John Nolan - Academia.edu (original) (raw)

Papers by John Nolan

Carolina Digital Repository (University of North Carolina at Chapel Hill), 1989

The path continuity of a symmetric p-stable process is examined in terms of any stochastic integr... more The path continuity of a symmetric p-stable process is examined in terms of any stochastic integral representation for the process. When 0 < p < 1, we give necessary and suflicient conditions for path continuity in terms of any (every) representation. When 1 &p<2, we extend the known sutliciency condition in terms of metric entropy and offer a conjecture for the stable version of the Dudley-Fernique theorem. Finally, necessary and sufficient conditions for path continuity are given in terms of continuity at a point for 0 < p < 2.

The following sections are a start on organizing references on stable distributions by topic. It ... more The following sections are a start on organizing references on stable distributions by topic. It is far from complete. Starting on page 18 there is an extensive list of papers on stable distributions, many of which are not included in the first section. Some of the papers there do not directly refer to stable distributions. Someday I may have the time to edit those out, but for now please ignore those references. This list includes a bibliography file provided by Gena Samorodnitsky from Cornell University. I would like to keep this list correct and up-to-date. If you have corrections or additions, please e-mail them to me at the above address, and suggest where to place your references in one of the sections below. A sentence or two summarizing the content would be useful. Please provide all references in BibTeX form, especially if you have more than one or two additions. (See http://en.wikipedia.org/wiki/BibTeX for basic information on BibT E X.) Please send a copy of your papers along. situations where there is impulsive, heavy-tailed noise. In such situations, linear Gaussian filters perform poorly. Using methods based on stable models gives robust non-linear signal processing methods.

$Research paper thumbnail of Truncated fractional moments of stable laws$

arXiv (Cornell University), Sep 4, 2017

Expressions are given for the truncated fractional moments EX p + of a general stable law. These ... more Expressions are given for the truncated fractional moments EX p + of a general stable law. These involve families of special functions that arose out of the study of multivariate stable densities and probabilities. As a particular case, an expression is given for E(X − a) + when α > 1.

$Research paper thumbnail of Random Variate Generation for the First Hit of a Ball for the Symmetric Stable Process in <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow></mrow><annotation encoding="application/x-tex"></annotation></semantics></math>\mathbb {R}^d$$$

Journal of statistical theory and practice, Feb 1, 2024

We provide uniformly efficient random variate generators for a collection of distributions for th... more We provide uniformly efficient random variate generators for a collection of distributions for the hits of the symmetric stable process in R d .

The R Journal

We introduce and showcase mvpd (an acronym for multivariate product distributions), a package tha... more We introduce and showcase mvpd (an acronym for multivariate product distributions), a package that uses a product distribution approach to calculating multivariate subgaussian stable distribution functions. The family of multivariate subgaussian stable distributions are elliptically contoured multivariate stable distributions that contain the multivariate Cauchy and the multivariate normal distribution. These distributions can be useful in modeling data and phenomena that have heavier tails than the normal distribution (more frequent occurrence of extreme values). Application areas include log returns for stocks, signal processing for radar and sonar data, astronomy, and hunting patterns of sharks.

Journal of Statistical Research of Iran, 2006

This paper addresses the problem of compressively sensing a set of temporally correlated sources,... more This paper addresses the problem of compressively sensing a set of temporally correlated sources, in order to achieve faithful sparse signal reconstruction from noisy multiple measurement vectors (MMV). To this end, a simple sensing mechanism is proposed, which does not require the restricted isometry property (RIP) to hold near the sparsity level, whilst it provides additional degrees of freedom to better capture and suppress the inherent sampling noise effects. In particular, a reduced set of MMVs is generated by projecting the source signals onto random vectors drawn from isotropic multivariate stable laws. Then, the correlated sparse signals are recovered from the random MMVs by means of a recently introduced sparse Bayesian learning algorithm. Experimental evaluations on synthetic data with varying number of sources, correlation values, and noise strengths, reveal the superiority of our proposed sensing mechanism, when compared against well-established RIP-based compressive sensing schemes.

Signal Processing, May 1, 2021

Conventional compressive sampling (CS) primarily assumes light-tailed models for the underlying s... more Conventional compressive sampling (CS) primarily assumes light-tailed models for the underlying signal and/or noise statistics. Nevertheless, this assumption is abolished when operating in impulsive environments, where non-Gaussian infinite-variance processes arise for the signal and/or noise components. This drives traditional linear sampling operators to failure, since the gross observation errors are spread uniformly over the generated compressed measurements, whilst masking the critical information content of the observed signal. To address this problem, this paper exploits the power of symmetric alpha-stable (S αS) distributions to design a robust nonlinear compressive sampling operator capable of suppressing the effects of infinite-variance additive observation noise. Specifically, a generalized alpha-stable matched filter is introduced for generating compressed measurements in a nonlinear fashion, which achieves increased robustness to impulsive observation noise, thus subsequently improving the accuracy of traditional sparse reconstruction algorithms. This filter emerges naturally in the case of additive observation noise modeled by S αS distributions, as an effective mechanism for downweighting gross outliers in the noisy signal. The theoretical justification along with the experimental evaluation demonstrate the improved performance of our nonlinear CS framework when compared against state-of-the-art CS techniques for a broad range of impulsive environments.

IEEE Transactions on Signal Processing, Feb 1, 2019

Traditional compressive sensing (CS) primarily assumes light-tailed models for the underlying sig... more Traditional compressive sensing (CS) primarily assumes light-tailed models for the underlying signal and/or noise statistics. Nevertheless, this assumption is not met in the case of highly impulsive environments, where non-Gaussian infinitevariance processes arise for the signal and/or noise components. This drives the traditional sparse reconstruction methods to failure, since they are incapable of suppressing the effects of heavy-tailed sampling noise. The family of symmetric alphastable (SαS) distributions, as a powerful tool for modeling heavytailed behaviors, is adopted in this paper to design a robust algorithm for sparse signal reconstruction from linear random measurements corrupted by infinite-variance additive noise. Specifically, a novel greedy reconstruction method is developed, which achieves increased robustness to impulsive sampling noise by solving a minimum dispersion (MD) optimization problem based on fractional lower-order moments. The MD criterion emerges naturally in the case of additive sampling noise modeled by SαS distributions, as an effective measure of the spread of reconstruction errors around zero, due to the lack of secondorder moments. The experimental evaluation demonstrates the improved reconstruction performance of the proposed algorithm when compared against state-of-the-art CS techniques for a broad range of impulsive environments.

$Research paper thumbnail of On the behavior of EMD and MEMD in presence of symmetric <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math>α-stable noise$

IEEE Signal Processing Letters, 2014

Empirical Mode Decomposition (EMD) and its extended versions such as Multivariate EMD (MEMD) are ... more Empirical Mode Decomposition (EMD) and its extended versions such as Multivariate EMD (MEMD) are datadriven techniques that represent nonlinear and non-stationary data as a sum of a f nite zero-mean AM-FM components referred to as Intrinsic Mode Functions (IMFs). The aim of this work is to analyze the behavior of EMD and MEMD in stochastic situations involving non-Gaussian noise, more precisely, we examine the case of Symmetric α-Stable (SαS) noise. We report numerical experiments supporting the claim that both EMD and MEMD act, essentially, as f lter banks on each channel of the input signal in the case of SαS noise. Reported results show that, unlike EMD, MEMD has the ability to align common frequency modes across multiple channels in same index IMFs. Further, simulations show that, contrary to EMD, for MEMD the stability property is well satisf ed for the modes of lower indices and this result is exploited for the estimation of the stability index of the SαS input signal.

Annals of Operations Research

Optimal sampling period selection for high-frequency data is at the core of financial instruments... more Optimal sampling period selection for high-frequency data is at the core of financial instruments based on algorithmic trading. The unique features of such data, absent in data measured at lower frequencies, raise significant challenges to their statistical analysis and econometric modelling, especially in the case of heavy-tailed data exhibiting outliers and rare events much more frequently. To address this problem, this paper proposes a new methodology for optimal sampling period selection, which better adapts to heavy-tailed statistics of high-frequency financial data. In particular, the novel concept of the degree of impulsiveness (DoI) is introduced first based on alpha-stable distributions, as an alternative source of information for characterising a broad range of impulsive behaviours. Then, a DoI-based generalised volatility signature plot is defined, which is further employed for determining the optimal sampling period. The performance of our method is evaluated in the case...

arXiv: Computation, 2015

A flexible model is developed for multivariate generalized spherical distributions, i.e. ones wit... more A flexible model is developed for multivariate generalized spherical distributions, i.e. ones with level sets that are star shaped. To work in dimension above 2 requires tools from computational geometry and multivariate numerical integration. In order to simulate from these star shaped contours, an algorithm to simulate from general tessellations has been developed that has applications in other situations. These techniques are implemented in an R package gensphere.

NAApproved DTIC ELECTE f SSEP 3 01987U 20L OISTRIGUTIONIASAiLABILIy OF ABSTRACT 21, ABST RACT SEC... more NAApproved DTIC ELECTE f SSEP 3 01987U 20L OISTRIGUTIONIASAiLABILIy OF ABSTRACT 21, ABST RACT SECuRITY CLASSIFICATION WNC .ASSiPIEO/UNLimiTEO SAMIE AS 11PT Z OTIC USERS UNCLASSI FIED 22&. NAME OF REFSPONSIBLE INDIVIDUAL. 22b. TELEPHONE NUMBER r)C, 22c OFFICE SYMBOL (/ncludo Alto Code,

Handbook of Heavy Tailed Distributions in Finance, 2003

Stable distributions are a class of probability distributions that allow heavy tails and skewness... more Stable distributions are a class of probability distributions that allow heavy tails and skewness. In addition to theoretical reasons for using stable laws, they are a rich family that can accurately model different kinds of financial data. We review the basic facts, describe programs that make it practical to use stable distributions, and give examples of these distributions in finance. A non-technical introduction to multivariate stable laws is also given.

Lévy Processes, 2001

A program for maximum likelihood estimation of general stable parameters is described. The Fisher... more A program for maximum likelihood estimation of general stable parameters is described. The Fisher information matrix is computed, making large sample estimation of stable parameters a practical tool. In addition, diagnostics are developed for assessing the stability of a data set. Applications to simulated data, stock price data, foreign exchange rate data, radar data and ocean wave energy are presented.

Journal of Multivariate Analysis, 2013

In this paper, we explore tail dependence modelling in multivariate extreme value distributions. ... more In this paper, we explore tail dependence modelling in multivariate extreme value distributions. The measure of dependence chosen is the scale function, which allows combinations of distributions in a very flexible way. The correspondences between the scale function and the spectral measure or the stable tail dependence function are given. Combining scale functions by simple operations, three parametric classes of laws are (re)constructed and analyzed, and resulting nested and structured models are discussed. Finally, the denseness of each of these classes is shown.

Stochastic Processes and their Applications, 1990

A sufficient condition is given for processes admitting a series expansion with partially depende... more A sufficient condition is given for processes admitting a series expansion with partially dependent components to have nonrandom oscillation. Included are infinitely divisible processes with no Gaussian component. This immediately gives information about the sample paths of such processes, e.g. a Belayev type dichotomy between path continuity and unboundedness for stationary or self-similar processes. The sufficient condition for nonrandom oscillation is shown to be necessary for a large class of infinitely divisible processes to have finite nonrandom oscillation. It is also used to relate path continuity to continuity at each point. Similar results are described for path differentiability. nonrandom oscillation * partial dependence * infinitely divisible processes * sample paths * path differentiability Research supported by AFOSR Contract No. F49620 85C 0144 and AFOSR Grant No. 87.0136.

Journal of Multivariate Analysis, 1989

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