Data Assimilation with Gaussian Mixture Models Using the Dynamically Orthogonal Field Equations. Part II: Applications

Abstract

The properties and capabilities of the GMM-DO filter are assessed and exemplified by applications to two dynamical systems: (1) the Double Well Diffusion and (2) Sudden Expansion flows; both of which admit far-from-Gaussian statistics. The former test case, or twin experiment, validates the use of the EM algorithm and Bayesian Information Criterion with Gaussian Mixture Models in a filtering context; the latter further exemplifies its ability to efficiently handle state vectors of non-trivial dimensionality and dynamics with jets and eddies. For each test case, qualitative and quantitative comparisons are made with contemporary filters. The sensitivity to input parameters is illustrated and discussed. Properties of the filter are examined and its estimates are described, including: the equation-based and adaptive prediction of the probability densities; the evolution of the mean field, stochastic subspace modes and stochastic coefficients; the fitting of Gaussian Mixture Models; and, the efficient and analytical Bayesian updates at assimilation times and the corresponding data impacts. The advantages of respecting nonlinear dynamics and preserving non-Gaussian statistics are brought to light. For realistic test cases admitting complex distributions and with sparse or noisy measurements, the GMM-DO filter is shown to fundamentally improve the filtering skill, outperforming simpler schemes invoking the Gaussian parametric distribution.

Bayesian decision and mixture models for AE monitoring of steel–concrete composite shear walls

“…Gaussian Mixture Model (GMM) has been used successfully in many fields including sound recognition [15], tracking dynamical systems [16][17][18], text recognition [19] forecasting [20]. However, its application in the AE community has been modest [6].…”

Data Assimilation with Gaussian Mixture Models Using the Dynamically Orthogonal Field Equations. Part I: Theory and Scheme

“…An important property of GMMs is that they are conjugate priors to the commonly used Gaussian observation models: their Bayesian update then remains a Gaussian mixture (Casella and Berger 2001;Sondergaard 2011). Specifically, for a prior multivariate GMM,…”

“…A derivation of this optimum M is given in Sondergaard (2011). In summary, Laplace's approximation is applied to the left hand of side of Bayes' Law (MacKay 2003),…”

“…The algorithm is summarized in the main body of the text, equations (10)-(13). For additional remarks on the EM algorithm and its application to GMMs, including the choice of starting parameters and the issue of convergence, we refer to Sondergaard (2011).…”

Learning uncertainty models for reliable operation of Autonomous Underwater Vehicles

“…ESSE has components of time-varying basis functions, multi-scale initializations, and stochastic ensemble predictions. ESSE was later combined with principled filtering techniques to derive the GMM-DO filter, a general data-assimilation technique [13]. Both ESSE and GMM-DO require knowledge of the approximate governing equations for the time-evolving error covariance bases, which is not the case for our approach.…”