Ensemble-based method for the inverse Frobenius–Perron operator problem: Data-driven global analysis from spatiotemporal “Movie” data

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Highlights

  • An ensemble-based approach improves the parameter inferences from density evolution.

  • Uncertainty around the optimal estimate can be approximated by the ensemble approach.

  • It enables assessment of the uncertainty of coherent structure identification

Abstract

Given a sequence of empirical distribution data (e.g. a movie of a spatiotemporal process such as a fluid flow), this work develops an ensemble data assimilation method to estimate the transition probability that represents a finite approximation of the Frobenius–Perron operator. This allows a dynamical systems knowledge to be incorporated into a prior ensemble, which provides sensible estimates in instances of limited observation. We demonstrate improved estimates over a constrained optimization approach based on a quadratic programming problem. The estimated transition probability then enables several probabilistic analysis of dynamical systems. We focus only on the identification of coherent patterns from the estimated Markov transition to demonstrate its application as a proof-of-concept. To the best of our knowledge, there have not been many works on data-driven methods to identify coherent patterns from this type of data. While here the results are presented only in the context of dynamical systems applications, this work we present here has the potential to make a contribution in wider application areas that require the estimation of transition probabilities from a time-ordered spatio-temporal distribution data.

Introduction

This work focuses on an ensemble-based technique to estimate a Markov transition matrix, which is a finite approximation of the Frobenius–Perron (or transfer) operator, from a sequence of empirical distribution data. For instances, in a fluid experiment where droplets of dye are introduced into the fluid surface and their evolution is recorded as a sequence of images or in some PIV imagery, a sequence of a large number of “unlabelled particles” are recorded, in which case it would be difficult or even infeasible to track individual particle trajectory. However, these particles can be aggregated (perhaps after experiments) to provide empirical distribution data. In particular, the conditional transition probability of the transition matrix is estimated and the ensemble of these parameters can be used to understand the uncertainty of the estimate as well. The algorithm in this work modifies the formulation of the ensemble Kalman filter (EnKF) but it is not used in the filtering application. Instead, the entire data is used at once for the estimation and, EnKF is iteratively applied to improve the estimate; hence iterative EnKF (IEnKF). This approach is a Bayesian framework in nature, so the prior ensemble of the parameters is updated according to Bayes’ rule to obtain the posterior ensemble. The flexibility of prior ensemble construction can be critical to obtaining realistic results in the case where available data is non-informative. Recent works with a similar goal [1], [2] have studied the approximation of the transition matrix in the context of the so-called inverse Frobenius–Perron problem [3] where the estimated transition matrix is subsequently used to reconstruct the underlying one-dimensional map. The estimation method is simply a gradient-based constraint optimization. However, the approach in [1], [2] was demonstrated only for a specific set of initial distributions, each of which is highly concentrated on individual Markov state, instead of arbitrary distributions. The estimation of the Markov transition probability is also of interest in a wider context such as econometric where the estimation problem is set up to allow for the quadratic programming [4], [5], [6]. A fully Bayesian approach was developed in [7] assuming that the rows of the transition matrix are independently distributed as the Dirichlet distribution. The posterior distribution is, however, analytically intractable. The structure of such a prior is also restrictive in the sense that the covariance between any two elements in the same row is always negative. Since the number of parameters is N2 for N Markov states, it can be practically infeasible to use a sampling method such as Markov chain Monte Carlo in this fully Bayesian setting. Therefore, uncertainty analysis is usually ignored and only the posterior mode is computed by solving a constrained nonlinear equation using, for example, a nonlinear programming approach. Note that in the limit of “large” samples, an approximation of the covariance matrix is available but for a large number of Markov states, the number of available observations is usually too small to be considered as large samples.

In obtaining an estimated transition matrix, several probabilistic analyses of the data are made possible. For example, it allows an identification of coherent patterns (e.g. [8], [9], [10], [11], [12]), an approximation of transition probabilities from a basin to another (e.g. the stochastic basin hopping in [13], [14] or computation of probabilistic transport pathway [15]. This work will focus only on the application of coherent structure or pattern identification. Understanding large-scale persistent patterns or coherent structures emerging from a chaotic dynamical system has been a subject of research interest since it is fundamental to gain insight both quantitatively and qualitatively into a study of transport problems. Thus several advanced algorithms to extract the large-scale persistent structure have been developed. Most recent algorithms are either geometric-based techniques [16], [17], [18], [19], [20], [21], [22], [23], [24], which rely on advanced theories of Lagrangian manifolds, or set-oriented methods, which can be considered as probabilistic approaches [8], [9], [10], [11], [12], [25]. In cases where equations of the underlying dynamical system are known or their (numerical) approximations are available, Lagrangian coherent structures, which are transport barriers roughly moving along with the flow, can be constructed from analytic geometric-based methods based on theoretical concepts such as lobe dynamics, finite-time material manifolds, shape coherence, and braiding. On the other hand, the set-oriented methods use transfer operator (or Frobenius–Perron operator) theories to identify a large-scale almost invariant set for autonomous or periodic systems or finite-time coherent sets for non-autonomous systems, which minimizes transport across their boundaries. An extensive comparison between the two approaches as well as highlights of recent applications can be found in [26].

In some situations, however, theoretical or computational models are not available. When only spatio-temporal observational data is available, coherent structures have to be inferred based solely on relevant data. Numerous works have been proposed to extract coherent structures from the Lagrangian trajectory data such as a collection of trajectories of ocean drifters. In [13], [22], [27], [28], [29], [30], [31], the coherent structures are motivated by the finite-time coherent sets represented by the eigenvector of the Frobenius–Perron operator, which are shown to approximately minimize the conditional probability of the transport in and out of the coherent sets. A variety of data-driven methods are used in these works such as those motivated by spectral clustering [22], [27], [29], [30], diffusion map method [28], and finite-element methods [31]. In [32], [33], the coherent structures are the slow-dynamics mode of the Koopman operator, which is the adjoint of the Frobenius–Perron operator. To our best knowledge, however, this is the first study that uses empirical distribution (or aggregate) data to infer coherent structures.

Section snippets

Approximation of Frobenius–Perron operator

This section briefly describes a basic formulation of the Frobenius–Perron operator approximation and how it can be used to find the coherent sets associated with an autonomous, deterministic dynamical system on a bounded domain. Thus the notion of coherent sets is simplified to the almost invariant sets as studied by several works [10], [12], [34], [35]. A generalization to the stochastically perturbed autonomous system or non-autonomous systems is given in [11], [31]. However, we will focus

Basic setup

To estimate the transition matrix, we make the following model assumption: yj(t)=iyi(t1)pij+ϵj(t),t=0,1,,T1,where the error from replacing the unconditional probabilities qj(t) in (2.8) by proportion data yj(t) is accounted for by a random variable ϵj(t). We assume that E[ϵj]=0, Var[ϵj]=σj2 and E[ϵiϵj]=0 for ij, hence the name “independent scheme”. The model (3.1) can be expressed as a linear model using the following notations: yj=[yj(1),yj(2),,yj(T)] is a (T×1) vector of the observed

Numerical experiments

Two numerical experiments are carried out in the subsequent sections. We will use the Hellinger distance to quantify the difference between two discrete probability measures. It is given by h(w1,w2)=12w1w22.The constant term is used for scaling so that the Hellinger distance varies between 0 and 1. In what follows, we evaluate the error for each row of the estimated transition matrix and report an average of all rows of the transition matrix.

Conclusions and future works

This paper presents an ensemble-based method to estimate the transition matrix representation of a dynamical system through a time-series of proportion data. The algorithm takes into account the Markov constraints on the transition matrix via the ALR transformation. Based on the transformed variable, the ensemble-based algorithm then follows a similar formulation to the EnKF but it is applied iteratively to improve the estimate. The ensemble-based approach in this work incorporates the prior

CRediT authorship contribution statement

Naratip Santitissadeekorn: Conceptualization, Methodology, Formal analysis, Visualization, Writing - original draft, Writing - review & editing. Erik M. Bollt: Conceptualization, Validation, Writing - review & editing.

References (45)

  • JudgeG.G. et al.

    Inequality restrictions in regression analysis

    J. Amer. Stat. Assoc.

    (1966)
  • TheilH. et al.

    A quadratic programming approach to the estimation of transition probabilities

    Manag. Sci.

    (1966)
  • KalbfleischJ.D. et al.

    Least squares estimation of transition probability from aggregate data

    Canadian J. Stats.

    (2008)
  • LeeT.C. et al.

    Estimating the Parameters of the Markov Probability Model From Aggregate Time Series Data

    (1970)
  • DellnitzM. et al.

    The algorithms behind GAIO – Set oriented numerical methods for dynamical systems

  • BillingsL. et al.

    Identifying almost invariant sets in stochastic dynamical systems

    Chaos

    (2008)
  • BolltE.M. et al.

    Applied and Computational Measurable Dynamics

    (2013)
  • BillingsL. et al.

    Phase-space transport of stochastic chaos in population dynamics of virus spread

    Phys. Rev. Lett.

    (2002)
  • Ser-GiacomiE. et al.

    Most probable paths in temporal weighted networks: An application to ocean transport

    Phys. Rev. E

    (2015)
  • MezićI. et al.

    A method fo visualization of invariant sets of dynamical systems based on the ergodic partition

    Chaos

    (1999)
  • HallerG.

    Finding finite-time invariant manifolds in two-dimensional velocity fields

    Chaos

    (2000)
  • WigginsS.

    The dynamical systems approach to Lagrangian tranport in oceanic flows

    Annu. Rev. Fluid Mech.

    (2005)
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