44,502 research outputs found
Entropy of Highly Correlated Quantized Data
This paper considers the entropy of highly correlated quantized samples. Two results are shown. The first concerns sampling and identically scalar quantizing a stationary continuous-time random process over a finite interval. It is shown that if the process crosses a quantization threshold with positive probability, then the joint entropy of the quantized samples tends to infinity as the sampling rate goes to infinity. The second result provides an upper bound to the rate at which the joint entropy tends to infinity, in the case of an infinite-level uniform threshold scalar quantizer and a stationary Gaussian random process. Specifically, an asymptotic formula for the conditional entropy of one quantized sample conditioned on the previous quantized sample is derived. At high sampling rates, these results indicate a sharp contrast between the large encoding rate (in bits/sec) required by a lossy source code consisting of a fixed scalar quantizer and an ideal, sampling-rate-adapted lossless code, and the bounded encoding rate required by an ideal lossy source code operating at the same distortion
Kernel conditional quantile estimation via reduction revisited
Quantile regression refers to the process of estimating the quantiles of a conditional distribution and has many important applications within econometrics and data mining, among other domains. In this paper, we show how to estimate these conditional quantile functions within a Bayes risk minimization framework using a Gaussian process prior. The resulting non-parametric probabilistic model is easy to implement and allows non-crossing quantile functions to be enforced. Moreover, it can directly be used in combination with tools and extensions of standard Gaussian Processes such as principled hyperparameter estimation, sparsification, and quantile regression with input-dependent noise rates. No existing approach enjoys all of these desirable properties. Experiments on benchmark datasets show that our method is competitive with state-of-the-art approaches.
A Method for the Combination of Stochastic Time Varying Load Effects
The problem of evaluating the probability that a structure becomes unsafe under a
combination of loads, over a given time period, is addressed. The loads and load effects
are modeled as either pulse (static problem) processes with random occurrence time, intensity and a specified shape or intermittent continuous (dynamic problem) processes which
are zero mean Gaussian processes superimposed 'on a pulse process. The load coincidence
method is extended to problems with both nonlinear limit states and dynamic responses,
including the case of correlated dynamic responses. The technique of linearization of a
nonlinear limit state commonly used in a time-invariant problem is investigated for timevarying
combination problems, with emphasis on selecting the linearization point. Results
are compared with other methods, namely the method based on upcrossing rate, simpler
combination rules such as Square Root of Sum of Squares and Turkstra's rule. Correlated
effects among dynamic loads are examined to see how results differ from correlated static
loads and to demonstrate which types of load dependencies are most important, i.e., affect'
the exceedance probabilities the most.
Application of the load coincidence method to code development is briefly discussed.National Science Foundation Grants CME 79-18053 and CEE 82-0759
Level crossings and other level functionals of stationary Gaussian processes
This paper presents a synthesis on the mathematical work done on level
crossings of stationary Gaussian processes, with some extensions. The main
results [(factorial) moments, representation into the Wiener Chaos, asymptotic
results, rate of convergence, local time and number of crossings] are
described, as well as the different approaches [normal comparison method, Rice
method, Stein-Chen method, a general -dependent method] used to obtain them;
these methods are also very useful in the general context of Gaussian fields.
Finally some extensions [time occupation functionals, number of maxima in an
interval, process indexed by a bidimensional set] are proposed, illustrating
the generality of the methods. A large inventory of papers and books on the
subject ends the survey.Comment: Published at http://dx.doi.org/10.1214/154957806000000087 in the
Probability Surveys (http://www.i-journals.org/ps/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Simplified model of statistically stationary spacecraft rotation and associated induced gravity environments
A stochastic model of spacecraft motion was developed based on the assumption that the net torque vector due to crew activity and rocket thruster firings is a statistically stationary Gaussian vector process. The process had zero ensemble mean value, and the components of the torque vector were mutually stochastically independent. The linearized rigid-body equations of motion were used to derive the autospectral density functions of the components of the spacecraft rotation vector. The cross-spectral density functions of the components of the rotation vector vanish for all frequencies so that the components of rotation were mutually stochastically independent. The autospectral and cross-spectral density functions of the induced gravity environment imparted to scientific apparatus rigidly attached to the spacecraft were calculated from the rotation rate spectral density functions via linearized inertial frame to body-fixed principal axis frame transformation formulae. The induced gravity process was a Gaussian one with zero mean value. Transformation formulae were used to rotate the principal axis body-fixed frame to which the rotation rate and induced gravity vector were referred to a body-fixed frame in which the components of the induced gravity vector were stochastically independent. Rice's theory of exceedances was used to calculate expected exceedance rates of the components of the rotation and induced gravity vector processes
The Excursion Set Theory of Halo Mass Functions, Halo Clustering, and Halo Growth
I review the excursion set theory (EST) of dark matter halo formation and
clustering. I recount the Press-Schechter argument for the mass function of
bound objects and review the derivation of the Press-Schechter mass function in
EST. The EST formalism is powerful and can be applied to numerous problems. I
review the EST of halo bias and the properties of void regions. I spend
considerable time reviewing halo growth in the EST. This section culminates
with descriptions of two Monte Carlo methods for generating halo mass accretion
histories. In the final section, I emphasize that the standard EST approach is
the result of several simplifying assumptions. Dropping these assumptions can
lead to more faithful predictions and a more versatile formalism. One such
assumption is the constant height of the barrier for nonlinear collapse. I
review implementations of the excursion set approach with arbitrary barrier
shapes. An application of this is the now well-known improvement to standard
EST that follows from the ellipsoidal-collapse barrier. Additionally, I
emphasize that the statement that halo accretion histories are independent of
halo environments is a simplifying assumption, rather than a prediction of the
theory. I review the method for constructing correlated random walks of the
density field in more general cases. I construct a simple toy model with
correlated walks and I show that excursion set theory makes a qualitatively
simple and general prediction for the relation between halo accretion histories
and halo environments: regions of high density preferentially contain
late-forming halos and conversely for regions of low density. I conclude with a
brief discussion of this prediction in the context of recent numerical studies
of the environmental dependence of halo properties. (Abridged)Comment: 62 pages, 19 figures. Review article based on lectures given at the
Sixth Summer School of the Helmholtz Institute for Supercomputational
Physics. Accepted for Publication in IJMPD. Comments Welcom
First Passage Time Densities in Resonate-and-Fire Models
Motivated by the dynamics of resonant neurons we discuss the properties of
the first passage time (FPT) densities for nonmarkovian differentiable random
processes. We start from an exact expression for the FPT density in terms of an
infinite series of integrals over joint densities of level crossings, and
consider different approximations based on truncation or on approximate
summation of this series. Thus, the first few terms of the series give good
approximations for the FPT density on short times. For rapidly decaying
correlations the decoupling approximations perform well in the whole time
domain.
As an example we consider resonate-and-fire neurons representing stochastic
underdamped or moderately damped harmonic oscillators driven by white Gaussian
or by Ornstein-Uhlenbeck noise. We show, that approximations reproduce all
qualitatively different structures of the FPT densities: from monomodal to
multimodal densities with decaying peaks. The approximations work for the
systems of whatever dimension and are especially effective for the processes
with narrow spectral density, exactly when markovian approximations fail.Comment: 11 pages, 8 figure
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