Simplicial Multivalued Maps and the Witness Complex for Dynamical Analysis of Time Series

Topology based analysis of time-series data from dynamical systems is powerful: it potentially allows for computer-based proofs of the existence of various classes of regular and chaotic invariant sets for high-dimensional dynamics. Standard methods are based on a cubical discretization of the dynamics and use the time series to construct an outer approximation of the underlying dynamical system. The resulting multivalued map can be used to compute the Conley index of isolated invariant sets of cubes. In this paper we introduce a discretization that uses instead a simplicial complex constructed from a witness-landmark relationship. The goal is to obtain a natural discretization that is more tightly connected with the invariant density of the time series itself. The time-ordering of the data also directly leads to a map on this simplicial complex that we call the witness map. We obtain conditions under which this witness map gives an outer approximation of the dynamics, and thus can be used to compute the Conley index of isolated invariant sets. The method is illustrated by a simple example using data from the classical H\'enon map.Comment: laTeX, 9 figures, 32 page

Accurate Memory Kernel Extraction from Discretized Time-Series Data

Memory effects emerge as a fundamental consequence of dimensionality reduction when low-dimensional observables are used to describe the dynamics of complex many-body systems. In the context of molecular dynamics (MD) data analysis, accounting for memory effects using the framework of the generalized Langevin equation (GLE) has proven efficient, accurate, and insightful, particularly when working with high-resolution time series data. However, in experimental systems, high-resolution data are often unavailable, raising questions about the impact of the data resolution on the estimated GLE parameters. This study demonstrates that direct memory extraction from time series data remains accurate when the discretization time is below the memory time. To obtain memory functions reliably, even when the discretization time exceeds the memory time, we introduce a Gaussian Process Optimization (GPO) scheme. This scheme minimizes the deviation of discretized two-point correlation functions between time series data and GLE simulations and is able to estimate accurate memory kernels as long as the discretization time stays below the longest time scale in the data, typically the barrier crossing time

Heterogeneous continuous dynamic Bayesian networks with flexible structure and inter-time segment information sharing

Classical dynamic Bayesian networks (DBNs) are based on the homogeneous Markov assumption and cannot deal with heterogeneity and non-stationarity in temporal processes. Various approaches to relax the homogeneity assumption have recently been proposed. The present paper aims to improve the shortcomings of three recent versions of heterogeneous DBNs along the following lines: (i) avoiding the need for data discretization, (ii) increasing the flexibility over a time-invariant network structure, (iii) avoiding over-flexibility and overfitting by introducing a regularization scheme based in inter-time segment information sharing. The improved method is evaluated on synthetic data and compared with alternative published methods on gene expression time series from Drosophila melanogaster. 1

Comparative study of discretization methods of microarray data for inferring transcriptional regulatory networks

<p>Abstract</p> <p>Background</p> <p>Microarray data discretization is a basic preprocess for many algorithms of gene regulatory network inference. Some common discretization methods in informatics are used to discretize microarray data. Selection of the discretization method is often arbitrary and no systematic comparison of different discretization has been conducted, in the context of gene regulatory network inference from time series gene expression data.</p> <p>Results</p> <p>In this study, we propose a new discretization method "bikmeans", and compare its performance with four other widely-used discretization methods using different datasets, modeling algorithms and number of intervals. Sensitivities, specificities and total accuracies were calculated and statistical analysis was carried out. Bikmeans method always gave high total accuracies.</p> <p>Conclusions</p> <p>Our results indicate that proper discretization methods can consistently improve gene regulatory network inference independent of network modeling algorithms and datasets. Our new method, bikmeans, resulted in significant better total accuracies than other methods.</p

Comments on “A selective overview of nonparametric methods in financial econometricsâ€Â

In recent years there has been increased interest in using nonparametric methods to deal with various aspects of financial data. The paper by Fan overviews some nonparametric techniques that have been used in the financial econometric literature, focusing on estimation and inference for diffusion models in continuous time and estimation of state price and transition density functions. Our comments on Fans paper will concentrate on two issues that relate in important ways to the papers focus on misspecification and discretization bias and the role of nonparametric methods in empirical finance. The first issue deals with the finite sample effects of various estimation methods and their implications for asset pricing. A good deal of recent attention in the econometric literature has focused on the benefits of full maximum likelihood (ML) estimation of diffusions and mechanisms for avoiding discretization bias in the construction of the likelihood. However, many of the problems of estimating dynamic models that are well known in discrete time series, such as the bias in ML estimation, also manifest in the estimation of continuous time systems and affect subsequent use of these estimates, for instance in derivative pricing. In consequence, a relevant concern is the relative importance of the estimation and discretization biases. As we will show below, the former often dominates the latter even when the sample size is large (at least 500 monthly observations, say). Moreover, it turns out that correction for the finite sample estimation bias continues to be more important when the diffusion component of the model is itself misspecified. Such corrections appear to be particularly important in models that are nonstationary or nearly nonstationary.nonparametric methods, financial data, Fan, empirical finance, discretization bias, misspecification, ML estimation