An Algorithm for Pattern Discovery in Time Series

We present a new algorithm for discovering patterns in time series and other sequential data. We exhibit a reliable procedure for building the minimal set of hidden, Markovian states that is statistically capable of producing the behavior exhibited in the data -- the underlying process's causal states. Unlike conventional methods for fitting hidden Markov models (HMMs) to data, our algorithm makes no assumptions about the process's causal architecture (the number of hidden states and their transition structure), but rather infers it from the data. It starts with assumptions of minimal structure and introduces complexity only when the data demand it. Moreover, the causal states it infers have important predictive optimality properties that conventional HMM states lack. We introduce the algorithm, review the theory behind it, prove its asymptotic reliability, use large deviation theory to estimate its rate of convergence, and compare it to other algorithms which also construct HMMs from data. We also illustrate its behavior on an example process, and report selected numerical results from an implementation.Comment: 26 pages, 5 figures; 5 tables; http://www.santafe.edu/projects/CompMech Added discussion of algorithm parameters; improved treatment of convergence and time complexity; added comparison to older method

Convergence and Perturbation Resilience of Dynamic String-Averaging Projection Methods

We consider the convex feasibility problem (CFP) in Hilbert space and concentrate on the study of string-averaging projection (SAP) methods for the CFP, analyzing their convergence and their perturbation resilience. In the past, SAP methods were formulated with a single predetermined set of strings and a single predetermined set of weights. Here we extend the scope of the family of SAP methods to allow iteration-index-dependent variable strings and weights and term such methods dynamic string-averaging projection (DSAP) methods. The bounded perturbation resilience of DSAP methods is relevant and important for their possible use in the framework of the recently developed superiorization heuristic methodology for constrained minimization problems.Comment: Computational Optimization and Applications, accepted for publicatio

Reconstructing propagation networks with natural diversity and identifying hidden sources

Our ability to uncover complex network structure and dynamics from data is fundamental to understanding and controlling collective dynamics in complex systems. Despite recent progress in this area, reconstructing networks with stochastic dynamical processes from limited time series remains to be an outstanding problem. Here we develop a framework based on compressed sensing to reconstruct complex networks on which stochastic spreading dynamics take place. We apply the methodology to a large number of model and real networks, finding that a full reconstruction of inhomogeneous interactions can be achieved from small amounts of polarized (binary) data, a virtue of compressed sensing. Further, we demonstrate that a hidden source that triggers the spreading process but is externally inaccessible can be ascertained and located with high confidence in the absence of direct routes of propagation from it. Our approach thus establishes a paradigm for tracing and controlling epidemic invasion and information diffusion in complex networked systems.Comment: 20 pages and 5 figures. For Supplementary information, please see http://www.nature.com/ncomms/2014/140711/ncomms5323/full/ncomms5323.html#

Exploring the Universe with Very High Energy Neutrinos

With the discovery of a high-energy neutrino flux in the 0.1 PeV to PeV range from beyond the Earth's atmosphere with the IceCube detector, neutrino astronomy has achieved a major breakthrough in the exploration of the high-energy universe. One of the main goals is the identification and investigation of the still mysterious sources of the cosmic rays which are observed at Earth with energies up to several $10^5$ PeV. In addition to being smoking-gun evidence for the presence of cosmic rays in a specific object, neutrinos escape even dense environments and can reach us from distant places in the universe, thereby providing us with a unique tool to explore cosmic accelerators. This article summarizes our knowledge about the observed astrophysical neutrino flux and current status of the search for individual cosmic neutrino sources. At the end, it gives an overview of plans for future neutrino telescope projects.Comment: 10 pages, 15 figures, to appear in the proceedings of ICHEP 201