wakefulness to deep sleep, revealing a monotonous relationship between entropy production and the level of consciousness. Our results constitute robust signatures of consciousness while also advancing our understanding of the link between consciousness and complexity from the fundamental perspective of statistical physics.

Model Selection Performance in Phylogenetic Comparative Methods Under Multivariate Ornstein-Uhlenbeck Models of Trait Evolution.

The advent of fast computational algorithms for phylogenetic comparative methods allows for considering multiple hypotheses concerning the co-adaptation of traits and also for studying if it is possible to distinguish between such models based on contemporary species measurements. Here we demonstrate how one can perform a study with multiple competing hypotheses using mvSLOUCH by analyzing two data sets, one concerning feeding styles and oral morphology in ungulates, and the other concerning fruit evolution in Ferula (Apiaceae). We also perform simulations to determine if it is possible to distinguish between various adaptive hypotheses. We find that Akaike's information criterion corrected for small sample size has the ability to distinguish between most pairs of considered models. However, in some cases there seems to be bias towards Brownian motion or simpler Ornstein-Uhlenbeck models. We also find that measurement error and forcing the sign of the diagonal of the drift matrix for an Ornstein-Uhlenbeck process influences identifiability capabilities. It is a cliché that some models, despite being imperfect, are more useful than others. Nonetheless, having a much larger repertoire of models will surely lead to a better understanding of the natural world, as it will allow for dissecting in what ways they are wrong. [Adaptation; AICc; model selection; multivariate Ornstein-Uhlenbeck process; multivariate phylogenetic comparative methods; mvSLOUCH.].

Graphical modeling of stochastic processes driven by correlated noise

We study a class of graphs that represent local independence structures in stochastic processes allowing for correlated noise processes. Several graphs may encode the same local independencies and we characterize such equivalence classes of graphs. In the worst case, the number of conditions in our characterizations grows superpolynomially as a function of the size of the node set in the graph. We show that deciding Markov equivalence of graphs from this class is coNP-complete which suggests that our characterizations cannot be improved upon substantially. We prove a global Markov property in the case of a multivariate Ornstein-Uhlenbeck process which is driven by correlated Brownian motions.

Ancestral sequence reconstruction for co-evolutionary models

The ancestral sequence reconstruction problem is the inference, back in time, of the properties of common sequence ancestors from the measured properties of contemporary populations. Standard algorithms for this problem assume independent (factorized) evolution of the characters of the sequences, which is generally wrong (e.g. proteins and genome sequences). In this work, we have studied this problem for sequences described by global co-evolutionary models, which reproduce the global pattern of cooperative interactions between the elements that compose it. For this, we first modeled the temporal evolution of correlated real valued characters by a multivariate Ornstein–Uhlenbeck process on a finite tree. This represents sequences as Gaussian vectors evolving in a quadratic potential, who describe the selection forces acting on the evolving entities. Under a Bayesian framework, we developed a reconstruction algorithm for these sequences and obtained an analytical expression to quantify the quality of our estimation. We extend this formalism to discrete valued sequences by applying our method to a Potts model. We showed that for both continuous and discrete configurations, there is a wide range of parameters where, to properly reconstruct the ancestral sequences, intra-species correlations must be taken into account. We also demonstrated that, for sequences with discrete elements, our reconstruction algorithm outperforms traditional schemes based on independent site approximations.

Pólya Urn with Memory Kernel and Asymptotic Behaviors of Autocorrelation Function

P\'{o}lya urn is a stochastic process in which balls are randomly drawn from an urn of red and blue balls, and balls of the same color as the drawn balls are added. The probability of a ball of a certain color being drawn is equal to the percentage of balls of that color in the urn. We introduce arbitrary memory kernels to modify this probability. If the memory kernel decays exponentially, it is a stationary process and is mean-reverting. If the memory kernel decays by a power-law, a phase transition occurs and the asymptotic behavior of the autocorrelation function changes. An auxiliary field variable is introduced to transform the process Markovian and the field obeys a multivariate Ornstein-Uhlenbeck process. The exponents of the power law are estimated for the decay of the leading and subleading terms of the autocorrelation function. It is shown that the power law exponents changes discontinuously at the critical point.

Likelihood theory for the graph Ornstein-Uhlenbeck process

We consider the problem of modelling restricted interactions between continuously-observed time series as given by a known static graph (or network) structure. For this purpose, we define a parametric multivariate Graph Ornstein-Uhlenbeck (GrOU) process driven by a general Lévy process to study the momentum and network effects amongst nodes, effects that quantify the impact of a node on itself and that of its neighbours, respectively. We derive the maximum likelihood estimators (MLEs) and their usual properties (existence, uniqueness and efficiency) along with their asymptotic normality and consistency. Additionally, an Adaptive Lasso approach, or a penalised likelihood scheme, infers both the graph structure along with the GrOU parameters concurrently and is shown to satisfy similar properties. Finally, we show that the asymptotic theory extends to the case when stochastic volatility modulation of the driving Lévy process is considered.

Sequential Dynamic Leadership Inference Using Bayesian Monte Carlo Methods

Hierarchy and leadership interactions commonly occur in animal groups, crowds of people, and in vehicle motions. Such interactions are often affected by one or more individuals who possess key domain information (e.g., final destination, environmental constraints, and best routes) or pertinent traits (e.g., better navigation, sensing, and decision making capabilities) compared with the rest of the group. This article presents a framework for the automatic identification of group structure and leadership from noisy sensory observations of tracked groups. Accordingly, a new leader-follower model is developed, which assumes the dynamics of the group to be a multivariate Ornstein-Uhlenbeck process with the designated leader(s) drifting to the destination and followers reverting to the leaders' state. Sequential Monte Carlo approaches, and specifically the sequential Markov chain Monte Carlo approach, are adopted to infer, probabilistically, the evolving leadership structure. A Rao-Blackwellisation scheme is employed such that the kinematic state of the objects in the group is inferred in closed form by Kalman filtering. Experiments show that the proposed techniques can successfully determine the leadership structures in challenging scenarios with a corresponding enhancement in tracking accuracy through direct consideration of the leadership interactions of the group.

Global multivariate model learning from hierarchically correlated data

Inverse statistical physics aims at inferring models compatible with a set of empirical averages estimated from a high-dimensional dataset of independently distributed equilibrium configurations of a given system. However, in several applications, such as biology, data result from stochastic evolutionary processes, and configurations are related through a hierarchical structure, typically represented by a tree, and therefore not independent. In turn, empirical averages of observables superpose intrinsic signals related to the equilibrium distribution of the studied system, and spurious historical (or phylogenetic) signals resulting from the structure underlying the data-generating process. The naive application of inverse statistical physics techniques therefore leads to systematic biases and an effective reduction of the sample size. To advance on the currently open task of extracting intrinsic signals from correlated data, we study a system described by a multivariate Ornstein–Uhlenbeck process defined on a finite tree. Using a Bayesian framework, we can disentangle covariances in the data corresponding to their multivariate Gaussian equilibrium distribution from those resulting from the historical correlations. Our approach leads to a clear gain in accuracy in the inferred equilibrium distribution, which corresponds to an effective two to fourfold increase in sample size.

Noise stability of synchronization and optimal network structures

We provide a theoretical framework for quantifying the expected level of synchronization in a network of noisy oscillators. Through linearization around the synchronized state, we derive the following quantities as functions of the eigenvalues and eigenfunctions of the network Laplacian using a standard technique for dealing with multivariate Ornstein-Uhlenbeck processes: the magnitude of the fluctuations around a synchronized state and the disturbance coefficients αi that represent how strongly node i disturbs the synchronization. With this approach, we can quantify the effect of individual nodes and links on synchronization. Our theory can thus be utilized to find the optimal network structure for accomplishing the best synchronization. Furthermore, when the noise levels of the oscillators are heterogeneous, we can also find optimal oscillator configurations, i.e., where to place oscillators in a given network depending on their noise levels. We apply our theory to several example networks to elucidate optimal network structures and oscillator configurations.

Model-based whole-brain effective connectivity to study distributed cognition in health and disease.

Neuroimaging techniques are now widely used to study human cognition. The functional associations between brain areas have become a standard proxy to describe how cognitive processes are distributed across the brain network. Among the many analysis tools available, dynamic models of brain activity have been developed to overcome the limitations of original connectivity measures such as functional connectivity. This goes in line with the many efforts devoted to the assessment of directional interactions between brain areas from the observed neuroimaging activity. This opinion article provides an overview of our model-based whole-brain effective connectivity to analyze fMRI data, while discussing the pros and cons of our approach with respect to other established approaches. Our framework relies on the multivariate Ornstein-Uhlenbeck (MOU) process and is thus referred to as MOU-EC. Once tuned, the model provides a directed connectivity estimate that reflects the dynamical state of BOLD activity, which can be used to explore cognition. We illustrate this approach using two applications on task-evoked fMRI data. First, as a connectivity measure, MOU-EC can be used to extract biomarkers for task-specific brain coordination, understood as the patterns of areas exchanging information. The multivariate nature of connectivity measures raises several challenges for whole-brain analysis, for which machine-learning tools present some advantages over statistical testing. Second, we show how to interpret changes in MOU-EC connections in a collective and model-based manner, bridging with network analysis. Our framework provides a comprehensive set of tools that open exciting perspectives to study distributed cognition, as well as neuropathologies.

Online Measurement-Based Estimation of Dynamic System State Matrix in Ambient Conditions

In this paper, a purely measurement-based method is proposed to estimate the dynamic system state matrix by applying the regression theorem of the multivariate Ornstein-Uhlenbeck process. The proposed method employs a recursive algorithm to minimize the required computational effort, making it applicable to the real-time environment. One main advantage of the proposed method is model independence, i.e., it is independent of the network model and the dynamic model of generators. Among various applications of the estimated matrix, detecting and locating unexpected network topology change is illustrated in details. Simulation studies have shown that the proposed measurement-based method can provide an accurate and efficient estimation of the dynamic system state matrix under the occurrence of unexpected topology change. Besides, various implementation conditions are tested to show that the proposed method can provide accurate approximation despite measurement noise, missing PMUs, and the implementation of higher-order generator models with control devices.

Network analysis of whole-brain fMRI dynamics: A new framework based on dynamic communicability

Neuroimaging techniques such as MRI have been widely used to explore the associations between brain areas. Structural connectivity (SC) captures the anatomical pathways across the brain and functional connectivity (FC) measures the correlation between the activity of brain regions. These connectivity measures have been much studied using network theory in order to uncover the distributed organization of brain structures, in particular FC for task-specific brain communication. However, the application of network theory to study FC matrices is often “static” despite the dynamic nature of time series obtained from fMRI. The present study aims to overcome this limitation by introducing a network-oriented analysis applied to whole-brain effective connectivity (EC) useful to interpret the brain dynamics. Technically, we tune a multivariate Ornstein-Uhlenbeck (MOU) process to reproduce the statistics of the whole-brain resting-state fMRI signals, which provides estimates for MOU-EC as well as input properties (similar to local excitabilities). The network analysis is then based on the Green function (or network impulse response) that describes the interactions between nodes across time for the estimated dynamics. This model-based approach provides time-dependent graph-like descriptor, named communicability, that characterize the roles that either nodes or connections play in the propagation of activity within the network. They can be used at both global and local levels, and also enables the comparison of estimates from real data with surrogates (e.g. random network or ring lattice). In contrast to classical graph approaches to study SC or FC, our framework stresses the importance of taking the temporal aspect of fMRI signals into account. Our results show a merging of functional communities over time, moving from segregated to global integration of the network activity. Our formalism sets a solid ground for the analysis and interpretation of fMRI data, including task-evoked activity.

Characterising the nonequilibrium stationary states of Ornstein–Uhlenbeck processes

We characterise the nonequilibrium stationary state of a generic multivariate Ornstein–Uhlenbeck process involving degrees of freedom. The irreversibility of the process is encoded in the antisymmetric part of the Onsager matrix. The linearity of the Langevin equations allows us to derive closed-form expressions in terms of the latter matrix for many quantities of interest, in particular the entropy production rate and the fluctuation–dissipation ratio matrix. This general setting is then illustrated by two classes of systems. First, we consider the one-dimensional ferromagnetic Gaussian spin model endowed with a stochastic dynamics where spatial asymmetry results in irreversibility. The stationary state on a ring is independent of the asymmetry parameter, whereas it depends continuously on the latter on an open chain. Much attention is also paid to finite-size effects, especially near the critical point. Second, we consider arrays of resistively coupled electrical circuits. The entropy production rate is evaluated in the regime where the local temperatures of the resistors have small fluctuations. For networks the entropy production rate grows linearly with the size of the array. For networks a quadratic growth law violating extensivity is predicted.

Fast Bayesian inference of the multivariate Ornstein-Uhlenbeck process.

The multivariate Ornstein-Uhlenbeck process is used in many branches of science and engineering to describe the regression of a system to its stationary mean. Here we present an O(N) Bayesian method to estimate the drift and diffusion matrices of the process from N discrete observations of a sample path. We use exact likelihoods, expressed in terms of four sufficient statistic matrices, to derive explicit maximum a posteriori parameter estimates and their standard errors. We apply the method to the Brownian harmonic oscillator, a bivariate Ornstein-Uhlenbeck process, to jointly estimate its mass, damping, and stiffness and to provide Bayesian estimates of the correlation functions and power spectral densities. We present a Bayesian model comparison procedure, embodying Ockham's razor, to guide a data-driven choice between the Kramers and Smoluchowski limits of the oscillator. These provide novel methods of analyzing the inertial motion of colloidal particles in optical traps.

Framework based on communicability and flow to analyze complex network dynamics.

Graph theory constitutes a widely used and established field providing powerful tools for the characterization of complex networks. The intricate topology of networks can also be investigated by means of the collective dynamics observed in the interactions of self-sustained oscillations (synchronization patterns) or propagationlike processes such as random walks. However, networks are often inferred from real-data-forming dynamic systems, which are different from those employed to reveal their topological characteristics. This stresses the necessity for a theoretical framework dedicated to the mutual relationship between the structure and dynamics in complex networks, as the two sides of the same coin. Here we propose a rigorous framework based on the network response over time (i.e., Green function) to study interactions between nodes across time. For this purpose we define the flow that describes the interplay between the network connectivity and external inputs. This multivariate measure relates to the concepts of graph communicability and the map equation. We illustrate our theory using the multivariate Ornstein-Uhlenbeck process, which describes stable and non-conservative dynamics, but the formalism can be adapted to other local dynamics for which the Green function is known. We provide applications to classical network examples, such as small-world ring and hierarchical networks. Our theory defines a comprehensive framework that is canonically related to directed and weighted networks, thus paving a way to revise the standards for network analysis, from the pairwise interactions between nodes to the global properties of networks including community detection.

EXPERT OPINIONS AND LOGARITHMIC UTILITY MAXIMIZATION FOR MULTIVARIATE STOCK RETURNS WITH GAUSSIAN DRIFT

This paper investigates optimal trading strategies in a financial market with multidimensional stock returns, where the drift is an unobservable multivariate Ornstein–Uhlenbeck process. Information about the drift is obtained by observing stock returns and expert opinions which provide unbiased estimates on the current state of the drift. The optimal trading strategy of investors maximizing expected logarithmic utility of terminal wealth depends on the filter which is the conditional expectation of the drift given the available information. We state filtering equations to describe its dynamics for different information settings. At information dates, the expert opinions lead to an update of the filter which causes a decrease in the conditional covariance matrix. We investigate properties of these conditional covariance matrices. First, we consider the asymptotic behavior of the covariance matrices for an increasing number of expert opinions on a finite time horizon. Second, we state conditions for convergence in infinite time with regularly-arriving expert opinions. Finally, we derive the optimal trading strategy of an investor. The optimal expected logarithmic utility of terminal wealth, the value function, is a functional of the conditional covariance matrices. Hence, our analysis of the covariance matrices allows us to deduce properties of the value function.

Parameter estimation of Ornstein–Uhlenbeck process generating a stochastic graph

Given Y a graph process defined by an incomplete information observation of a multivariate Ornstein–Uhlenbeck process X, we investigate whether we can estimate the parameters of X. We define two statistics of Y. We prove convergence properties and show how these can be used for parameter inference. Finally, numerical tests illustrate our results and indicate possible extensions and applications.

Estimation of Bounded and Unbounded Trajectories in Diffusion MRI

Disentangling the tissue microstructural information from the diffusion magnetic resonance imaging (dMRI) measurements is quite important for extracting brain tissue specific measures. The autocorrelation function of diffusing spins is key for understanding the relation between dMRI signals and the acquisition gradient sequences. In this paper, we demonstrate that the autocorrelation of diffusion in restricted or bounded spaces can be well approximated by exponential functions. To this end, we propose to use the multivariate Ornstein-Uhlenbeck (OU) process to model the matrix-valued exponential autocorrelation function of three-dimensional diffusion processes with bounded trajectories. We present detailed analysis on the relation between the model parameters and the time-dependent apparent axon radius and provide a general model for dMRI signals from the frequency domain perspective. For our experimental setup, we model the diffusion signal as a mixture of two compartments that correspond to diffusing spins with bounded and unbounded trajectories, and analyze the corpus-callosum in an ex-vivo data set of a monkey brain.