Application Of Extreme Value Statistics Research Articles

Application of extreme value statistics to internal pore distribution in ceramics and prediction of size dependency of strength scatter

To implement ceramics in high-reliability components, it is necessary to understand the strength scatter caused by the stochastic distribution of defects in ceramics. In this study, we propose a numerical simulation method to predict the strength scatter of ceramics and its size dependency using microstructural data obtained using X-ray computed tomography and scanning electron microscopy. The generalized Pareto (GP) model, a type of extreme value statistic, was applied to treat the three-dimensional distribution of the pore size. Further, we predicted the bending strengths of the four types of bending tests with different effective volumes, and the results corresponded with the experimental results. Additionally, the method was applied to specimen models with different discretization sizes to verify the advantages of the GP model. The results show the effectiveness of the proposed method for the specimen size dependence of the bending strength and specimen discretization size independence.

The most massive Population III stars

ABSTRACT Recent data from the JWST suggest that there are realistic prospects for detecting the earliest generation of stars at redshift ∼20. These metal-poor, gaseous Population III (Pop III) stars are likely in the mass range $10\!-\!10^3\, {\rm M}_{\odot }$ . We develop a framework for calculating the abundances of Pop III stars as well as the distribution of the most massive Pop III stars based on an application of extreme-value statistics. Our calculations use the star formation rate density from a recent simulation to calibrate the star formation efficiency from which the Pop III stellar abundances are derived. Our extreme-value modelling suggests that the most massive Pop III stars at redshifts 10 < z < 20 are likely to be $\gtrsim 10^3\!-\!10^4\, {\rm M}_\odot$ . Such extreme Pop III stars were sufficiently numerous to be the seeds of supermassive black holes at high redshifts and possibly source detectable gravitational waves. We conclude that the extreme-value formalism provides an effective way to constrain the stellar initial mass function.

Probabilistic and defect tolerant fatigue assessment of AM materials under size effect

Basic fatigue properties captured by laboratory testing of small specimens are usually applied to safety design of actual components. Nevertheless, their fatigue performances are significantly different due to size effect attributing a fault to possibility of large defects occur. Accordingly, extreme value statistics and weakest-link theory are applicable to defect tolerant fatigue assessment under size effect in this study. Firstly, the basic theory of extreme value statistics and weakest-link are explicated. Then, a probabilistic framework is developed to estimate the fatigue strength/life of Additive manufacturing (AM) material, wherein size effect and fatigue scatter from statistical viewpoint are illustrated and quantitative analysis. In particular, different types of size effect are simplified and unified as defect size variation, and the quantile of the critical defect derived by extreme value statistics is implemented to estimate the bounds of fatigue performance. Finally, three groups of additive manufactured specimens with different volume and processing parameters are utilized for method validation and comparison, which show great capability to estimate the fatigue scatter of AM material under size effect. The goal of this study is also to clarify the potential application of extreme value statistics and weakest-link theory from the probability and defect tolerant perspective.

Debris-Flow Hazard Assessments: A Practitioner's View

ABSTRACTSubstantial advances have been achieved in various aspects of debris-flow hazard assessments over the past decade. These advances include sophisticated ways to date previous events, two- and three-dimensional runout models including multi-phase flows and debris entrainment options, and applications of extreme value statistics to assemble frequency–magnitude analyses. Pertinent questions have remained the same: How often, how big, how fast, how deep, how intense, and how far? Similarly, although major life loss attributable to debris flows can often, but not always, be avoided in developed nations, debris flows remain one of the principal geophysical killers in mountainous terrains. Substantial differences in debris-flow hazard persist between nations. Some rely on a design magnitude associated with a specific return period; others use relationships between intensity and frequency; and some allow for, but do not mandate, in-depth quantitative risk assessments. Differences exist in the management of debris-flow risks, from highly sophisticated and nation-wide applied protocols to retroaction in which catastrophic debris flows occur before they are considered for mitigation. Two factors conspire to challenge future generations of debris-flow researchers, practitioners, and decision makers: Population growth and climate change, which are increasingly manifested by augmenting hydroclimatic extremes. While researchers will undoubtedly finesse future remote sensing, dating, and runout techniques and models, practitioners will need to focus on translating those advances into practical cost-efficient tools and integrating those tools into long-term debris-flow risk management.

Last-Passage Time for Linear Diffusions and Application to the Emptying Time of a Box

We study the statistics of last-passage time for linear diffusions. First we present an elementary derivation of the Laplace transform of the probability density of the last-passage time, thus recovering known results from the mathematical literature. We then illustrate them on several explicit examples. In a second step we study the spectral properties of the Schr\"{o}dinger operator associated to such diffusions in an even potential $U(x) = U(-x)$, unveiling the role played by the so-called Weyl coefficient. Indeed, in this case, our approach allows us to relate the last-passage times for dual diffusions (i.e., diffusions driven by opposite force fields) and to obtain new explicit formulae for the mean last-passage time. We further show that, for such even potentials, the small time $t$ expansion of the mean last-passage time on the interval $[0,t]$ involves the Korteveg-de Vries invariants, which are well known in the theory of Schr\"odinger operators. Finally, we apply these results to study the emptying time of a one-dimensional box, of size $L$, containing $N$ independent Brownian particles subjected to a constant drift. In the scaling limit where both $N \to \infty$ and $L \to \infty$, keeping the density $\rho = N/L$ fixed, we show that the limiting density of the emptying time is given by a Gumbel distribution. Our analysis provides a new example of the applications of extreme value statistics to out-of-equilibrium systems.

Extreme value estimation for discretely sampled continuous processes

In environmental applications of extreme value statistics, the underlying stochastic process is often modeled either as a max-stable process in continuous time/space or as a process in the domain of attraction of such a max-stable process. In practice, however, the processes are typically only observed at discrete points and one has to resort to interpolation to fill in the gaps. We discuss the influence of such an interpolation on estimators of marginal parameters as well as estimators of the exponent measure. In particular, natural conditions on the fineness of the observational scheme are developed which ensure that asymptotically the interpolated estimators behave in the same way as the estimators which use fully observed continuous processes.

Pitting corrosion of aluminium in the tropical climate of Mauritius

ABSTRACTAluminium is a commonly used metal in Mauritius. However, the tropical Mauritian atmosphere is quite corrosive with the highest measured corrosivity, according to ISO 9223, falling in the category C4. Hence, this study was performed to investigate the atmospheric corrosion behaviour of aluminium AA1050 in the island. The aluminium samples were exposed outdoors at three sites, according to ISO 8565. Upon removal after specific time intervals over a 1½ years period, pit depth, pit diameter and pit density of the samples were determined. The maximum pit depth varied with the time of exposure, according to the power law function. Pit diameter increased linearly with the time of exposure. Chart rating, according to ISO 11463, was also performed. It was found that the corrosivity of the Mauritian atmosphere falls in categories A4-B1-C1 and A5-B1-C1. Application of extreme value statistics showed that Gumbel type 1 distribution fits maximum pit depth data at the three sites.

Abell 2744: too much substructure for ΛCDM?

The massive substructures found in Abell 2744 by Jauzac et al. (2016) present a challenge to the cold dark matter paradigm due to their number and proximity to the cluster centre. We use one of the biggest N-body simulations, the Millennium XXL, to investigate the substructure in a large sample of massive dark matter haloes. A range of effects which influence the comparison with the observations is considered, extending the preliminary evaluation carried out by Jauzac et al. (2016). There are many tens of haloes in the simulation with a total mass comparable with or larger than that of Abell 2744. However, we find no haloes with a number and distribution of massive substructures (> 5 10^13 Msun) that is close to that inferred from the observations of Abell 2744. The application of extreme value statistics suggests that we would need a simulation of at least ten times the volume of the Millennium XXL to find a single dark matter halo with a similar internal structure to Abell 2744. Explaining the distribution of massive substructures in clusters is a new hurdle for hierarchical models to negotiate, which is not weakened by appeals to baryonic physics or uncertainty over the nature of the dark matter particle.

Extreme amplitudes of a periodically forced Duffing oscillator

A novel development in the theory of deterministic dynamical systems is the application of extreme value statistics. The idea is to evaluate a scalar observable along an evolution of the system. Under appropriate conditions the large values of this time series can be interpreted as realizations of a Generalized Extreme Value (GEV) distribution. The purpose of this paper is to illustrate the methodology with a time series of amplitudes of a periodically forced Duffing oscillator. The parameters of the GEV distribution are estimated by means of a block maximum method. It will be shown that a numerical estimation of these parameters can pose serious problems due to the fractal nature of the attractor of the system.

Assessing the Potential Impact of Heat Waves in Cities: Implications for Hazard Preparation and Planning

This paper summarizes the hazards of high temperature extremes in the built environment and considers the challenges of managing risk with uncertainty of prediction of the magnitude and frequency of events. The application of extreme value statistics on a typical time series of temperature data has provided some insight into the observed and predicted variability. Using a long temperature time series as an example it is shown that return periods and the increased risk associated with climate change can be interpolated from the analysis. Considering the chaotic nature of cities, their rapid growth and a warming climate, it is suggested that cities could be made more resilient to heat waves if governments were be better able to appropriately allocate resources to manage the hazard.

An application of extreme value statistics to the most massive galaxy clusters at low and high redshifts

In this work we present an application of general extreme value statistics (GEV) to very massive single clusters at high and low redshifts. After introducing the formalism, we apply this statistics to four very massive high redshift clusters. Those clusters comprise ACT-CL J0102-4915 with a mass of M_200m=(2.16+/-0.32)x10^{15} M_sun at a redshift of z=0.87, SPT-CL J2106-5844 with a mass of M_200m=(1.27+/-0.21)x10^{15} M_sun at z=1.132 and two clusters found by the XMM-Newton Distant Cluster Project survey: XMMU J2235.32557 with a mass of M_200c= (7.3+/-1.3)x10^{14} M_sun located at a redshift of z=1.4 and XMMU J0044.0-2033 having a mass in the range of M_200c= (3.5-5.0)x10^{14} M_sun at z=1.579. By relating those systems to their corresponding distribution functions of being the most massive system in a given survey area, we find that none of the systems alone is in extreme tension with LCDM. We confront these results with a GEV analysis of four very massive low redshift clusters: A2163, A370, RXJ1347-1145 and 1E0657-558, finding no tendency of the high-z systems to be more extreme than the low-z ones. In addition, we study the extreme quantiles of single clusters at high-z and present contour plots for fixed quantiles in the mass vs. survey area plane for four redshift intervals, finding that, in order to be significantly in conflict with LCDM, cluster masses would have to be substantially higher than the currently observed ones.

Statistical Inferences for Generalized Pareto Distribution Based on Interior Penalty Function Algorithm and Bootstrap Methods and Applications in Analyzing Stock Data

This paper studies the application of extreme value statistics (EVS) theory on analysis for stock data, based on interior penalty function algorithm and Bootstrap methods. The generalized Pareto distribution (GPD) models are considered in analyzing the closing price data of Shanghai stock market. The maximum likelihood estimates (MLEs) are obtained by using the interior penalty function algorithm. Correspondingly, the bias and standard errors of MLEs, and the hypothesis test on the shape parameter are concerned through Bootstrap methods. Some simulations are performed to demonstrate the efficacy of parameter estimation and the power of the test. The estimates of the tail index in this paper are compared with those obtained via classical methods. At last, the model is diagnosed by numerical and graphical methods and the Value-at-Risk (VaR) is estimated.

Reduction of diffusion broadening in flow by analysis of time-gated single-molecule data

We describe the application of Extreme Value Statistics to the analysis of discrete species that possess distinguishable properties (fluorescence wavelength, fluorescence intensity, light scattering, etc.) as they cross a well-defined observation/probe region. Time-gated selection and extreme value data analysis result in increased resolution in analytical determinations. When only the data corresponding to the smallest crossing times are selected for analysis, the width of the diffusion band decreases for the measured parameter. The molecules with the smallest crossing times diffuse preferentially along the flow direction. A Monte Carlo technique and the probability density function (pdf) for a freely diffusing species are used to generate data streams to provide a theoretical basis for the aforementioned phenomenon. These calculations are included to characterize the effect of the average flow rate and the diffusion constant. We have also included a procedure for extracting the normal diffusion constant (D) from the Extreme Value Distribution. In contrast to standard flow analysis, which requires long path lengths, our approach is particularly suited for measurements in picolitre and nanolitre volumes and provides another dimension to single-molecule measurements in cellular size volumes. We believe that this is a general phenomenon that depends upon the details of the pdf, which can be complex.

The size distribution of innovations revisited: An application of extreme value statistics to citation and value measures of patent significance

This paper focuses on the analysis of size distributions of innovations, which are known to be highly skewed. We use patent citations as one indicator of innovation significance, constructing two large datasets from the European and US Patent Offices at a high level of aggregation, and the Trajtenberg [1990, A penny for your quotes: patent citations and the value of innovations. Rand Journal of Economics 21(1), 172–187] dataset on CT scanners at a very low one. We also study self-assessed reports of patented innovation values using two very recent patent valuation datasets from the Netherlands and the UK, as well as a small dataset of patent licence revenues of Harvard University. Statistical methods are applied to analyse the properties of the empirical size distributions, where we put special emphasis on testing for the existence of ‘heavy tails’, i.e., whether or not the probability of very large innovations declines more slowly than exponentially. While overall the distributions appear to resemble a lognormal, we argue that the tails are indeed fat. We invoke some recent results from extreme value statistics and apply the Hill [1975. A simple general approach to inference about the tails of a distribution. The Annals of Statistics 3, 1163–1174] estimator with data-driven cut-offs to determine the tail index for the right tails of all datasets except the NL and UK patent valuations. On these latter datasets we use a maximum likelihood estimator for grouped data to estimate the tail index for varying definitions of the right tail. We find significantly and consistently lower tail estimates for the returns data than the citation data (around 0.6–1 vs. 3–5). The EPO and US patent citation tail indices are roughly constant over time, but the latter estimates are significantly lower than the former. The heaviness of the tails, particularly as measured by value indicators, we argue, has significant implications for technology policy and growth theory, since the second and possibly even the first moments of these distributions may not exist.

Highway bridge live loading assessment and load carrying capacity estimation using a health monitoring system

The Land Transport Authority of Singapore has a continuing program of highway bridge upgrading, to refurbish and strengthen bridges to allow for increasing vehicle traffic and increasing axle loads. One subject of this program has been a short span bridge taking a busy highway across a coastal inlet near a major port facility. Experiment-based structural assessments of the bridge were conducted before and after upgrading works including strengthening. Each assessment exercise comprised two separate components; a strain and acceleration monitoring exercise lasting approximately one month, and a full-scale dynamic test carried out in a single day. This paper reports the application of extreme value statistics to estimate bridge live loads using strain measurements.

Benefits of toughening a vinyl ester resin matrix on structural materials

A commonplace vinyl ester resin blended with a core-shell polymer additive has been used as a matrix for some typical structural commercial materials to determine the benefits of increased matrix toughness. In a simulation of polymer concrete, the tougher matrix was found to increase the toughness of the concrete by a small amount. Unexpectedly the flexural strength was increased by 30% which has been ascribed to the greater damage tolerance of the matrix. In composites with fibre glass cloth, the interlaminar toughness is also improved. An application of extreme value statistics showed that the change in resin toughness due to blending was fully transferred to the composite, and also allowed an estimation of the effect of the reinforcement on toughness.

Application of Extreme Value Statistics to Annual Maximum Magnitudes in Jordan Employing a Mixture Distribution

The concept of extreme value mixture distribution (EVmix) has been implemented in this study to estimate maximum earthquake magnitude occurrence. The EVmix model is applied to annual maximum earthquake magnitude occurrence in Jordan and conterminous regions spanning over the period 1918 to 1997. The maximum likelihood method, in conjunction with the two optimization methods, was employed for determining the statistical parameters of the Gumbel's asymptotic distribution, i.e., GI, and the extreme value distributions EVIII and EVmix. The Simplex method of Nelder and Mead (1965) was found to be more successful in obtaining the maximum likelihood estimators of the three given distributions than the Newton-Raphson method. The difficulties inherent to the Newton-Raphson method were overcome by the Simplex method. It is shown in this study that the EVmix model fits the observed annual maxima far better than GI and EVIII models. In addition, the maximum likelihood estimators obtained using the Simplex method were used to calculate the earthquake risk for a given return period and a design lifetime of structures.

Statistics for modeling heavy tailed distributions in geology : Part II. Applications

In this paper, we present three diverse types of applications of extreme value statistics in geology, namely: earthquakes magnitudes, diamond values, and impact crater size distribution on terrestrial planets. Each of these applications has a different perspective toward tail modeling, yet many of these phenomena exhibit heavy or long tails which can be modeled by power laws. It is shown that the estimation of important tail characteristics, such as the extreme value index, is directly linked to the interpretation of the underlying geological process. Only the most extreme data are useful for studying such phenomena, so thresholds must be selected above which the data become power laws. In the case of earthquake magnitudes, we investigate the use of extreme value statistics in predicting large events on the global scale and for shallow intracontinental earthquakes in Asia. Large differences are found between estimates obtained from extreme value statistics and the usually applied standard statistical techniques. In the case of diamond deposits, we investigate the impact of the most precious stones in the global valuation of primary deposits. It is shown that in the case of Pareto-type behavior, the expected value of few extreme stones in the entire deposit has considerable influence on the global valuation. In the case of impact crater distributions, we study the difference between craters distributions on Earth and Mars and distributions occurring on other planets or satellites within the solar system. A striking result is that all planets display the same distributional tail except for Earth and Mars. In a concluding account, we demonstrate the apparent loghyperbolic variation in all of the above-mentioned examples.

Tail Index Estimation, Pareto Quantile Plots Regression Diagnostics

Abstract Successful application of extreme value statistics for estimating the Pareto tail index relies heavily on the choice of the number of extreme values taken into account. It is shown that these tail index estimators can be considered estimates of the slope at the right upper tail of a Pareto quantile plot, obtained using a weighted least squares algorithm. From this viewpoint, based on classical ideas on regression diagnostics, algorithms can be constructed searching for that order statistic to the right of which one obtains an optimal linear fit of the quantile plot.

Application of Extreme Value Statistics to Corrosion.

Application of extreme value statistics to corrosion is reviewed. It is emphasized that the concept of corrosion probability is important for a quantitative evaluation of corrosion failure and its prediction, especially for localized corrosion. Extreme value statistics is quite useful for assessing the maximum pit depth and/or the minimum time for crack generation. The maximum pit depth depending on the surface area can be evaluated by using the Gumbel distribution with the concept of return period. A standardized procedure is proposed for estimating the maximum pit depth. Examples of corrosion failure analysis using extreme value statistics, which were reported mainly in Japan, are shown. Accumulated experiences suggest that an appropriate classification of data based on the corrosion mechanism is required before applying extreme value analysis.