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Ulysses’ Compass
Published in Richard McElreath, Statistical Rethinking, 2020
Smuggling a bunch of mathematical details under the carpet, this strategy results in a useful approximation of the cross-validation score. The approximation goes by the awkward name of Pareto-smoothed importance sampling cross-validation.115 We’ll call it PSIS for short, and the PSIS function will compute it. PSIS uses importance sampling, which just means that it uses the importance weights approach described in the previous paragraph. The Pareto-smoothing is a technique for making the importance weights more reliable. Pareto is the name of a small town in northern Italy. But it is also the name of an Italian scientist, Vilfredo Pareto (1848–1923), who made many important contributions. One of these is known as the Pareto distribution. PSIS uses this distribution to derive more reliable cross-validation score, without actually doing any cross-validation. If you want a little more detail, see the Overthinking box below.
3 Distributions for X-Band Maritime Surveillance Radar Clutter
Published in Graham V. Weinberg, Radar Detection Theory of Sliding Window Processes, 2017
Suppose that X is a random variable with a Pareto distribution with shape parameter α, and scale and location parameters β = γ. Then for any k ∊ R+, the random variable Xkalso has a Pareto distribution, with shape parameterαk $ \frac{\alpha }{k} $ and scale and location parameter βk.
A heuristic master planning algorithm that includes fairness and flexibility
Published in International Journal of Management Science and Engineering Management, 2023
Ching-Chin Chern, Chun-Ying Leng, Bo Hsiao
When α is a variable, the Pareto distribution is sometimes referred to as the Pareto principle, or the 80–20 rule, which implies that 80% of a society’s wealth is held by 20% of its population. It arises if the Pareto index α is set to 1.16 (α = log45 ≈ 1.16) in the Pareto distribution. The 80–20 rule also applies to various cases, including natural phenomena and most human activities. Therefore, it can be inferred that roughly 80% of the weights to meet demand are oriented toward 20% of the periods, which must be those nearest to the current period. The Pareto distribution ratio for each period is calculated, with xmin as the current period and x as the due period for the demand, and the results are obtained in Table 4.
A robust methodology for predicting extreme structural responses of offshore wind turbines
Published in Ships and Offshore Structures, 2021
In order to establish the extreme value distribution by utilising the peak over threshold (POT) method, N load extremes, Lr (r = 1, 2, N) can be extracted from the aforementioned turbine response joint time series (the maximum load from each segment of the time series that lies between two successive upcrossings of a chosen threshold is retained as a load extreme). These extremes can then be used to establish an empirical distribution function, to which a parametric probability distribution model may be fit if desired. The POT method is based on what is called the Generalized Pareto distribution in the following manner: It has been shown that asymptotically, the excess values above a high level will follow a Generalized Pareto distribution if and only if the parent distribution belongs to the domain of attraction of one of the extreme value distributions. The cumulative distribution function of the Generalized Pareto distribution is (Coles (2001); Dargahi-Noubary (1989); Wang (2013, 2016)):for some a, c, L, , , . This distribution is call a Generalized Pareto distribution in which a is called the scale parameter and c is called the shape parameter.
Bayesian and Classical Estimation of the Inverse Pareto Distribution and Its Application to Strength-Stress Models
Published in American Journal of Mathematical and Management Sciences, 2018
The Pareto distribution, named after the Italian civil engineer and economist Vilfredo Pareto, is a power law probability distribution that is used in the description of social, scientific, geophysical, actuarial, and many other types of observable phenomena. Klass et al. (2006) considered statistical regularities at the top end of the wealth distribution using the Forbes 400 lists during 1988–2003, and showed that wealth is distributed according to a Pareto distribution. Vidondo et al. (1997) used the Pareto distribution for modeling size spectra data in aquatic ecology. For more applications of this distribution, see Arnold (1985).