Analysis of persistent nonstationary time series and applications

Abstract

We give an alternative and unified derivation of the general framework developed in the last few years for analyzing nonstationary time series. A different approach for handling the resulting variational problem numerically is introduced. We further expand the framework by employing adaptive finite element algorithms and ideas from information theory to solve the problem of finding the most adequate model based on a maximum-entropy ansatz, thereby reducing the number of underlying probabilistic assumptions. In addition, we formulate and prove the result establishing the link between the optimal parametrizations of the direct and the inverse problems and compare the introduced algorithm to standard approaches like Gaussian mixture models, hidden Markov models, artificial neural networks and local kernel methods. Furthermore, based on the introduced general framework, we show how to create new data analysis methods for specific practical applications. We demonstrate the application of the framework to data samples from toy models as well as to real-world problems such as biomolecular dynamics, DNA sequence analysis and financial applications.

Systematic attribution of observed Southern Hemisphere circulation trends to external forcing and internal variability

“…The FEM-BV-VARX approach is a general variational framework that is reduced to the well-known methods of linear regression, autoregressive models, k means and hidden Markov approaches when more restrictive assumptions are made on the nature of the underlying data-generating process (Lean and Rind, 2008;Bromwich et al, 2013;Roscoe and Haigh, 2007;Metzner et al, 2012). Furthermore, as VARX is a tool for inferring the Granger causality (Granger, 1988) (causation between time series variables in terms of predictability and not correlation), FEM-BV-VARX is a more general approach which allows going beyond the standard stationarity assumption of the usual methods currently used for inferring cause-response relationships, e.g., in ecology (Sugihara et al, 2012), economics (Granger, 1988) and climate science (Mosedale et al, 2006;Wang et al, 2004).…”

“…This family of time series analysis techniques, which is reviewed concisely by Metzner et al (2012), allows for systematic timedependent model identification when assumptions of temporal stationarity or spatial homogeneity of some underlying statistics are not justifiable. The main idea is based on regularized variational minimization of a scalar-valued functional describing the error g (x(t), u(t), θ (t)) of some model for a given observation x(t) subject to available external impacts/covariates u(t) and characterized by the timedependent set of model parameters θ (t)…”

“…For more detailed information please see Sect. 2h "Relation to classical methods of unsupervised learning" in Metzner et al (2012). It is an important feature of the deployed methodology, since it allows us to test different standards or more advanced methods in the context of the same theoretical and algorithmic FEM-BV framework.…”

“…We tested this assumption using a nonparametric information-theoretic algorithm from Metzner et al (2012) and found that for all of the model errors t the most optimal parametric family was indeed the log-normal distribution. Additionally, the log-normal distribution was fitted to the model errors, the respective log-likelihoods computed and used to calculate the AIC for the non-stationary models.…”

On inference of causality for discrete state models in a multiscale context

“…For « 2 it is a special case of the nonstationary and nonparametric model family called finite element models with bounded variation of parameters (FEM-BV) that has been introduced recently (16,17) (more details in SI Text). As explained in SI Text, an iterative numerical scheme can be deployed to solve [5,6] w.r.t.…”