9,756 research outputs found
Covariant Lyapunov vectors
The recent years have witnessed a growing interest for covariant Lyapunov
vectors (CLVs) which span local intrinsic directions in the phase space of
chaotic systems. Here we review the basic results of ergodic theory, with a
specific reference to the implications of Oseledets' theorem for the properties
of the CLVs. We then present a detailed description of a "dynamical" algorithm
to compute the CLVs and show that it generically converges exponentially in
time. We also discuss its numerical performance and compare it with other
algorithms presented in literature. We finally illustrate how CLVs can be used
to quantify deviations from hyperbolicity with reference to a dissipative
system (a chain of H\'enon maps) and a Hamiltonian model (a Fermi-Pasta-Ulam
chain)
Sparse Identification and Estimation of Large-Scale Vector AutoRegressive Moving Averages
The Vector AutoRegressive Moving Average (VARMA) model is fundamental to the
theory of multivariate time series; however, in practice, identifiability
issues have led many authors to abandon VARMA modeling in favor of the simpler
Vector AutoRegressive (VAR) model. Such a practice is unfortunate since even
very simple VARMA models can have quite complicated VAR representations. We
narrow this gap with a new optimization-based approach to VARMA identification
that is built upon the principle of parsimony. Among all equivalent
data-generating models, we seek the parameterization that is "simplest" in a
certain sense. A user-specified strongly convex penalty is used to measure
model simplicity, and that same penalty is then used to define an estimator
that can be efficiently computed. We show that our estimator converges to a
parsimonious element in the set of all equivalent data-generating models, in a
double asymptotic regime where the number of component time series is allowed
to grow with sample size. Further, we derive non-asymptotic upper bounds on the
estimation error of our method relative to our specially identified target.
Novel theoretical machinery includes non-asymptotic analysis of infinite-order
VAR, elastic net estimation under a singular covariance structure of
regressors, and new concentration inequalities for quadratic forms of random
variables from Gaussian time series. We illustrate the competitive performance
of our methods in simulation and several application domains, including
macro-economic forecasting, demand forecasting, and volatility forecasting
Closed-form expression for finite predictor coefficients of multivariate ARMA processes
We derive a closed-form expression for the finite predictor coefficients of
multivariate ARMA (autoregressive moving-average) processes. The expression is
given in terms of several explicit matrices that are of fixed sizes independent
of the number of observations. The significance of the expression is that it
provides us with a linear-time algorithm to compute the finite predictor
coefficients. In the proof of the expression, a correspondence result between
two relevant matrix-valued outer functions plays a key role. We apply the
expression to determine the asymptotic behavior of a sum that appears in the
autoregressive model fitting and the autoregressive sieve bootstrap. The
results are new even for univariate ARMA processes.Comment: Journal of Multivariate Analysis, to appea
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