The moments of the non-central t-distribution

This study considers the application of the Ignorance Score (IS, also known as the Logarithmic Score) for ensemble verification. In particular, we consider the case where an ensemble forecast is transformed to a normal forecast distribution, and this distribution is evaluated by the IS. It is shown that the IS systematically depends on the ensemble size, such that larger ensembles yield better expected scores. An ensemble-adjusted IS is proposed, which extrapolates the score of an m-member ensemble to the score that the ensemble would achieve if it had fewer or more than m members. Using the ensemble adjustment, a fair version of the IS is derived, which is optimized if ensembles are statistically consistent with the observations. The benefit of the ensemble adjustment is illustrated by comparing ISs of ensembles of different sizes in a seasonal climate forecasting context and a medium-range weather forecasting context. An ensemble-adjusted score can be used for a fair comparison between ensembles of different sizes, and to accurately estimate the expected score of a large operational ensemble by running a much smaller hindcast ensemble.

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“…The raw moments of a random variable T ∼ t ν , z are given by Hogben et al. ():

double-struckE ([], T^{k}) = {((), \frac{ν}{2})}^{\frac{k}{2}} \frac{normalΓ ((), \frac{ν - k}{2})}{normalΓ ((), \frac{ν}{2})} \exp ((), - \frac{z^{2}}{2}) \frac{\partial^{k}}{\partial z^{k}} \exp ((), \frac{z^{2}}{2}) .

…”

Section: Proofsmentioning

confidence: 99%

The ensemble‐adjusted Ignorance Score for forecasts issued as normal distributions

Siegert

Ferro

Stephenson

et al. 2019

Quart J Royal Meteoro Soc

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“…In addition, our ESCI software is intended to assist understanding of noncentral t and its use, and to provide a computation facility for simple cases. Owen (1968) surveyed the properties and applications of noncentral t. Hogben, Pinkham, and Wilk (1961) studied the mean and variance: The mean is a∆, where the factor a depends on df and approaches 1 as df increases. For df = 2, a = 1.77; for df = 15, a = 1.05; and for df = 60, a = 1.013.…”

Section: Noncentral T Distributionsmentioning

confidence: 99%

A Primer on the Understanding, Use, and Calculation of Confidence Intervals that are Based on Central and Noncentral Distributions

Cumming

Finch

2001

Educational and Psychological Measurement

412

252

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Reform of statistical practice in the social and behavioral sciences requires wider use of confidence intervals (CIs), effect size measures, and meta-analysis. The authors discuss four reasons for promoting use of CIs: They (a) are readily interpretable, (b) are linked to familiar statistical significance tests, (c) can encourage meta-analytic thinking, and (d) give information about precision. The authors discuss calculation of CIs for a basic standardized effect size measure, Cohen’s δ (also known as Cohen’s d), and contrast these with the familiar CIs for original score means. CIs for δ require use of noncentral t distributions, which the authors apply also to statistical power and simple meta-analysis of standardized effect sizes. They provide the ESCI graphical software, which runs under Microsoft Excel, to illustrate the discussion. Wider use of CIs for δ and other effect size measures should help promote highly desirable reform of statistical practice in the social sciences.

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“…The analytical PDF that best fitted the greatest number of observed PDFs (≈ 35%) was the non-central Student's T (NCT) PDF, which is a complicated four-parameter function [31] and thus is impractical.…”

Section: Shoreline Height Pdfsmentioning

confidence: 99%

LiDAR Observations of Multi-modal Swash Probability Distributions on a Dissipative Beach

Stringari¹,

Power²

2020

Preprint

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Understanding swash zone dynamics is of crucial importance for coastal management as the swash motion, consisting of the uprush of the wave on the beach face and the subsequent downrush, is responsible for driving changes the beach morphology trough sediment exchanges between the sub-aerial and sub-aqueous beach. Improved understanding of the probabilistic characteristics of these motions has the potential to allow coastal engineers to develop improved sediment transport models which, in turn, can be further developed into coastal management tools. In this paper, novel descriptors of swash motions are obtained by combining field data and statistical modelling. Our results indicate that the probability distribution function (PDF) of shoreline height (p(ζ)) and trough-to-peak swash heights (p(ρ)) measured at a high energy, sandy beach were both inherently multimodal. Based on the observed multimodality of these PDFs, Gaussian Mixtures are shown to be the best method to statistically model them. Further, our results show that both offshore and surf zone dynamics are responsible for driving swash zone dynamics, which indicates unsaturated swash. The novel methods and results developed in this paper, both data collection and analysis, could aid coastal managers to develop improved swash zone models in the future.

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The moments of the non-central t-distribution

Cited by 30 publications

References 5 publications

The ensemble‐adjusted Ignorance Score for forecasts issued as normal distributions

The ensemble‐adjusted Ignorance Score for forecasts issued as normal distributions

A Primer on the Understanding, Use, and Calculation of Confidence Intervals that are Based on Central and Noncentral Distributions

LiDAR Observations of Multi-modal Swash Probability Distributions on a Dissipative Beach

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