Topic

# Laplace distribution

About: Laplace distribution is a research topic. Over the lifetime, 1403 publications have been published within this topic receiving 28107 citations.

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TL;DR: In this article, a time-frequency distribution of L. Cohen's (1966) class is introduced, which is called exponential distribution (ED) after its exponential kernel function, and the authors interpret the ED from the spectral density-estimation point of view.

Abstract: The authors introduce a time-frequency distribution of L. Cohen's (1966) class and examines its properties. This distribution is called exponential distribution (ED) after its exponential kernel function. First, the authors interpret the ED from the spectral-density-estimation point of view. They then show how the exponential kernel controls the cross terms as represented in the generalized ambiguity function domain, and they analyze the ED for two specific types of multicomponent signals: sinusoidal signals and chirp signals. Next, they define the ED for discrete-time signals and the running windowed exponential distribution (RWED), which is computationally efficient. Finally, the authors present numerical examples of the RWED using the synthetically generated signals. It is found that the ED is very effective in diminishing the effects of cross terms while retaining most of the properties which are useful for a time-frequency distribution. >

1,306 citations

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01 Jan 1997

TL;DR: The Laplace Principle for the Random Walk Model with Discontinuous Statistics as mentioned in this paper has been extended to the continuous-time Markov Processes with continuous statistics, and the Laplace principle has been used for the continuous time Markov Chain model as well.

Abstract: Formulation of Large Deviation Theory in Terms of the Laplace Principle. First Example: Sanov's Theorem. Second Example: Mogulskii's Theorem. Representation Formulas for Other Stochastic Processes. Compactness and Limit Properties for the Random Walk Model. Laplace Principle for the Random Walk Model with Continuous Statistics. Laplace Principle for the Random Walk Model with Discontinuous Statistics. Laplace Principle for the Empirical Measures of a Markov Chain. Extensions of the Laplace Principle for the Empirical Measures of a Markov Chain. Laplace Principle for Continuous-Time Markov Processes with Continuous Statistics. Appendices. Bibliography. Indexes.

1,130 citations

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TL;DR: This article introduced the idea of Bayesian quantile regression employing a likelihood function that is based on the asymmetric Laplace distribution, which is a natural and effective way to model quantile regressions.

Abstract: The paper introduces the idea of Bayesian quantile regression employing a likelihood function that is based on the asymmetric Laplace distribution. It is shown that irrespective of the original distribution of the data, the use of the asymmetric Laplace distribution is a very natural and effective way for modelling Bayesian quantile regression. The paper also demonstrates that improper uniform priors for the unknown model parameters yield a proper joint posterior. The approach is illustrated via a simulated and two real data sets.

773 citations

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TL;DR: This work offers a rigorous mathematical analysis using a doubly stochastic model of the images, which not only provides the theoretical explanations necessary, but also leads to insights about various other observations from the literature.

Abstract: Over the past two decades, there have been various studies on the distributions of the DCT coefficients for images. However, they have concentrated only on fitting the empirical data from some standard pictures with a variety of well-known statistical distributions, and then comparing their goodness of fit. The Laplacian distribution is the dominant choice balancing simplicity of the model and fidelity to the empirical data. Yet, to the best of our knowledge, there has been no mathematical justification as to what gives rise to this distribution. We offer a rigorous mathematical analysis using a doubly stochastic model of the images, which not only provides the theoretical explanations necessary, but also leads to insights about various other observations from the literature. This model also allows us to investigate how certain changes in the image statistics could affect the DCT coefficient distributions.

743 citations

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TL;DR: A novel computer algorithm for fast two-dimensional inverse Laplace transformation to obtain T(1)--T(2) correlation functions is demonstrated, producing a result that is found to be stable in the presence of noise.

Abstract: Spin relaxation is a sensitive probe of molecular structure and dynamics. Correlation of relaxation time constants, such as T(1) and T(2), conceptually similar to the conventional multidimensional spectroscopy, have been difficult to determine primarily due to the absense of an efficient multidimensional Laplace inversion program. We demonstrate the use of a novel computer algorithm for fast two-dimensional inverse Laplace transformation to obtain T(1)--T(2) correlation functions. The algorithm efficiently performs a least-squares fit on two-dimensional data with a nonnegativity constraint. We use a regularization method to find a balance between the residual fitting errors and the known noise amplitude, thus producing a result that is found to be stable in the presence of noise. This algorithm can be extended to include functional forms other than exponential kernels. We demonstrate the performance of the algorithm at different signal-to-noise ratios and with different T(1)--T(2) spectral characteristics using several brine-saturated rock samples.

643 citations