Hermite Polynomials and Measures of Non-gaussianity

Puuronen, Jouni; Hyvärinen, Aapo

doi:10.1007/978-3-642-21738-8_27

Cited by 5 publications

(3 citation statements)

References 7 publications

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“…In this research we focus on one expansion, termed the Gauss-Hermite expansion following the terminology of [1]. This expansion has good convergence properties in practice and is robust to outliers [21]. We define this expansion below:…”

Section: Gauss-hermite Expansionmentioning

confidence: 99%

“…A few further comments are in order. In practice, the Gauss-Hermite expansion provides a good fit to a wide variety of probability density functions [21]. For completeness however, the following shortcomings should be noted as they may be important depending on the application of the Gauss-Hermite estimate of the density.…”

Section: Truncated Gauss-hermite Expansions and Nonparametric Density...mentioning

confidence: 99%

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Sequential quantiles via Hermite series density estimation

Stephanou¹,

Varughese²,

MacDonald³

2017

Electron. J. Statist.

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Sequential quantile estimation refers to incorporating observations into quantile estimates in an incremental fashion thus furnishing an online estimate of one or more quantiles at any given point in time. Sequential quantile estimation is also known as online quantile estimation. This area is relevant to the analysis of data streams and to the one-pass analysis of massive data sets. Applications include network traffic and latency analysis, real time fraud detection and high frequency trading. We introduce new techniques for online quantile estimation based on Hermite series estimators in the settings of static quantile estimation and dynamic quantile estimation. In the static quantile estimation setting we apply the existing Gauss-Hermite expansion in a novel manner. In particular, we exploit the fact that Gauss-Hermite coefficients can be updated in a sequential manner. To treat dynamic quantile estimation we introduce a novel expansion with an exponentially weighted estimator for the Gauss-Hermite coefficients which we term the Exponentially Weighted Gauss-Hermite (EWGH) expansion. These algorithms go beyond existing sequential quantile estimation algorithms in that they allow arbitrary quantiles (as opposed to pre-specified quantiles) to be estimated at any point in time. In doing so we provide a solution to online distribution function and online quantile function estimation on data streams. In particular we derive an analytical expression for the CDF and prove consistency results for the CDF under certain conditions. In addition we analyse the associated quantile estimator. Simulation studies and tests on real data reveal the Gauss-Hermite based algorithms to be competitive with a leading existing algorithm.

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Section: Gauss-hermite Expansionmentioning

confidence: 99%

Section: Truncated Gauss-hermite Expansions and Nonparametric Density...mentioning

confidence: 99%

Sequential quantiles via Hermite series density estimation

Stephanou¹,

Varughese²,

MacDonald³

2017

Electron. J. Statist.

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“…Here, we conduct a thorough investigation to assess whether a Hermite functions-based measure of non-Gaussianity is useful as a normality test and as an ICA contrast function. We note that similar approaches using Hermite functions have been previously proposed for moment-based normality tests (Almuzara et al, 2019;Amengual et al, 2022) and distribution shape-sensitive ICA (Puuronen and Hyvärinen, 2011), but a comparison to other normality tests or applicability as an ICA contrast function had not been undertaken. We test the proposed method by using datasets constructed to have simple distributions as well as realistic EEG simulations.…”

Section: Introductionmentioning

confidence: 99%

Application of a Hermite-based measure of non-Gaussianity to normality tests and independent component analysis

Jain

Knight

Victor

2023

Front. Neuroinform.

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In the analysis of neural data, measures of non-Gaussianity are generally applied in two ways: as tests of normality for validating model assumptions and as Independent Component Analysis (ICA) contrast functions for separating non-Gaussian signals. Consequently, there is a wide range of methods for both applications, but they all have trade-offs. We propose a new strategy that, in contrast to previous methods, directly approximates the shape of a distribution via Hermite functions. Applicability as a normality test was evaluated via its sensitivity to non-Gaussianity for three families of distributions that deviate from a Gaussian distribution in different ways (modes, tails, and asymmetry). Applicability as an ICA contrast function was evaluated through its ability to extract non-Gaussian signals in simple multi-dimensional distributions, and to remove artifacts from simulated electroencephalographic datasets. The measure has advantages as a normality test and, for ICA, for heavy-tailed and asymmetric distributions with small sample sizes. For other distributions and large datasets, it performs comparably to existing methods. Compared to standard normality tests, the new method performs better for certain types of distributions. Compared to contrast functions of a standard ICA package, the new method has advantages but its utility for ICA is more limited. This highlights that even though both applications—normality tests and ICA—require a measure of deviation from normality, strategies that are advantageous in one application may not be advantageous in the other. Here, the new method has broad merits as a normality test but only limited advantages for ICA.

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Detection algorithm of non-Gaussian character of sample<sup>1</sup> distribution and its implementation in radar data processing

Прокопенко

Babanska

2015

2015 Signal Processing Symposium (SPSympo)

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Hermite Polynomials and Measures of Non-gaussianity

Cited by 5 publications

References 7 publications

Sequential quantiles via Hermite series density estimation

Sequential quantiles via Hermite series density estimation

Application of a Hermite-based measure of non-Gaussianity to normality tests and independent component analysis

Detection algorithm of non-Gaussian character of sample<sup>1</sup> distribution and its implementation in radar data processing

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