Modeling extreme value statistics

Korb el al. proposed a model for dynamics of water molecules at protein interfaces, characterized by the occurrence of variable-strength water binding sites. They used extreme-value statistics of rare events, which led to a Pareto distribution of the reorientational correlation times and a power law in the Larmor frequency for spin-lattice relaxation in D2O at low magnetic fields. The method was applied to the analysis of multiple-field relaxation measurements on D2O in cross-linked protein systems (see section 3.4). The reorientational dynamics of interfacial water molecules next to surfaces of varying hydrophobicity was investigated by Stirnemann and co-workers. Making use of MD simulations and analytical models, they were able to explain non-monotonous variation of water reorientational dynamics with surface hydrophobicity. In a similar study, Laage and Thompson modelled reorientation dynamics of water confined in hydrophilic and hydrophobic nanopores. 

Section 1.6.2 discussed some theoretical distributions which are defined by more or less complicated mathematical formulae they aim at modeling real empirical data distributions or are used in statistical tests. There are some reasons to believe that phenomena observed in nature indeed follow such distributions. The normal distribution is the most widely used distribution in statistics, and it is fully determined by the mean value p. and the standard deviation a. For practical data these two parameters have to be estimated using the data at hand. This section discusses some possibilities to estimate the mean or central value, and the next section mentions different estimators for the standard deviation or spread the described criteria are fisted in Table 1.2. The choice of the estimator depends mainly on the data quality. Do the data really follow the underlying hypothetical distribution Or are there outliers or extreme values that could influence classical estimators and call for robust counterparts ... 

Harris has developed several models based on statistical time series analysis. At the one extreme, we have a stationary or homeostatic model suitable for analytes showing relatively fast, random fluctuations around a constant mean (set point). The set point is estimated from past values that are given equal weights. Another... 

Fig. 3. Statistical models a Correlation between the product of sets sizes and the mean of the raw score. The fitted function typically corresponds to an equation of the formula = mxn + p with n = 1. b Correlation between the product of sets sizes and the standard deviation of the raw score. The fitted function typically corresponds to an equation of the formula ya=qxr+ s, with 0.6

S. Coles, An introduction to statistical modelling of extreme values. Springer Series in Statistics (Springer, Berlin, 2001). 

Coles, S., 2001. An Introduction to Statistical Modeling of Extreme Values. Springer Verlag, London. 

The bias-correction is necessary to correct both the absolute magnitude and the seasonal cycle to that of the observations. This approach assumes that the same model biases persist in the future climate and thus GCMs more accurately simulate relative change than absolute values. It provides a correction of monthly mean climate only and does not correct biases in higher order statistics including the simulation of extreme events and persistence. 

The simple collision theory for bimolecular gas phase reactions is usually introduced to students in the early stages of their courses in chemical kinetics. They learn that the discrepancy between the rate constants calculated by use of this model and the experimentally determined values may be interpreted in terms of a steric factor, which is defined to be the ratio of the experimental to the calculated rate constants Despite its inherent limitations, the collision theory introduces the idea that molecular orientation (molecular shape) may play a role in chemical reactivity. We now have experimental evidence that molecular orientation plays a crucial role in many collision processes ranging from photoionization to thermal energy chemical reactions. Usually, processes involve a statistical distribution of orientations, and information about orientation requirements must be inferred from indirect experiments. Over the last 25 years, two methods have been developed for orienting molecules prior to collision (1) orientation by state selection in inhomogeneous electric fields, which will be discussed in this chapter, and (2) bmte force orientation of polar molecules in extremely strong electric fields. Several chemical reactions have been studied with one of the reagents oriented prior to collision. ... 

An estimator (or more specifically an optimal state estimator ) in this usage is an algorithm for obtaining approximate values of process variables which cannot be directly measured. It does this by using knowledge of the system and measurement dynamics, assumed statistics of measurement noise, and initial condition information to deduce a minimum error state estimate. The basic algorithm is usually some version of the Kalman filter.14 In extremely simple terms, a stochastic process model is compared to known process measurements, the difference is minimized in a least-squares sense, and then the model values are used for unmeasurable quantities. Estimators have been tested on a variety of processes, including mycelial fermentation and fed-batch penicillin production,13 and baker s yeast fermentation.15 The... 

P-values from the PPC method that are -0.5 indicate that the model adequately describes the data with approximately 50% of the predictions being more extreme or equal to the observed test statistic. P-values close to 0 or 1 indicate some bias in the model predictions and in some circumstances may be used as evidence to reject the candidate model. There is no fixed value of the P-value that indicates poor model performance, although values more extreme than 0.1 or 0.9 may confer reasonable evidence against a model. 

---