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On the Predictability of Seizures
Published in Andrea Varsavsky, Iven Mareels, Mark Cook, Epileptic Seizures and the EEG, 2016
Andrea Varsavsky, Iven Mareels, Mark Cook
Traditional methods of estimating the fractal exponent a include the identification of power-laws constructed in a variety of ways. These methods are described in great detail in references such as [18], [36] and [130], and typically involve higher order statistics (e.g., variance and higher) because first order statistics (e.g., mean) do not reveal the differences between random and LRD processes [130]. Variance is the most often used second order statistic. Other typical methods include: Auto-correlation: Defined in Section 3.3.1 in Chapter 3. The auto-correlation not only gives an idea of how far in time there is non-trivial coupling, but can also be used directly to estimate ci. If a power-law is evident in the log-log plot of absolute auto-correlation magnitude versus delay then its gradient β = 1 — α [7].Power Spectral Density (PSD): Defined in Section 3.3.2 in Chapter 3. Since the auto-correlation can be related to the PSD, it follows that α can also be estimated by analyzing the frequency content of the signal. Long memory relates to low frequencies (and slow time scales). The power-law generated at low frequencies in the PSD scales as β = α.Fano Factor: This is an estimation of the variance in the number of events observed in a time period T. Each T is representative of a different time scale. The gradient of the Fano factor scales as β = α +1 for increasing T. In the random case, the Fano factor does not depend on the duration of the observation, resulting in an β = 1 (α = 0) as expected. Any deviation from β = 1 indicates a complexity or richness of information not present in a process with no memory [7].
Semiconductor Detectors
Published in Douglas S. McGregor, J. Kenneth Shultis, Radiation Detection, 2020
Douglas S. McGregor, J. Kenneth Shultis
The low Z number and low volume density of electrons causes the γ-ray absorption coefficient to be small for Si. Further, the energy at which the photoelectric absorption and the Compton scattering are equal is relatively low at only 60 keV. The Compton scattering coefficient is much lower than most semiconductors used for radiation detection (see Fig. 16.1). Hence, Si is a poor choice for high energy γ-ray spectroscopy. However, its K absorption edge appears at 1.838 keV, meaning that the absorption edge discontinuity does not adversely affect x-ray absorption at higher energies, nor does the appearance of x-ray escape peaks cause significant issues in spectra. By comparison, the K absorption edge for Ge is 11.103 keV. Because higher energy γ rays have less chance of interacting in Si, this property serves to reduce background effects. For these reasons, Si does have importance as an x-ray spectrometer for applications such as x-ray fluorescence, x-ray microanalysis, particle induced x-ray emission (PIXE), x-ray absorption spectroscopy (XAS), x-ray diffraction, and Mössbauer spectroscopy at energies generally below 50 keV. Energy resolution is the full width at half the maximum (FWHM) of a spectral full energy peak. Silicon detectors deliver excellent energy resolution, and the FWHM, reported in units of energy, is FWHMeh=22ln(2)wEγF, where w is the average energy to produce an electron-hole pair, Eγ is the photon energy, and F is the Fano factor (typically 0.12). The Fano factor is a correction factor that accounts for the typically higher energy resolution than predicted from pure Gaussian statistics. With the inclusion of noise sources, the energy resolution becomes FWHM=[(FWHMnoise)2+(FWHMeh)2]1/2. where the FWHMnoise contribution includes electronic noise and fluctuations in the leakage current.
A robust methodology for predicting extreme structural responses of offshore wind turbines
Published in Ships and Offshore Structures, 2021
In applying the peak over threshold method one should guarantee that the peaks over the selected threshold should occur randomly in time according to an approximate Poisson process, and the exceedances should have an approximate Generalized Pareto distribution and be approximately independent. In practice, one does not always find a Poisson distribution for the number of exceedances. Because extreme values sometimes have a tendency to cluster, some declustering algorithm can be applied to identify the largest value in each of the clusters, and then use a Poisson distribution for the number of clusters. To select the clusters and check the Poisson character one can use the Fano factor, which is the ratio between the variance and the mean of the number of peaks (Cox and Lewis (1966); Wang (2016, 2017)):The aforementioned Fano factor is a measure used to quantify whether a set of observed occurrences are clustered or dispersed compared to a standard statistical model. It measures the homogenity of data and the purpose of Fano factor is to determine the threshold where the number of exceedances in a fixed period (Tb) is consistent with a Poisson process (Please note that the mean in equation (19) is the mean number of exceedances in a fixed period (Tb). The variance in equation (19) is the variance of the numbers of exceedances in a fixed period (Tb).). The threshold should be chosen to be high enough so that the Fano factor is not significantly different from 1. The Poisson hypothesis is not rejected if the estimated Fano factor (F) is between:where M is the total number of fixed period/blocks (Tb) and is a significant level (default value is 5%). In the above formula (20), is defined as the upper 100 percentage point of a chi-squared random variable u (whose probability density function is f(u)) with M-1 degrees of freedom such that the probability that exceeds this value is . is defined as the lower 100 percentage point of a chi-squared random variable u (whose probability density function is f(u)) with M-1 degrees of freedom such that the probability that is less than this value is .