Extreme value cumulative

A number of statistical transformations have since then been proposed to quantify the distributions in pitting variables. Gumbel is given the credit for the original development of extreme value statistics (EVS) for the characterization of pit depth distribution [10]. The EVS procedure is to measure maximum pit depths on several replicate specimens that have pitted, then arrange the pit depth values in order of increasing rank. The Gumbel or extreme value cumulative probability function [f(x)] is shown in Eq (6.1), where A and a are the location and scale parameters, respectively. This probability function can be used to characterize data sets and estimate the extreme pit depth that possibly can affect a system. 

Display and/or print tables that contain bin number, number of events, individual, cumulative % population, and the normalized bin boundaries (z (bb-x f.3n)/s ), X "-components, well as various statistical indicators (extreme values, number of events outside bin-range, mean, SD). 

The extreme value of n trials from a probability density f(W) with the cumulant distribution F(W) satisfying the large behavior... 

Figure 5a and Figure 5h show typical lifecycle cost and cumulative utility for a Monte-Carlo simulation of our Markov models for both architectures. The time horizon Xpps for this simulation was set to 35 years. Each point in Fig. 5a and Fig. 5b represents the final values of the lifecycle cost and utility after 35 years for one single run (i.e., one single state trajectory of the Markov model). The set of points results from 5000 runs of our Markov models, and thus constitutes a cost-utility cloud for each architecture for the given time-horizon considered. These clouds facihtate the visualization of extreme values and dispersion of cost and utility along both dimensions simultaneously. It is the dynamics of these clouds, their comparative expected values and dispersions, and their dependence on the three critical parameters in our models, a, p, and Y (the failure probabihty, the cost intensity, and... 

Figure 20-3 Plot of the points defined by the cumulative probabilities and losses depicted in Figure 20-3 and examples of some of the types of information that can be obtained from extreme value projection.

When the phenomenon is known only poorly, the collected data may contain some degree of randomness or even reflect beliefs and bias on the part of the collector. To mitigate this problem, analysts may rely on probability, which relates statistical concepts to stochastic variables by mean of large data sets. An example of the use of one statistical and probabilistic method is the use of extreme value distributions applied to cumulative distribution functions to analyze pitting corrosion as performed by Macdonald and colleagues and shown in Figure 3.5. ... 

The normal probability relationship and its familiar beU-shaped curve represent a totahty of data, all of the scores on a test, average soil resistivities, or all pit depths form the basis for the curve. Application of the cumulative probability function for an exponential extreme value distribution of a standard variate to practical situations requires statistically valid collection of data. A practical and consistent sample size must be selected and enough samples must be taken to attain reliable results. 

The resulting cumulative distribution functions for the long-term extreme values are shown in Figure 18(a) for the all peak amplitude approach and in Figure 18(b) for the m-nautical... 

Fig. 4.33 Cumulative frequency of the extreme values of defects in cast iron...

The severity and the cumulative probability of each sub-period are plotted in a special probability diagram, Fignres 15.8 and 15.9. In extreme-value projection Type I, the Y-axis (severity) is linear and in Type II it is logarithmic. The vertical scale in the diagram is selected in a way that allows for extrapolations to two to three times the maximum severity of the material that was analysed. 

The median is defined as the diameter for which one-half the total munber of particles are smaller and one-half are larger. The median is also the diameter that divides the frequency distribution curve into equal areas, and the diameter corresponding to a cumulative fraction of 0.5. The mode is the most frequent size, or the diameter associated with the highest point on the frequency function curve. The mode can be determined by. setting the derivative of the frequency function equal to zero and solving for d. For symmetrical distributions such as the normal distribution, the mean, median, and mode will have the same value, which is the diameter of the axis of synunetry. For an asymmetrical or skewed distribution, these quantities will have different values. The median is conunonly used with skewed distributions, because extreme values in the tail have less effect on the median than on the mean. Most aerosol size distributions are skewed, with a long tail to the right, as shown in Fig. 4.4. For such a distribution,... 

We now discuss (ii), the evaluation of operator expectation values with the reference ho- We are interested in multireference problems, where may be extremely complicated (i.e., a very long Slater determinant expansion) or a compact but complex wavefunction, such as the DMRG wavefunction. By using the cumulant decomposition, we limit the terms that appear in the effective Hamiltonian to only low-order (e.g., one- and two-particle operators), and thus we only need the one- and two-particle density matrices of the reference wavefunction to evaluate the expectation value of the energy in the energy expression (7). To solve the amplitude equations, we further require the commutator of which, for a two-particle effective Hamiltonian and two-particle operator y, again involves the expectation value of three-particle operators. We therefore invoke the cumulant decomposition once more, and solve instead the modihed amplitude equation... 

Figure 30. Effect of extreme static disorder in EXAFS analysis. Left Gaussian distributions of equal area but with much different widths. Both the sum of the two functions and the single narrow distribution function are shown. Right Simulated EXAFS functions for the functions at left. It is seen that there is no detectable difference in the EXAFS except at low k-values. This difference would overlap with XANES and be extremely difficult to analyze. Hence physical distributions with a broad tail will have re duced coordination numbers via standard EXAFS analysis, as well as an artificially produced distance contraction . For cases not as severe as this, cumulate analysis can quantify the degree of static disorder and allow more correct results. After Kortright et al. (1983).

In days gone by this was achieved using probability paper, specially ruled graph paper which took care of the normal pdf. Nowadays, spreadsheets have functions to perform this calculation in Excel it is NORMSINV(x), where x is the normalized cumulative frequency. If the data are normally distributed this graph should be linear. Obvious outliers are seen as points at the extremes of the x-axis, that is, at values much greater than would be expected. Example 3.1 shows how to determine whether data are normally distributed using a Rankit plot in Excel. 

In Fig. 14 the correlation function (q (t)q (O) is shown for the nonlinear potential in Eq. (3.85) at /3 = 10. This correlation function presents another nontrivial test of the various approximate methods because, classically, it can have no negative values while, quantum mechanically, it can be negative due to interference effects. Clearly, only the cumulant method can describe the latter effects. The classical result is extremely poor for this low-temperature correlation function. The CMD with semiclassical operators method also cannot give a correlation function with negative values in this case. This feature of the latter method arises because the correlation of the two operators at different times is ignored when the Gaussian averages are performed. Consequently, the semiclassical operator approximation underestimates the quantum real-time interference of the two operators and thus fails to... 

To test the methods outlined in Section III.B.3 for calculating general correlation functions in the phase-space centroid perspective, the correlation function >l(f)B(0)), where A= pq and B = qp, was studied [5], The results of this calculation are shown in Fig. 15 for the nonlinear potential in Eq. (3.85) at /8 = 10. The classical MD result is, as expected, extremely inaccurate for this low temperature. The CMD with semiclassical operators method does not reproduce the amplitude and negative values of this correlation function as well. On the other hand, the cumulant method can describe the quantum interference effects for this correlation function, and it appears to do so quite well. 

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