Heavy-tailed distributions

Impact of disruptive events is the financial loss due to those events. In general, the impact of such extreme events is modeled using heavy-tailed distributions such as Weibull, Gumbel, and Frechet distributions. Heavy tails of these distributions are appropriate for maximum or minimum... 

Frequently, the measurement error distributions arising in a practical data set deviate from the assumed Gaussian model, and they are often characterized by heavier tails (due to the presence of outliers). A typical heavy-tailed noise record is given in Fig. 7, while Fig. 8 shows the QQ-plots of this record, based on the hypothesized standard normal distribution. 

As discussed before, the outliers generated by the heavy-tails of the underlying distribution have a considerable influence on the OLS problem arising in a conventional data reconciliation procedure. To solve this problem, a limiting transformation, which operates on the data set, is defined to eliminate or reduce the influence of outliers on the performance of a conventional rectification scheme. 

There is often a particular concern for the effects of outliers or heavy-tailed distributions when using standard statistical techniques. To address this type of a situation, a parametric approach would be to use ML estimation assuming a heavy-tailed distribution (perhaps a Student t distribution with few degrees of freedom). However, simple ad hoc methods such as trimmed means may be useful. There is a large statistical literature on robust and outlier-resistant methods, (e.g., Hoaglin et al. 1983 Barnett and Lewis 1994). 

If data are normally distributed, the mean and standard deviation are the best description possible of the data. Modern analytical chemistry is often automated to the extent that data are not individually scrutinized, and parameters of the data are simply calculated with a hope that the assumption of normality is valid. Unfortunately, the odd bad apple, or outlier, can spoil the calculations. Data, even without errors, may be more or less normal but with more extreme values than would be expected. These are known has heavy-tailed distributions, and the values at the extremes are called outliers. In interlaboratory studies designed to assess proficiency, the data often have outliers, which cannot be rejected out of hand. It would be a misrepresentation for a proficiency testing body to announce that all its laboratories give results within 2 standard deviations (except the ones that were excluded from the calculations). 

The normal mixture prior does not need to represent a true mixture of two different types of effect, active and inactive. Such a distinction may be convenient for interpretation, but is artificial in the modeling and should be introduced later at the interpretation stage. Instead the use of a normal mixture can be just a convenient way of representing prior beliefs with a heavy-tailed distribution. A scaled t, double exponential, or Cauchy prior distribution could be used instead, but the mixture of two normal distributions is more flexible. 

The parameters in these distributions are tabulated in Table 3. The first one of them is actually the normal distribution and the remaining ones have different heavy tails and skewness. 

Although anomalous diffusion is expected in fractal pore systems, the presence of anomalous diffusion does not prove that the porous media is fractal. A heterogeneity along transport pathways may result in an anomalous transport regardless of the presence or the absence of self-similarity of the pore space (Beven et al., 1993). The physical interpretation of Levy motions does not presume the presence of fractal scaling in the porous media in which the motions occur (Klafter et al, 1990). The applicability of the FADE may be closely related to the distribution of pore-water velocities. In saturated media, the presence of heavy-tailed distributions of the hydraulic conductivity directly implies the validity of the FADE (Meer-schaert et al., 1999 Benson et al., 1999). The heavy-tailed hydraulic conductivity distributions were found in geologic media (Painter, 1996 Benson et al., 1999). Heavy-tailed velocity distributions can also be expected in unsaturated and structured soils, and therefore the FADE may be a useful model in these conditions. 

Heavy-tailed normal distribution where 80% of the deviations from true values were normally distributed with a standard deviation of 10 and the remainder were normally distributed with a standard deviation of 20. 

The use of mixture models is not limited to identification of important subpopulations. A common assumption in modeling pharmacokinetic parameters is that the distribution of a random effect is log-normal, or approximately normal on a log-scale. Sometimes, the distribution of a random effect is heavy tailed and when examined on a log-scale, is skewed and not exactly normal. A mixture distribution can be used to account for the large skewness in the distribution. However, the mixture used in this way does not in any way imply the distribution consists of two populations, but acts solely to account for heavy tails in the distribution of the parameter. 

Anomalous diffusion is often caused by memory effects and Levy-type statistics [185, 53], Specifically, superdiffusion is observed for random walks with heavytailed jump length distributions and subdiffusion for heavy-tailed waiting time distributions, see Sect. 3.4. The latter type of distribution can be caused by traps that have an infinite mean waiting time [185]. For reviews of anomalous diffusion see, e.g., [298,299, 229]. 

The symmetric Riesz fractional derivative (3.92) is the pseudo-differential operator with symbol - k ". Such a derivative describes a redistribution of particles in the whole space according to the heavy-tailed distribution of the jumps... 

The results of a single round of a PT scheme are frequently summarized as shown in Figure 4.11. If the results follow a normal distribution with mean and standard deviation <7, the z-scores will be a sample from the standard normal distribution, i.e. a normal distribution with mean zero and variance 1. Thus a laboratory with a z value of <2 is generally regarded as having performed satisfactorily, a z value between 2 and 3 is questionable (two successive values in this range from one laboratory would be regarded as unsatisfactory), and z values >3 are unacceptable. Of course even the laboratories with satisfactory scores will strive to improve their values in the subsequent rounds of the PT. In practice it is not uncommon to find heavy-tailed distributions, i.e. more results than expected with z > 2. 

Robust statistical methods can be applied to samples from symmetrical but heavy-tailed distributions, or when outliers may occur. They should not be applied in situations where the underlying distribution is bi-modal, multimodal, or very asymmetrical, e.g. log-normal distributions. 

In experimental science we normally have no advance knowledge of whether or not our data might come from a heavy-tailed distribution or might contain outliers. So ideally we would like to use an estimate of location (the mean and the median are location estimates) that behaves like the mean when the underlying distribution is truly normal, but has the robust properties of the median when outliers or heavy tails do occur. Analogous arguments apply to measures of spread. Over 30 years ago Huber and others showed that these desirable properties are available. 

The mean and standard deviations obtained by the MC simulations are summarized in Table 3. Considering the uncertainties, the mean of the robustness index predicted by the different methods is more or less the same. However, the normal approximations yield a higher standard deviation due to the heavy tail of the simulations using normal distributions. 

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