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48
The twoparameter PoissonDirichlet distribution derived from a stable subordinator.
, 1995
"... The twoparameter PoissonDirichlet distribution, denoted pd(ff; `), is a distribution on the set of decreasing positive sequences with sum 1. The usual PoissonDirichlet distribution with a single parameter `, introduced by Kingman, is pd(0; `). Known properties of pd(0; `), including the Markov ..."
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Cited by 364 (33 self)
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The twoparameter PoissonDirichlet distribution, denoted pd(ff; `), is a distribution on the set of decreasing positive sequences with sum 1. The usual PoissonDirichlet distribution with a single parameter `, introduced by Kingman, is pd(0; `). Known properties of pd(0; `), including the Markov chain description due to VershikShmidtIgnatov, are generalized to the twoparameter case. The sizebiased random permutation of pd(ff; `) is a simple residual allocation model proposed by Engen in the context of species diversity, and rediscovered by Perman and the authors in the study of excursions of Brownian motion and Bessel processes. For 0 ! ff ! 1, pd(ff; 0) is the asymptotic distribution of ranked lengths of excursions of a Markov chain away from a state whose recurrence time distribution is in the domain of attraction of a stable law of index ff. Formulae in this case trace back to work of Darling, Lamperti and Wendel in the 1950's and 60's. The distribution of ranked lengths of e...
Nonlinear source separation: the postnonlinear mixtures
 In: Proceedings of the ESANN’97
, 1997
"... Abstract—In this paper, we address the problem of separation of mutually independent sources in nonlinear mixtures. First, we propose theoretical results and prove that in the general case, it is not possible to separate the sources without nonlinear distortion. Therefore, we focus our work on speci ..."
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Cited by 150 (26 self)
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Abstract—In this paper, we address the problem of separation of mutually independent sources in nonlinear mixtures. First, we propose theoretical results and prove that in the general case, it is not possible to separate the sources without nonlinear distortion. Therefore, we focus our work on specific nonlinear mixtures known as postnonlinear mixtures. These mixtures constituted by a linear instantaneous mixture (linear memoryless channel) followed by an unknown and invertible memoryless nonlinear distortion, are realistic models in many situations and emphasize interesting properties i.e., in such nonlinear mixtures, sources can be estimated with the same indeterminacies as in instantaneous linear mixtures. The separation structure of nonlinear mixtures is a twostage system, namely, a nonlinear stage followed by a linear stage, the parameters of which are updated to minimize an output independence criterion expressed as a mutual information criterion. The minimization of this criterion requires knowledge or estimation of source densities or of their logderivatives. A first algorithm based on a Gram–Charlier expansion of densities is proposed. Unfortunately, it fails for hard nonlinear mixtures. A second algorithm based on an adaptive estimation of the logderivative of densities leads to very good performance, even with hard nonlinearities. Experiments are proposed to illustrate these results. Index Terms—Entropy, neural networks, nonlinear mixtures, source separation, unsupervised adaptive algorithms. I.
Advances in nonlinear blind source separation
 In Proc. of the 4th Int. Symp. on Independent Component Analysis and Blind Signal Separation (ICA2003
, 2003
"... Abstract — In this paper, we briefly review recent advances in blind source separation (BSS) for nonlinear mixing models. After a general introduction to the nonlinear BSS and ICA (independent Component Analysis) problems, we discuss in more detail uniqueness issues, presenting some new results. A f ..."
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Cited by 41 (2 self)
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Abstract — In this paper, we briefly review recent advances in blind source separation (BSS) for nonlinear mixing models. After a general introduction to the nonlinear BSS and ICA (independent Component Analysis) problems, we discuss in more detail uniqueness issues, presenting some new results. A fundamental difficulty in the nonlinear BSS problem and even more so in the nonlinear ICA problem is that they are nonunique without extra constraints, which are often implemented by using a suitable regularization. Postnonlinear mixtures are an important special case, where a nonlinearity is applied to linear mixtures. For such mixtures, the ambiguities are essentially the same as for the linear ICA or BSS problems. In the later part of this paper, various separation techniques proposed for postnonlinear mixtures and general nonlinear mixtures are reviewed. I. THE NONLINEAR ICA AND BSS PROBLEMS Consider Æ samples of the observed data vector Ü, modeled by
ON THE MARKOV–KREIN IDENTITY AND QUASIINVARIANCE OF THE GAMMA PROCESS
, 2004
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Source Separation: From Dusk Till Dawn
"... The first part of this paper is concerned by the history of source separation. It include our comments and those of a few other researchers on the development of this new research field. The second part is focused on recent developments of the separation in nonlinear mixtures. ..."
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Cited by 17 (5 self)
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The first part of this paper is concerned by the history of source separation. It include our comments and those of a few other researchers on the development of this new research field. The second part is focused on recent developments of the separation in nonlinear mixtures.
Detection of the Number of Signals Using the BenjaminiHochberg Procedure
"... This work presents a novel approach to detect multiple signals embedded in noisy observations from a sensor array. We formulate the detection problem as a multiple hypothesis test. To control the global level of the multiple test, we apply the false discovery rate (FDR) criterion proposed by Benjami ..."
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Cited by 14 (1 self)
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This work presents a novel approach to detect multiple signals embedded in noisy observations from a sensor array. We formulate the detection problem as a multiple hypothesis test. To control the global level of the multiple test, we apply the false discovery rate (FDR) criterion proposed by Benjamini and Hochberg. Compared to the classical familywise error rate (FWE) criterion, the FDRcontrolling procedure leads to a significant gain in power for large size problems. In addition, we apply the bootstrap technique to estimate the observed significance level required by the FDRcontrolling procedure. Simulations show that the FDRcontrolling procedure always provides higher probability of correct detection than the FWEcontrolling procedure. Furthermore, the reliability of the proposed test procedure is not affected by the gain in power of the test.
Distinguished properties of the gamma processes and related properties
, 2000
"... We study fundamental properties of the gamma process and their relation to various topics such as Poisson–Dirichlet measures and stable processes. We prove the quasiinvariance of the gamma process with respect to a large group of linear transformations. We also show that it is a renormalized limit ..."
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Cited by 9 (1 self)
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We study fundamental properties of the gamma process and their relation to various topics such as Poisson–Dirichlet measures and stable processes. We prove the quasiinvariance of the gamma process with respect to a large group of linear transformations. We also show that it is a renormalized limit of the stable processes and has an equivalent sigmafinite measure (quasiLebesgue) with important invariance properties. New properties of the gamma process can be applied to the Poisson—Dirichlet measures. We also emphasize the deep similarity between the gamma process and the Brownian motion. The connection of the above topics makes more transparent some old and new facts about stable and gamma processes, and the PoissonDirichlet measures.
On signal detection using the benjaminihochberg procedure
 In Proc. IEEE Workshop on Statistical and Signal Processing
, 2005
"... We investigate a multiple hypothesis test designed for detecting signals embedded in noisy observations of a sensor array. The global level of the multiple test is controlled by the false discovery rate (FDR) criterion recently suggested by Benjamini and Hochberg instead of the classical familywise ..."
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Cited by 8 (4 self)
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We investigate a multiple hypothesis test designed for detecting signals embedded in noisy observations of a sensor array. The global level of the multiple test is controlled by the false discovery rate (FDR) criterion recently suggested by Benjamini and Hochberg instead of the classical familywise error rate (FWE) criterion. In the previous study [3], the suggested procedure has shown promising results on simulated data. Here we carefully examine the independence condition required by the Benjamini Hochberg procedure to ensure the control of FDR. Based on the properties of beta distribution, we proved the applicability of the Benjamini Hochberg procedure to the underlying test. Further simulation results show that the false alarm rate is less than 0.02 for a choice of FDR as high as 0.1, implying the reliability of the test has not been affected by the increase in power. 1.
Laws and likelihoods for Ornstein UhlenbeckGamma and other BNS OU stochastic volatilty models with extensions
, 2006
"... In recent years there have been many proposals as flexible alternatives to Gaussian based continuous time stochastic volatility models. A great deal of these models employ positive Lévy processes. Among these are the attractive nonGaussian positive OrnsteinUhlenbeck (OU) processes proposed by Barn ..."
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Cited by 4 (1 self)
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In recent years there have been many proposals as flexible alternatives to Gaussian based continuous time stochastic volatility models. A great deal of these models employ positive Lévy processes. Among these are the attractive nonGaussian positive OrnsteinUhlenbeck (OU) processes proposed by BarndorffNielsen and Shephard (BNS) in a series of papers. One current problem of these approaches is the unavailability of a tractable likelihood based statistical analysis for the returns of financial assets. This paper, while focusing on the BNS models, develops general theory for the implementation of statistical inference for a host of models. Specifically we show how to reduce the infinitedimensional process based models to finite, albeit high, dimensional ones. Inference can then be based on Monte Carlo methods. As highlights, specific to BNS we show that an OU process driven by an infinite activity Gamma process, that is an OUΓ, exhibits unique features which allows one to exactly sample from relevant joint distributions. We show that this is a consequence of the OU structure and the unique calculus of Gamma and Dirichlet processes. Owing to another connection between Gamma/Dirichlet processes and the theory of Generalized Gamma Convolutions (GGC) we identify a large class of models, we call (FGGC), where one can perfectly sample marginal distributions relevant to option pricing and Monte Carlo likelihood analysis. This involves a curious result, we establish as Theorem 6.1. We also