Sparse Median Graphs Estimation in a High Dimensional Semiparametric Model

In this manuscript a unified framework for conducting inference on complex aggregated data in high dimensional settings is proposed. The data are assumed to be a collection of multiple non-Gaussian realizations with underlying undirected graphical structures. Utilizing the concept of median graphs in summarizing the commonality across these graphical structures, a novel semiparametric approach to modeling such complex aggregated data is provided along with robust estimation of the median graph, which is assumed to be sparse. The estimator is proved to be consistent in graph recovery and an upper bound on the rate of convergence is given. Experiments on both synthetic and real datasets are conducted to illustrate the empirical usefulness of the proposed models and methods

Concavity of Mutual Information Rate for Input-Restricted Finite-State Memoryless Channels at High SNR

We consider a finite-state memoryless channel with i.i.d. channel state and the input Markov process supported on a mixing finite-type constraint. We discuss the asymptotic behavior of entropy rate of the output hidden Markov chain and deduce that the mutual information rate of such a channel is concave with respect to the parameters of the input Markov processes at high signal-to-noise ratio. In principle, the concavity result enables good numerical approximation of the maximum mutual information rate and capacity of such a channel.Comment: 26 page

Analyticity of Entropy Rate of Hidden Markov Chains

We prove that under mild positivity assumptions the entropy rate of a hidden Markov chain varies analytically as a function of the underlying Markov chain parameters. A general principle to determine the domain of analyticity is stated. An example is given to estimate the radius of convergence for the entropy rate. We then show that the positivity assumptions can be relaxed, and examples are given for the relaxed conditions. We study a special class of hidden Markov chains in more detail: binary hidden Markov chains with an unambiguous symbol, and we give necessary and sufficient conditions for analyticity of the entropy rate for this case. Finally, we show that under the positivity assumptions the hidden Markov chain {\em itself} varies analytically, in a strong sense, as a function of the underlying Markov chain parameters.Comment: The title has been changed. The new main theorem now combines the old main theorem and the remark following the old main theorem. A new section is added as an introduction to complex analysis. General principle and an example to determine the domain of analyticity of entropy rate have been added. Relaxed conditions for analyticity of entropy rate and the corresponding examples are added. The section about binary markov chain corrupted by binary symmetric noise is taken out (to be part of another paper

Joint Estimation of Multiple Graphical Models from High Dimensional Time Series

In this manuscript we consider the problem of jointly estimating multiple graphical models in high dimensions. We assume that the data are collected from n subjects, each of which consists of T possibly dependent observations. The graphical models of subjects vary, but are assumed to change smoothly corresponding to a measure of closeness between subjects. We propose a kernel based method for jointly estimating all graphical models. Theoretically, under a double asymptotic framework, where both (T,n) and the dimension d can increase, we provide the explicit rate of convergence in parameter estimation. It characterizes the strength one can borrow across different individuals and impact of data dependence on parameter estimation. Empirically, experiments on both synthetic and real resting state functional magnetic resonance imaging (rs-fMRI) data illustrate the effectiveness of the proposed method.Comment: 40 page

Derivatives of Entropy Rate in Special Families of Hidden Markov Chains

Consider a hidden Markov chain obtained as the observation process of an ordinary Markov chain corrupted by noise. Zuk, et. al. [13], [14] showed how, in principle, one can explicitly compute the derivatives of the entropy rate of at extreme values of the noise. Namely, they showed that the derivatives of standard upper approximations to the entropy rate actually stabilize at an explicit finite time. We generalize this result to a natural class of hidden Markov chains called ``Black Holes.'' We also discuss in depth special cases of binary Markov chains observed in binary symmetric noise, and give an abstract formula for the first derivative in terms of a measure on the simplex due to Blackwell.Comment: The relaxed condtions for entropy rate and examples are taken out (to be part of another paper). The section about general principle and an example to determine the domain of analyticity is taken out (to be part of another paper). A section about binary Markov chains corrupted by binary symmetric noise is adde

A Modeling Framework for Schedulability Analysis of Distributed Avionics Systems

This paper presents a modeling framework for schedulability analysis of distributed integrated modular avionics (DIMA) systems that consist of spatially distributed ARINC-653 modules connected by a unified AFDX network. We model a DIMA system as a set of stopwatch automata (SWA) in UPPAAL to analyze its schedulability by classical model checking (MC) and statistical model checking (SMC). The framework has been designed to enable three types of analysis: global SMC, global MC, and compositional MC. This allows an effective methodology including (1) quick schedulability falsification using global SMC analysis, (2) direct schedulability proofs using global MC analysis in simple cases, and (3) strict schedulability proofs using compositional MC analysis for larger state space. The framework is applied to the analysis of a concrete DIMA system.Comment: In Proceedings MARS/VPT 2018, arXiv:1803.0866