7,348 research outputs found
Clustering piecewise stationary processes
The problem of time-series clustering is considered in the case where each data-point is a sample generated by a piecewise stationary process. While stationary processes comprise one of the most general classes of processes in nonparametric statistics, and in particular, allow for arbitrary long-range dependencies, their key assumption of stationarity remains restrictive for some applications. We address this shortcoming by considering piecewise stationary processes, studied here for the first time in the context of clustering. It turns out that this problem allows for a rather natural definition of consistency of clustering algorithms. Efficient algorithms are proposed which are shown to be asymptotically consistent without any additional assumptions beyond piecewise stationarity. The theoretical results are complemented with experimental evaluations
Estimating Time-Varying Effective Connectivity in High-Dimensional fMRI Data Using Regime-Switching Factor Models
Recent studies on analyzing dynamic brain connectivity rely on sliding-window
analysis or time-varying coefficient models which are unable to capture both
smooth and abrupt changes simultaneously. Emerging evidence suggests
state-related changes in brain connectivity where dependence structure
alternates between a finite number of latent states or regimes. Another
challenge is inference of full-brain networks with large number of nodes. We
employ a Markov-switching dynamic factor model in which the state-driven
time-varying connectivity regimes of high-dimensional fMRI data are
characterized by lower-dimensional common latent factors, following a
regime-switching process. It enables a reliable, data-adaptive estimation of
change-points of connectivity regimes and the massive dependencies associated
with each regime. We consider the switching VAR to quantity the dynamic
effective connectivity. We propose a three-step estimation procedure: (1)
extracting the factors using principal component analysis (PCA) and (2)
identifying dynamic connectivity states using the factor-based switching vector
autoregressive (VAR) models in a state-space formulation using Kalman filter
and expectation-maximization (EM) algorithm, and (3) constructing the
high-dimensional connectivity metrics for each state based on subspace
estimates. Simulation results show that our proposed estimator outperforms the
K-means clustering of time-windowed coefficients, providing more accurate
estimation of regime dynamics and connectivity metrics in high-dimensional
settings. Applications to analyzing resting-state fMRI data identify dynamic
changes in brain states during rest, and reveal distinct directed connectivity
patterns and modular organization in resting-state networks across different
states.Comment: 21 page
Extreme Value Theory for Piecewise Contracting Maps with Randomly Applied Stochastic Perturbations
We consider globally invertible and piecewise contracting maps in higher
dimensions and we perturb them with a particular kind of noise introduced by
Lasota and Mackey. We got random transformations which are given by a
stationary process: in this framework we develop an extreme value theory for a
few classes of observables and we show how to get the (usual) limiting
distributions together with an extremal index depending on the strength of the
noise.Comment: 16 pages. arXiv admin note: text overlap with arXiv:1407.041
Laws of rare events for deterministic and random dynamical systems
The object of this paper is twofold. From one side we study the dichotomy, in
terms of the Extremal Index of the possible Extreme Value Laws, when the rare
events are centred around periodic or non periodic points. Then we build a
general theory of Extreme Value Laws for randomly perturbed dynamical systems.
We also address, in both situations, the convergence of Rare Events Point
Processes. Decay of correlations against observables will play a central
role in our investigations
An analytical model for Loc/ID mappings caches
Concerns regarding the scalability of the interdomain routing have encouraged researchers to start elaborating a more robust Internet architecture. While consensus on the exact form of the solution is yet to be found, the need for a semantic decoupling of a node's location and identity is generally accepted as a promising way forward. However, this typically requires the use of caches that store temporal bindings between the two namespaces, to avoid hampering router packet forwarding speeds. In this article, we propose a methodology for an analytical analysis of cache performance that relies on the working-set theory. We first identify the conditions that network traffic must comply with for the theory to be applicable and then develop a model that predicts average cache miss rates relying on easily measurable traffic parameters. We validate the result by emulation, using real packet traces collected at the egress points of a campus and an academic network. To prove its versatility, we extend the model to consider cache polluting user traffic and observe that simple, low intensity attacks drastically reduce performance, whereby manufacturers should either overprovision router memory or implement more complex cache eviction policies.Peer ReviewedPostprint (author's final draft
Diffusion Maps, Spectral Clustering and Eigenfunctions of Fokker-Planck operators
This paper presents a diffusion based probabilistic interpretation of
spectral clustering and dimensionality reduction algorithms that use the
eigenvectors of the normalized graph Laplacian. Given the pairwise adjacency
matrix of all points, we define a diffusion distance between any two data
points and show that the low dimensional representation of the data by the
first few eigenvectors of the corresponding Markov matrix is optimal under a
certain mean squared error criterion. Furthermore, assuming that data points
are random samples from a density p(\x) = e^{-U(\x)} we identify these
eigenvectors as discrete approximations of eigenfunctions of a Fokker-Planck
operator in a potential 2U(\x) with reflecting boundary conditions. Finally,
applying known results regarding the eigenvalues and eigenfunctions of the
continuous Fokker-Planck operator, we provide a mathematical justification for
the success of spectral clustering and dimensional reduction algorithms based
on these first few eigenvectors. This analysis elucidates, in terms of the
characteristics of diffusion processes, many empirical findings regarding
spectral clustering algorithms.Comment: submitted to NIPS 200
Functional Limit Theorems for Dynamical Systems with Correlated Maximal Sets
In order to obtain functional limit theorems for heavy tailed stationary
processes arising from dynamical systems, one needs to understand the
clustering patterns of the tail observations of the process. These patterns are
well described by means of a structure called the pilling process introduced
recently in the context of dynamical systems. So far, the pilling process has
been computed only for observable functions maximised at a single repelling
fixed point. Here, we study richer clustering behaviours by considering
correlated maximal sets, in the sense that the observable is maximised in
multiple points belonging to the same orbit, and we work out explicit
expressions for the pilling process when the dynamics is piecewise linear and
expanding (-dimensional and -dimensional)
Reconciliation of Waiting Time Statistics of Solar Flares Observed in Hard X-Rays
We study the waiting time distributions of solar flares observed in hard
X-rays with ISEE-3/ICE, HXRBS/SMM, WATCH/GRANAT, BATSE/CGRO, and RHESSI.
Although discordant results and interpretations have been published earlier,
based on relatively small ranges ( decades) of waiting times, we find that
all observed distributions, spanning over 6 decades of waiting times ( hrs), can be reconciled with a single distribution
function, , which
has a powerlaw slope of at large waiting times ( hrs) and flattens out at short waiting times \Delta t \lapprox
\Delta t_0 = 1/\lambda_0. We find a consistent breakpoint at hours from the WATCH, HXRBS, BATSE, and RHESSI data.
The distribution of waiting times is invariant for sampling with different flux
thresholds, while the mean waiting time scales reciprocically with the number
of detected events, . This waiting time
distribution can be modeled with a nonstationary Poisson process with a flare
rate that varies as . This flare rate distribution represents a highly
intermittent flaring productivity in short clusters with high flare rates,
separated by quiescent intervals with very low flare rates.Comment: Preprint also available at
http://www.lmsal.com/~aschwand/eprints/2010_wait.pd
Graph Signal Processing: Overview, Challenges and Applications
Research in Graph Signal Processing (GSP) aims to develop tools for
processing data defined on irregular graph domains. In this paper we first
provide an overview of core ideas in GSP and their connection to conventional
digital signal processing. We then summarize recent developments in developing
basic GSP tools, including methods for sampling, filtering or graph learning.
Next, we review progress in several application areas using GSP, including
processing and analysis of sensor network data, biological data, and
applications to image processing and machine learning. We finish by providing a
brief historical perspective to highlight how concepts recently developed in
GSP build on top of prior research in other areas.Comment: To appear, Proceedings of the IEE
- …