Observing and tracking bandlimited graph processes from sampled measurements

A critical challenge in graph signal processing is the sampling of bandlimited graph signals; signals that are sparse in a well-defined graph Fourier domain. Current works focused on sampling time-invariant graph signals and ignored their temporal evolution. However, time can bring new insights on sampling since sensor, biological, and financial network signals are correlated in both domains. Hence, in this work, we develop a sampling theory for time varying graph signals, named graph processes, to observe and track a process described by a linear state-space model. We provide a mathematical analysis to highlight the role of the graph, process bandwidth, and sample locations. We also propose sampling strategies that exploit the coupling between the topology and the corresponding process. Numerical experiments corroborate our theory and show the proposed methods trade well the number of samples with accuracy

Getting Started with Particle Metropolis-Hastings for Inference in Nonlinear Dynamical Models

This tutorial provides a gentle introduction to the particle Metropolis-Hastings (PMH) algorithm for parameter inference in nonlinear state-space models together with a software implementation in the statistical programming language R. We employ a step-by-step approach to develop an implementation of the PMH algorithm (and the particle filter within) together with the reader. This final implementation is also available as the package pmhtutorial in the CRAN repository. Throughout the tutorial, we provide some intuition as to how the algorithm operates and discuss some solutions to problems that might occur in practice. To illustrate the use of PMH, we consider parameter inference in a linear Gaussian state-space model with synthetic data and a nonlinear stochastic volatility model with real-world data.Comment: 41 pages, 7 figures. In press for Journal of Statistical Software. Source code for R, Python and MATLAB available at: https://github.com/compops/pmh-tutoria

Phenotypic evolution studied by layered stochastic differential equations

Time series of cell size evolution in unicellular marine algae (division Haptophyta; Coccolithus lineage), covering 57 million years, are studied by a system of linear stochastic differential equations of hierarchical structure. The data consists of size measurements of fossilized calcite platelets (coccoliths) that cover the living cell, found in deep-sea sediment cores from six sites in the world oceans and dated to irregular points in time. To accommodate biological theory of populations tracking their fitness optima, and to allow potentially interpretable correlations in time and space, the model framework allows for an upper layer of partially observed site-specific population means, a layer of site-specific theoretical fitness optima and a bottom layer representing environmental and ecological processes. While the modeled process has many components, it is Gaussian and analytically tractable. A total of 710 model specifications within this framework are compared and inference is drawn with respect to model structure, evolutionary speed and the effect of global temperature.Comment: Published in at http://dx.doi.org/10.1214/12-AOAS559 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org