Concept# Kurtosis

Summary

In probability theory and statistics, kurtosis (from κυρτός, kyrtos or kurtos, meaning "curved, arching") is a measure of the "tailedness" of the probability distribution of a real-valued random variable. Like skewness, kurtosis describes a particular aspect of a probability distribution. There are different ways to quantify kurtosis for a theoretical distribution, and there are corresponding ways of estimating it using a sample from a population. Different measures of kurtosis may have different interpretations.
The standard measure of a distribution's kurtosis, originating with Karl Pearson, is a scaled version of the fourth moment of the distribution. This number is related to the tails of the distribution, not its peak; hence, the sometimes-seen characterization of kurtosis as "peakedness" is incorrect. For this measure, higher kurtosis corresponds to greater extremity of deviations (or outliers), and not the configuration of data near the mean.
It is common to compare the exce

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John David Scott Dines, Philip Neil Garner, Weifeng Li

This paper presents an adaptive beamforming application based on the capture of far-field speech data from a real single speaker in a real meeting room. After the position of a speaker is estimated by a speaker tracking system, we construct a subband-domain beamformer in generalized sidelobe canceller (GSC) configuration. In contrast to conventional practice, we then optimize the active weight vectors of the GSC so that kurtosis of output signals is maximized. Our beamforming algorithms can suppress noise and reverberation without the signal cancellation problems encountered in conventional beamforming algorithms. We demonstrate the effectiveness of our proposed techniques through a series of automatic speech recognition experiments on the Multi-Channel Wall Street Journal Audio Visual Corpus (MC-WSJ-AV). The beamforming algorithm proposed here achieved a 13.6% WER, whereas the simple delay-and-sum beamformer provided a WER of 17.8%.

2008Sibi Raj Bhaskaran Pillai, Emre Telatar

The kurtosis of a signal is a quantitative measure of how

2006`peaky' it is. In this paper we consider two scenarios of communication over fading channels with kurtosis constraints: in the first, we analyze a non-coherent Rayleigh fading channel where the input signal is required to satisfy a kurtosis constraint in addition to a power constraint. In the second, we find the `

worst' fading process that satisfies a kurtosis constraint and has a given second moment, while the fading coefficients are assumed to be known at the receiver. In both cases the transmitter is assumed ignorant of the instantaneous fading realization. The technique that enables our analysis is based on bounding mutual information between random variables which satisfy kurtosis and second moment constraints; the bound is tight in the low second moment regime and can be extended to multi-antenna communications.Generalized Additive Models (GAM) are a widely popular class of regression models to forecast electricity demand, due to their high accuracy, flexibility and interpretability. However, the residuals of the fitted GAM are typically heteroscedastic and leptokurtic caused by the nature of energy data. In this paper we propose a novel approach to estimate the time-varying conditional variance of the GAM residuals, which we call the GAM^2 algorithm. It allows utility companies and network operators to assess the uncertainty of future electricity demand and incorporate it into their planning processes. The basic idea of our algorithm is to apply another GAM to the squared residuals to explain the dependence of uncertainty on exogenous variables. Empirical evidence shows that the residuals rescaled by the estimated conditional variance are approximately normal. We combine our modeling approach with online learning algorithms that adjust for dynamic changes in the distributions of demand. We illustrate our method by a case study on data from RTE, the operator of the French transmission grid.

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