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Aug 14 – 18, 2017
Uppsala University
Europe/Stockholm timezone

Joint Bayesian nonparametric reconstruction of dynamical equations

Aug 17, 2017, 9:30 AM
Ångströmslaboratoriet (Uppsala University)


Uppsala University



Mr Christos Merkatas (Department of Mathematics, University of the Aegean, Greece)


We propose a Bayesian nonparametric mixture model for the joint full reconstruction of $m$ dynamical equations, given $m$ observed dynamically-noisy-corrupted chaotic time series. The method of reconstruction is based on the Pairwise Dependent Geometric Stick Breaking Processes mixture priors (PDGSBP) first proposed by Hatjispyros et al. (2017). We assume that each set of dynamical equations has a deterministic part with a known functional form i.e. $$ x_{ji} = g_{j}(\vartheta_j, x_{j,i-1},\ldots,x_{j,i-l_j}) + \epsilon_{x_{ji}},\,\,\, 1\leq j \leq m,\,\, 1\leq i \leq n_{j}. $$ under the assumption that the noise processes $(\epsilon_{x_{ji}})$ are independent and identically distributed for all $j$ and $i$ from some unknown zero mean process $f_j(\cdot)$. Additionally, we assume that a-priori we have the knowledge that the processes $(\epsilon_{x_{ji}})$ for $j=1,\ldots,m$ have common characteristics, e.g. they may have common variances or even have similar tail behavior etc. For a full reconstruction, we would like to jointly estimate the following quantities $$ (\vartheta_{j})\in\Theta\subseteq{\cal R}^{k_j},\quad (x_{j,0},\ldots, x_{j,l_j-1})\in{\cal X}_j\subseteq{\cal R}^{l_j}, $$ and perform density estimation to the $m$ noise components $(f_j)$. Our contention is that whenever there is at least one sufficiently large data set, using carefully selected informative borrowing-of-strength-prior-specifications we are able to reconstruct those dynamical processes that are responsible for the generation of time series with small sample sizes; namely sample sizes that are inadequate for an independent reconstruction. We illustrate the joint estimation process for the case $m=2$, when the two time series are coming from a quadratic and a cubic stochastic process of lag one and the noise processes are zero mean normal mixtures with common components.

Primary authors

Mr Christos Merkatas (Department of Mathematics, University of the Aegean, Greece) Prof. Spyridon Hatjispyros (Department of Mathematics, University of the Aegean, Greece)

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