199 research outputs found
Cascades and Dissipative Anomalies in Relativistic Fluid Turbulence
We develop first-principles theory of relativistic fluid turbulence at high
Reynolds and P\'eclet numbers. We follow an exact approach pioneered by
Onsager, which we explain as a non-perturbative application of the principle of
renormalization-group invariance. We obtain results very similar to those for
non-relativistic turbulence, with hydrodynamic fields in the inertial-range
described as distributional or "coarse-grained" solutions of the relativistic
Euler equations. These solutions do not, however, satisfy the naive
conservation-laws of smooth Euler solutions but are afflicted with dissipative
anomalies in the balance equations of internal energy and entropy. The
anomalies are shown to be possible by exactly two mechanisms, local cascade and
pressure-work defect. We derive "4/5th-law"-type expressions for the anomalies,
which allow us to characterize the singularities (structure-function scaling
exponents) required for their non-vanishing. We also investigate the Lorentz
covariance of the inertial-range fluxes, which we find is broken by our
coarse-graining regularization but which is restored in the limit that the
regularization is removed, similar to relativistic lattice quantum field
theory. In the formal limit as speed of light goes to infinity, we recover the
results of previous non-relativistic theory. In particular, anomalous heat
input to relativistic internal energy coincides in that limit with anomalous
dissipation of non-relativistic kinetic energy
Cascades and Dissipative Anomalies in Compressible Fluid Turbulence
We investigate dissipative anomalies in a turbulent fluid governed by the
compressible Navier-Stokes equation. We follow an exact approach pioneered by
Onsager, which we explain as a non-perturbative application of the principle of
renormalization-group invariance. In the limit of high Reynolds and P\'eclet
numbers, the flow realizations are found to be described as distributional or
"coarse-grained" solutions of the compressible Euler equations, with standard
conservation laws broken by turbulent anomalies. The anomalous dissipation of
kinetic energy is shown to be due not only to local cascade, but also to a
distinct mechanism called pressure-work defect. Irreversible heating in
stationary, planar shocks with an ideal-gas equation of state exemplifies the
second mechanism. Entropy conservation anomalies are also found to occur by two
mechanisms: an anomalous input of negative entropy (negentropy) by
pressure-work and a cascade of negentropy to small scales. We derive
"4/5th-law"-type expressions for the anomalies, which allow us to characterize
the singularities (structure-function scaling exponents) required to sustain
the cascades. We compare our approach with alternative theories and empirical
evidence. It is argued that the "Big Power-Law in the Sky" observed in electron
density scintillations in the interstellar medium is a manifestation of a
forward negentropy cascade, or an inverse cascade of usual thermodynamic
entropy
Overlearning in marginal distribution-based ICA: analysis and solutions
The present paper is written as a word of caution, with users of
independent component analysis (ICA) in mind, to overlearning
phenomena that are often observed.\\
We consider two types of overlearning, typical to high-order
statistics based ICA. These algorithms can be seen to maximise the
negentropy of the source estimates. The first kind of overlearning
results in the generation of spike-like signals, if there are not
enough samples in the data or there is a considerable amount of
noise present. It is argued that, if the data has power spectrum
characterised by curve, we face a more severe problem, which
cannot be solved inside the strict ICA model. This overlearning is
better characterised by bumps instead of spikes. Both overlearning
types are demonstrated in the case of artificial signals as well as
magnetoencephalograms (MEG). Several methods are suggested to
circumvent both types, either by making the estimation of the ICA
model more robust or by including further modelling of the data
Different Estimation Methods for the Basic Independent Component Analysis Model
Inspired by classic cocktail-party problem, the basic Independent Component Analysis (ICA) model is created. What differs Independent Component Analysis (ICA) from other kinds of analysis is the intrinsic non-Gaussian assumption of the data. Several approaches are proposed based on maximizing the non-Gaussianity of the data, which is measured by kurtosis, mutual information, and others. With each estimation, we need to optimize the functions of expectations of non-quadratic functions since it can help us to access the higher-order statistics of non-Gaussian part of the data. In this thesis, our goal is to review the one of the most efficient estimation methods, that is, the Fast Fixed-Point Independent Component Analysis (FastICA) algorithm, illustrate it with some examples using an R package
A Canonical Genetic Algorithm for Blind Inversion of Linear Channels
It is well known the relationship between source separation and blind
deconvolution: If a filtered version of an unknown i.i.d. signal is observed, temporal
independence between samples can be used to retrieve the original signal,
in the same manner as spatial independence is used for source separation. In
this paper we propose the use of a Genetic Algorithm (GA) to blindly invert
linear channels. The use of GA is justified in the case of small number of samples,
where other gradient-like methods fails because of poor estimation of statistics
A multivariate generalized independent factor GARCH model with an application to financial stock returns
We propose a new multivariate factor GARCH model, the GICA-GARCH model ,
where the data are assumed to be generated by a set of independent components (ICs).
This model applies independent component analysis (ICA) to search the conditionally
heteroskedastic latent factors. We will use two ICA approaches to estimate the ICs. The
first one estimates the components maximizing their non-gaussianity, and the second
one exploits the temporal structure of the data. After estimating the ICs, we fit an
univariate GARCH model to the volatility of each IC. Thus, the GICA-GARCH reduces
the complexity to estimate a multivariate GARCH model by transforming it into a small
number of univariate volatility models. We report some simulation experiments to show
the ability of ICA to discover leading factors in a multivariate vector of financial data.
An empirical application to the Madrid stock market will be presented, where we
compare the forecasting accuracy of the GICA-GARCH model versus the orthogonal
GARCH one
A multivariate generalized independent factor GARCH model with an application to financial stock returns
We propose a new multivariate factor GARCH model, the GICA-GARCH model , where the data are assumed to be generated by a set of independent components (ICs). This model applies independent component analysis (ICA) to search the conditionally heteroskedastic latent factors. We will use two ICA approaches to estimate the ICs. The first one estimates the components maximizing their non-gaussianity, and the second one exploits the temporal structure of the data. After estimating the ICs, we fit an univariate GARCH model to the volatility of each IC. Thus, the GICA-GARCH reduces the complexity to estimate a multivariate GARCH model by transforming it into a small number of univariate volatility models. We report some simulation experiments to show the ability of ICA to discover leading factors in a multivariate vector of financial data. An empirical application to the Madrid stock market will be presented, where we compare the forecasting accuracy of the GICA-GARCH model versus the orthogonal GARCH one.ICA, Multivariate GARCH, Factor models, Forecasting volatility
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