A folded Laplace distribution

Liu, Yucui; Kozubowski, Tomasz J.

doi:10.1186/s40488-015-0033-9

Cited by 14 publications

(8 citation statements)

References 38 publications

(25 reference statements)

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“…Let W 1 2 = |X|, where X is distributed according to a Laplace distribution with mean µ and scale parameter σ. The density of W 1 2 equals f (y) = 1 σ e −µ/σ cosh(y/σ) for 0 ≤ y < µ e −y/σ cosh(µ/σ) for 0 ≤ µ ≤ y , see also Liu and Kozubowski (2015). The density of log W 1 2 equals e y f (e y ), such that log-concavity of log W 1 2 requires that the second derivative of this is non-positive.…”

Section: Bivariate Pareto Distributions and Emtpmentioning

confidence: 96%

Total positivity in multivariate extremes

Röttger¹,

Engelke²,

Zwiernik³

2021

Preprint

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Positive dependence is present in many real world data sets and has appealing stochastic properties. In particular, the notion of multivariate total positivity of order 2 (MTP 2 ) is a convex constraint and acts as an implicit regularizer in the Gaussian case. We study positive dependence in multivariate extremes and introduce EMTP 2 , an extremal version of MTP 2 . This notion turns out to appear prominently in extremes and, in fact, it is satisfied by many classical models. For a Hüsler-Reiss distribution, the analogue of a Gaussian distribution in extremes, we show that it is EMTP 2 if and only if its precision matrix is a Laplacian of a connected graph. We propose an estimator for the parameters of the Hüsler-Reiss distribution under EMTP 2 as the solution of a convex optimization problem with Laplacian constraint. We prove that this estimator is consistent and typically yields a sparse model with possibly non-decomposable extremal graphical structure. At the example of two data sets, we illustrate this regularization and the superior performance compared to existing methods.

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Section: Bivariate Pareto Distributions and Emtpmentioning

confidence: 96%

Total positivity in multivariate extremes

Röttger¹,

Engelke²,

Zwiernik³

2021

Preprint

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“…A folded Laplace distribution is also accomplished via transform (31), and the pdf of the transformed random variable X becomes (32). Placing (45) in (32), we have the pdf of a folded Laplace distribution [26]:…”

Section: B Prior Distributions Of β and α In Mscd-lmentioning

confidence: 99%

“…We call these two methods MSCD-l 2 and MSCD-l 1 . Equally important, we derive the proposed MSCD from the Bayesian perspective, showing that MSCD-l 2 and MSCD-l 1 can be derived if a multivariate half-Gaussian distribution [25] and a multivariate half-Laplace distribution [26] are assumed as the prior distributions of the coefficient vectors. To our knowledge, it is the first time that the cone representations with the l 2 -norm and l 1 -norm regularisations are derived from the Bayesian perspective, as well as the prior distributions identified.…”

mentioning

confidence: 99%

Matched Shrunken Cone Detector (MSCD): Bayesian Derivations and Case Studies for Hyperspectral Target Detection

Wang

Zhu

Fukui

et al. 2017

IEEE Trans. on Image Process.

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Hyperspectral images (HSIs) possess non-negative properties for both hyperspectral signatures and abundance coefficients, which can be naturally modeled using cone-based representation. However, in hyperspectral target detection, cone-based methods are barely studied. In this paper, we propose a new regularized cone-based representation approach to hyperspectral target detection, as well as its two working models by incorporating into the cone representation l-norm and l-norm regularizations, respectively. We call the new approach the matched shrunken cone detector (MSCD). Also important, we provide principled derivations of the proposed MSCD from the Bayesian perspective: we show that MSCD can be derived by assuming a multivariate half-Gaussian distribution or a multivariate half-Laplace distribution as the prior distribution of the coefficients of the models. In the experimental studies, we compare the proposed MSCD with the subspace methods and the sparse representation-based methods for HSI target detection. Two real hyperspectral data sets are used for evaluating the detection performances on sub-pixel targets and full-pixel targets, respectively. Results show that the proposed MSCD can outperform other methods in both cases, demonstrating the competitiveness of the regularized cone-based representation.

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“…The Laplace distribution with location parameter zero and scale parameter one is called the classical Laplace distribution and its p.d.f is given by Several modifications of the Laplace distribution are currently available in the literature. Some recent studies in this respect were made by Cordeiro and Lemonte (2011), Jose and Thomas (2014), Liu and Kozubowski (2015), Mahmoudvand et al (2015), Kozubowski et al (2016), Nassar (2016) and Li (2017). Laplace distribution has wide range of applications in real life to model and analyze data sets in engineering, financial, industrial, environmental and biological fields.…”

Section: Introductionmentioning

confidence: 99%

An alternative Laplace distribution

Kumar

Jose

2020

JSR

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In this paper, we propose an alternative version to the Laplace distribution which we named as “alternative Laplace distribution (ALD)” and discuss some of its important properties. A location-scale extension of the ALD is considered and the maximum likelihood estimation procedures for estimating its parameters is described. Further, the distribution is ﬁtted to certain real life data sets for illustrating the utility of the model. A simulation study is carried out to examine the performance of likelihood estimators of the parameters of the distribution.

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A folded Laplace distribution

Cited by 14 publications

References 38 publications

Total positivity in multivariate extremes

Total positivity in multivariate extremes

Matched Shrunken Cone Detector (MSCD): Bayesian Derivations and Case Studies for Hyperspectral Target Detection

An alternative Laplace distribution

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