19,872 research outputs found
Statistical Fourier Analysis: Clarifications and Interpretations
This paper expounds some of the results of Fourier theory that are essential to the statistical analysis of time series. It employs the algebra of circulant matrices to expose the structure of the discrete Fourier transform and to elucidate the filtering operations that may be applied to finite data sequences. An ideal filter with a gain of unity throughout the pass band and a gain of zero throughout the stop band is commonly regarded as incapable of being realised in finite samples. It is shown here that, to the contrary, such a filter can be realised both in the time domain and in the frequency domain. The algebra of circulant matrices is also helpful in revealing the nature of statistical processes that are band limited in the frequency domain. In order to apply the conventional techniques of autoregressive moving-average modelling, the data generated by such processes must be subjected to antialiasing filtering and sub sampling. These techniques are also described. It is argued that band-limited processes are more prevalent in statistical and econometric time series than is commonly recognised.
Computation of the asymptotic states of modulated open quantum systems with a numerically exact realization of the quantum trajectory method
Quantum systems out of equilibrium are presently a subject of active
research, both in theoretical and experimental domains. In this work we
consider time-periodically modulated quantum systems which are in contact with
a stationary environment. Within the framework of a quantum master equation,
the asymptotic states of such systems are described by time-periodic density
operators. Resolution of these operators constitutes a non-trivial
computational task. To go beyond the current size limits, we use the quantum
trajectory method which unravels master equation for the density operator into
a set of stochastic processes for wave functions. The asymptotic density matrix
is calculated by performing a statistical sampling over the ensemble of quantum
trajectories, preceded by a long transient propagation. We follow the ideology
of event-driven programming and construct a new algorithmic realization of the
method. The algorithm is computationally efficient, allowing for long 'leaps'
forward in time, and is numerically exact in the sense that, being given the
list of uniformly distributed (on the unit interval) random numbers, , one could propagate a quantum trajectory (with 's
as norm thresholds) in a numerically exact way. %Since the quantum trajectory
method falls into the class of standard sampling problems, performance of the
algorithm %can be substantially improved by implementing it on a computer
cluster. By using a scalable -particle quantum model, we demonstrate that
the algorithm allows us to resolve the asymptotic density operator of the model
system with states on a regular-size computer cluster, thus reaching
the scale on which numerical studies of modulated Hamiltonian systems are
currently performed
A survey of exemplar-based texture synthesis
Exemplar-based texture synthesis is the process of generating, from an input
sample, new texture images of arbitrary size and which are perceptually
equivalent to the sample. The two main approaches are statistics-based methods
and patch re-arrangement methods. In the first class, a texture is
characterized by a statistical signature; then, a random sampling conditioned
to this signature produces genuinely different texture images. The second class
boils down to a clever "copy-paste" procedure, which stitches together large
regions of the sample. Hybrid methods try to combine ideas from both approaches
to avoid their hurdles. The recent approaches using convolutional neural
networks fit to this classification, some being statistical and others
performing patch re-arrangement in the feature space. They produce impressive
synthesis on various kinds of textures. Nevertheless, we found that most real
textures are organized at multiple scales, with global structures revealed at
coarse scales and highly varying details at finer ones. Thus, when confronted
with large natural images of textures the results of state-of-the-art methods
degrade rapidly, and the problem of modeling them remains wide open.Comment: v2: Added comments and typos fixes. New section added to describe
FRAME. New method presented: CNNMR
Geometrical families of mechanically stable granular packings
We enumerate and classify nearly all of the possible mechanically stable (MS)
packings of bidipserse mixtures of frictionless disks in small sheared systems.
We find that MS packings form continuous geometrical families, where each
family is defined by its particular network of particle contacts. We also
monitor the dynamics of MS packings along geometrical families by applying
quasistatic simple shear strain at zero pressure. For small numbers of
particles (N < 16), we find that the dynamics is deterministic and highly
contracting. That is, if the system is initialized in a MS packing at a given
shear strain, it will quickly lock into a periodic orbit at subsequent shear
strain, and therefore sample only a very small fraction of the possible MS
packings in steady state. In studies with N>16, we observe an increase in the
period and random splittings of the trajectories caused by bifurcations in
configuration space. We argue that the ratio of the splitting and contraction
rates in large systems will determine the distribution of MS-packing
geometrical families visited in steady-state. This work is part of our
long-term research program to develop a master-equation formalism to describe
macroscopic slowly driven granular systems in terms of collections of small
subsystems.Comment: 18 pages, 23 figures, 5 table
Quantum field tomography
We introduce the concept of quantum field tomography, the efficient and
reliable reconstruction of unknown quantum fields based on data of correlation
functions. At the basis of the analysis is the concept of continuous matrix
product states, a complete set of variational states grasping states in quantum
field theory. We innovate a practical method, making use of and developing
tools in estimation theory used in the context of compressed sensing such as
Prony methods and matrix pencils, allowing us to faithfully reconstruct quantum
field states based on low-order correlation functions. In the absence of a
phase reference, we highlight how specific higher order correlation functions
can still be predicted. We exemplify the functioning of the approach by
reconstructing randomised continuous matrix product states from their
correlation data and study the robustness of the reconstruction for different
noise models. We also apply the method to data generated by simulations based
on continuous matrix product states and using the time-dependent variational
principle. The presented approach is expected to open up a new window into
experimentally studying continuous quantum systems, such as encountered in
experiments with ultra-cold atoms on top of atom chips. By virtue of the
analogy with the input-output formalism in quantum optics, it also allows for
studying open quantum systems.Comment: 31 pages, 5 figures, minor change
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