23,882 research outputs found
On the behavior of EMD and MEMD in presence of symmetric alpha-stable noise
EmpiricalMode Decomposition (EMD) and its extended versions such as Multivariate EMD (MEMD) are data-driven techniques that represent nonlinear and non-stationary data as a sum of a finite zero-mean AM-FM components referred to as Intrinsic Mode Functions (IMFs). The aim of this work is to analyze the behavior of EMD and MEMD in stochastic situations involving non-Gaussian noise, more precisely, we examine the case of Symmetric Alpha-Stable noise. We report numerical experiments supporting the claim that both EMD and MEMD act, essentially, as filter banks on each channel of the input signal in the case of Symmetric Alpha Stable noise. Reported results show that, unlike EMD, MEMD has the ability to align common frequency modes across multiple channels in same index IMFs. Further, simulations show that, contrary to EMD, for MEMD the stability property is well satisfied for the modes of lower indices and this result is exploited for the estimation of the stability index of the Symmetric Alpha Stable input signal
Feller Processes: The Next Generation in Modeling. Brownian Motion, L\'evy Processes and Beyond
We present a simple construction method for Feller processes and a framework
for the generation of sample paths of Feller processes. The construction is
based on state space dependent mixing of L\'evy processes.
Brownian Motion is one of the most frequently used continuous time Markov
processes in applications. In recent years also L\'evy processes, of which
Brownian Motion is a special case, have become increasingly popular.
L\'evy processes are spatially homogeneous, but empirical data often suggest
the use of spatially inhomogeneous processes. Thus it seems necessary to go to
the next level of generalization: Feller processes. These include L\'evy
processes and in particular Brownian motion as special cases but allow spatial
inhomogeneities.
Many properties of Feller processes are known, but proving the very existence
is, in general, very technical. Moreover, an applicable framework for the
generation of sample paths of a Feller process was missing. We explain, with
practitioners in mind, how to overcome both of these obstacles. In particular
our simulation technique allows to apply Monte Carlo methods to Feller
processes.Comment: 22 pages, including 4 figures and 8 pages of source code for the
generation of sample paths of Feller processe
Chance, long tails, and inference: a non-Gaussian, Bayesian theory of vocal learning in songbirds
Traditional theories of sensorimotor learning posit that animals use sensory
error signals to find the optimal motor command in the face of Gaussian sensory
and motor noise. However, most such theories cannot explain common behavioral
observations, for example that smaller sensory errors are more readily
corrected than larger errors and that large abrupt (but not gradually
introduced) errors lead to weak learning. Here we propose a new theory of
sensorimotor learning that explains these observations. The theory posits that
the animal learns an entire probability distribution of motor commands rather
than trying to arrive at a single optimal command, and that learning arises via
Bayesian inference when new sensory information becomes available. We test this
theory using data from a songbird, the Bengalese finch, that is adapting the
pitch (fundamental frequency) of its song following perturbations of auditory
feedback using miniature headphones. We observe the distribution of the sung
pitches to have long, non-Gaussian tails, which, within our theory, explains
the observed dynamics of learning. Further, the theory makes surprising
predictions about the dynamics of the shape of the pitch distribution, which we
confirm experimentally
Spectral densities of Wishart-Levy free stable random matrices: Analytical results and Monte Carlo validation
Random matrix theory is used to assess the significance of weak correlations
and is well established for Gaussian statistics. However, many complex systems,
with stock markets as a prominent example, exhibit statistics with power-law
tails, that can be modelled with Levy stable distributions. We review
comprehensively the derivation of an analytical expression for the spectra of
covariance matrices approximated by free Levy stable random variables and
validate it by Monte Carlo simulation.Comment: 10 pages, 1 figure, submitted to Eur. Phys. J.
"Infinite-variance, Alpha-stable Shocks in Monetary SVAR: Final Working Paper Version"
This paper adumbrates a theory of what might be going wrong in the monetary SVAR literature and provides supporting empirical evidence. The theory is that macroeconomists may be attempting to identify structural forms that do not exist, given the true distribution of the innovations in the reduced-form VAR. The paper shows that this problem occurs whenever (1) some innovation in the VAR has an infinite-variance distribution and (2) the matrix of coefficients on the contemporaneous terms in the VAR's structural form is nonsingular. Since (2) is almost always required for SVAR analysis, it is germane to test hypothesis (1). Hence, in this paper, we fit a-stable distributions to VAR residuals and, using a parametric-bootstrap method, test the hypotheses that each of the error terms has finite variance.Vector Autoregression; Levy-stable Distribution; Infinite Variance; Monetary Policy Shocks; Heavy-tailed Error Terms; Factorization; Impulse-Response Function
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