A scalable approach to the computation of invariant measures for high-dimensional Markovian systems

Abstract

The Markovian invariant measure is a central concept in many disciplines. Conventional numerical techniques for data-driven computation of invariant measures rely on estimation and further numerical processing of a transition matrix. Here we show how the quality of data-driven estimation of a transition matrix crucially depends on the validity of the statistical independence assumption for transition probabilities. Moreover, the cost of the invariant measure computation in general scales cubically with the dimension - and is usually unfeasible for realistic high-dimensional systems. We introduce a method relaxing the independence assumption of transition probabilities that scales quadratically in situations with latent variables. Applications of the method are illustrated on the Lorenz-63 system and for the molecular dynamics (MD) simulation data of the α-synuclein protein. We demonstrate how the conventional methodologies do not provide good estimates of the invariant measure based upon the available α-synuclein MD data. Applying the introduced approach to these MD data we detect two robust meta-stable states of α-synuclein and a linear transition between them, involving transient formation of secondary structure, qualitatively consistent with previous purely experimental reports.

Direct Bayesian model reduction of smaller scale convective activity conditioned on large scale dynamics

“…The obtained models are also less subject to overfitting issues and are more advantageous in terms of the model quality measures (Gerber and Horenko, 2017;Gerber et al, 2018). This manifests in the variance of the estimated parameter λ * ik , which shows a K/n-times smaller uncertainty than Λ ij , cf.…”

A deeper look into natural sciences with physics-based and data-driven measures

“…Step 1: Compute the transition matrices L K based on the direct Bayesian model reduction (Hofmann, 1999(Hofmann, , 2001Ding et al, 2006;Gerber & Horenko, 2015Gerber et al, 2018), as well as the quantities S K = À 1 N logL K for every K going from 1 to n (To be precise, S K = À…”

“…Given this setting, the algorithm to compute the latent entropy and latent dimension as described in the study by ( Horenko et al., 2019 ) is then given as follows (The MATLAB code is available on https://www.dropbox.com/s/w3few6elo9soegz/MATLAB_Code.zip?dl=0 ). Step 1: Compute the transition matrices based on the direct Bayesian model reduction ( Hofmann, 1999 , 2001 ; Ding et al., 2006 ; Gerber & Horenko, 2015 , 2017 ; Gerber et al., 2018 ), as well as the quantities for every K going from 1 to n (To be precise, , where with χ being an indicator function, is the average contingency table of the data X and Y .). Step 2: Determine the posterior probabilities p K for the different latent dimensions K = 1, …, n by means of the Akaike information criterion ( Hurvich and Tsai, 1989 ) as follows: where and .…”

“…Step 1: Compute the transition matrices based on the direct Bayesian model reduction ( Hofmann, 1999 , 2001 ; Ding et al., 2006 ; Gerber & Horenko, 2015 , 2017 ; Gerber et al., 2018 ), as well as the quantities for every K going from 1 to n (To be precise, , where with χ being an indicator function, is the average contingency table of the data X and Y .).…”

Markov State Models of protein-protein encounters

“…There are other alternatives to defining lower-dimensional models to facilitate analysis of slow dynamics in terms of a few meta-stable states. However, their performance in the context of protein-protein encounters is currently unknown [79,80].…”