6,854 research outputs found
Factorial graphical lasso for dynamic networks
Dynamic networks models describe a growing number of important scientific
processes, from cell biology and epidemiology to sociology and finance. There
are many aspects of dynamical networks that require statistical considerations.
In this paper we focus on determining network structure. Estimating dynamic
networks is a difficult task since the number of components involved in the
system is very large. As a result, the number of parameters to be estimated is
bigger than the number of observations. However, a characteristic of many
networks is that they are sparse. For example, the molecular structure of genes
make interactions with other components a highly-structured and therefore
sparse process.
Penalized Gaussian graphical models have been used to estimate sparse
networks. However, the literature has focussed on static networks, which lack
specific temporal constraints. We propose a structured Gaussian dynamical
graphical model, where structures can consist of specific time dynamics, known
presence or absence of links and block equality constraints on the parameters.
Thus, the number of parameters to be estimated is reduced and accuracy of the
estimates, including the identification of the network, can be tuned up. Here,
we show that the constrained optimization problem can be solved by taking
advantage of an efficient solver, logdetPPA, developed in convex optimization.
Moreover, model selection methods for checking the sensitivity of the inferred
networks are described. Finally, synthetic and real data illustrate the
proposed methodologies.Comment: 30 pp, 5 figure
Active Classification for POMDPs: a Kalman-like State Estimator
The problem of state tracking with active observation control is considered
for a system modeled by a discrete-time, finite-state Markov chain observed
through conditionally Gaussian measurement vectors. The measurement model
statistics are shaped by the underlying state and an exogenous control input,
which influence the observations' quality. Exploiting an innovations approach,
an approximate minimum mean-squared error (MMSE) filter is derived to estimate
the Markov chain system state. To optimize the control strategy, the associated
mean-squared error is used as an optimization criterion in a partially
observable Markov decision process formulation. A stochastic dynamic
programming algorithm is proposed to solve for the optimal solution. To enhance
the quality of system state estimates, approximate MMSE smoothing estimators
are also derived. Finally, the performance of the proposed framework is
illustrated on the problem of physical activity detection in wireless body
sensing networks. The power of the proposed framework lies within its ability
to accommodate a broad spectrum of active classification applications including
sensor management for object classification and tracking, estimation of sparse
signals and radar scheduling.Comment: 38 pages, 6 figure
Adaptive Smoothing for Trajectory Reconstruction
Trajectory reconstruction is the process of inferring the path of a moving
object between successive observations. In this paper, we propose a smoothing
spline -- which we name the V-spline -- that incorporates position and velocity
information and a penalty term that controls acceleration. We introduce a
particular adaptive V-spline designed to control the impact of irregularly
sampled observations and noisy velocity measurements. A cross-validation scheme
for estimating the V-spline parameters is given and we detail the performance
of the V-spline on four particularly challenging test datasets. Finally, an
application of the V-spline to vehicle trajectory reconstruction in two
dimensions is given, in which the penalty term is allowed to further depend on
known operational characteristics of the vehicle.Comment: 25 pages, submitte
Non-Gaussian bias: insights from discrete density peaks
Corrections induced by primordial non-Gaussianity to the linear halo bias can
be computed from a peak-background split or the widespread local bias model.
However, numerical simulations clearly support the prediction of the former, in
which the non-Gaussian amplitude is proportional to the linear halo bias. To
understand better the reasons behind the failure of standard Lagrangian local
bias, in which the halo overdensity is a function of the local mass overdensity
only, we explore the effect of a primordial bispectrum on the 2-point
correlation of discrete density peaks. We show that the effective local bias
expansion to peak clustering vastly simplifies the calculation. We generalize
this approach to excursion set peaks and demonstrate that the resulting
non-Gaussian amplitude, which is a weighted sum of quadratic bias factors,
precisely agrees with the peak-background split expectation, which is a
logarithmic derivative of the halo mass function with respect to the
normalisation amplitude. We point out that statistics of thresholded regions
can be computed using the same formalism. Our results suggest that halo
clustering statistics can be modelled consistently (in the sense that the
Gaussian and non-Gaussian bias factors agree with peak-background split
expectations) from a Lagrangian bias relation only if the latter is specified
as a set of constraints imposed on the linear density field. This is clearly
not the case of standard Lagrangian local bias. Therefore, one is led to
consider additional variables beyond the local mass overdensity.Comment: 24 pages. no figure (v2): minor clarification added. submitted to
JCAP (v3): 1 figure added. in Press in JCA
Variational approach for learning Markov processes from time series data
Inference, prediction and control of complex dynamical systems from time
series is important in many areas, including financial markets, power grid
management, climate and weather modeling, or molecular dynamics. The analysis
of such highly nonlinear dynamical systems is facilitated by the fact that we
can often find a (generally nonlinear) transformation of the system coordinates
to features in which the dynamics can be excellently approximated by a linear
Markovian model. Moreover, the large number of system variables often change
collectively on large time- and length-scales, facilitating a low-dimensional
analysis in feature space. In this paper, we introduce a variational approach
for Markov processes (VAMP) that allows us to find optimal feature mappings and
optimal Markovian models of the dynamics from given time series data. The key
insight is that the best linear model can be obtained from the top singular
components of the Koopman operator. This leads to the definition of a family of
score functions called VAMP-r which can be calculated from data, and can be
employed to optimize a Markovian model. In addition, based on the relationship
between the variational scores and approximation errors of Koopman operators,
we propose a new VAMP-E score, which can be applied to cross-validation for
hyper-parameter optimization and model selection in VAMP. VAMP is valid for
both reversible and nonreversible processes and for stationary and
non-stationary processes or realizations
Sampling from a system-theoretic viewpoint
This paper studies a system-theoretic approach to the problem of reconstructing an analog signal from its samples. The idea, borrowed from earlier treatments in the control literature, is to address the problem as a hybrid model-matching problem in which performance is measured by system norms. \ud
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The paper is split into three parts. In Part I we present the paradigm and revise the lifting technique, which is our main technical tool. In Part II optimal samplers and holds are designed for various analog signal reconstruction problems. In some cases one component is fixed while the remaining are designed, in other cases all three components are designed simultaneously. No causality requirements are imposed in Part II, which allows to use frequency domain arguments, in particular the lifted frequency response as introduced in Part I. In Part III the main emphasis is placed on a systematic incorporation of causality constraints into the optimal design of reconstructors. We consider reconstruction problems, in which the sampling (acquisition) device is given and the performance is measured by the -norm of the reconstruction error. The problem is solved under the constraint that the optimal reconstructor is -causal for a given i.e., that its impulse response is zero in the time interval where is the sampling period. We derive a closed-form state-space solution of the problem, which is based on the spectral factorization of a rational transfer function
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