Statistical characterization of active sonar reverberation using extreme value theory

Investigation on drought characteristics such as severity, duration, and frequency is crucial for water resources planning and management in a river basin. While the methodology for multivariate drought frequency analysis is well established by applying the copulas, the estimation on the associated parameters by various parameter estimation methods and the effects on the obtained results have not yet been investigated. This research aims at conducting a comparative analysis between the maximum likelihood parametric and non-parametric method of the Kendall \(\tau \) estimation method for copulas parameter estimation. The methods were employed to study joint severity–duration probability and recurrence intervals in Karkheh River basin (southwest Iran) which is facing severe water-deficit problems. Daily streamflow data at three hydrological gauging stations (Tang Sazbon, Huleilan and Polchehr) near the Karkheh dam were used to draw flow duration curves (FDC) of these three stations. The \(Q_{75}\) index extracted from the FDC were set as threshold level to abstract drought characteristics such as drought duration and severity on the basis of the run theory. Drought duration and severity were separately modeled using the univariate probabilistic distributions and gamma–GEV, LN2–exponential, and LN2–gamma were selected as the best paired drought severity–duration inputs for copulas according to the Akaike Information Criteria (AIC), Kolmogorov–Smirnov and chi-square tests. Archimedean Clayton, Frank, and extreme value Gumbel copulas were employed to construct joint cumulative distribution functions (JCDF) of droughts for each station. Frank copula at Tang Sazbon and Gumbel at Huleilan and Polchehr stations were identified as the best copulas based on the performance evaluation criteria including AIC, BIC, log-likelihood and root mean square error (RMSE) values. Based on the RMSE values, nonparametric Kendall-\(\tau \) is preferred to the parametric maximum likelihood estimation method. The results showed greater drought return periods by the parametric ML method in comparison to the nonparametric Kendall \(\tau \) estimation method. The results also showed that stations located in tributaries (Huleilan and Polchehr) have close return periods, while the station along the main river (Tang Sazbon) has the smaller return periods for the drought events with identical drought duration and severity.

ABSTRACTThe generalized Pareto distribution (GPD) is widely used for extreme values over a threshold. Most existing methods for parameter estimation either perform unsatisfactorily when the shape parameter k is larger than 0.5, or they suffer from heavy computation as the sample size increases. In view of the fact that k > 0.5 is occasionally seen in numerous applications, including two illustrative examples used in this study, we remedy the deficiencies of existing methods by proposing two new estimators for the GPD parameters. The new estimators are inspired by the minimum distance estimation and the M-estimation in the linear regression. Through comprehensive simulation, the estimators are shown to perform well for all values of k under small and moderate sample sizes. They are comparable to the existing methods for k < 0.5 while perform much better for k > 0.5.

As part of the flood risk assessment in The Netherlands, extreme hydraulic boundary conditions (HBC) at coastal water defenses are calculated to assess the safety and sufficiency of the defenses.For this purpose, extreme offshore values of wave height and wave period with a frequency of occurrence of 1/10,000 years need to be estimated.The estimation is carried out using measurements at nine offshore locations in the North Sea, with a record length of 30 years.To estimate extreme wave heights and wave periods, the Generalized Pareto Distribution (GPD), which follows from extreme value theory, was fitted to peaks-over-threshold (POT) at each of the nine measurement stations.The fitted GPD parameters and consequently the extrapolated extreme values are particularly sensitive to the choice of threshold.A reproducible method was employed for threshold selection based on the anticipated behavior of the GPD shape parameter as the threshold approaches the domain of attraction.A modified regional frequency analysis (MRFA) was subsequently carried out to spatially smooth the GPD shape parameters.Final GPD fits were made using the shape-parameter output of the MRFA to produce final extrapolated extreme value estimates.The extreme values were compared with those resulting from an operationally-used distribution function in The Netherlands known as the conditional Weibull distribution.

A pioneering body of work in the area of mobile opportunistic networks has shown that characterizing aggregated inter-contact time between mobile nodes is crucial. The most common approaches to study aggregated inter-contact time (ICT) in literatures are focused on a certain type of mobile wireless networks. Up to now, it still lacks a general approach to model the aggregated ICT effectively in different type of networks. This paper proposes a new approach to model aggregated ICT based on extreme value theory (EVT), which uses the generalized Pareto distribution as the unique asymptotic model of the tail distribution. Mean excess function is proposed to choose the threshold value and also other parameter estimation methods to the statistical model are discussed and applied to real mobility traces collected from different opportunistic networks, such as social pocket switched networks and vehicular ad hoc network. A decision rule based on confidence level of parameter is proposed to determine the statistical characteristics of aggregated ICT. By performing extensive experiments using EVT-based model, the aggregated ICT distributions for different type of networks are obtained. To show how well the EVT-based model actually fits the data in the tails, the complementary cumulative distribution function in a logarithmic scale is used to validate the model. Also according to the Kolmogorov---Smirnov test, it illustrates that EVT-based distributions with different parameter values fit the tails well at the desired significance level 0.05 for the taxi mobility in VANET and the human node in the other type of opportunistic networks. Finally, some prospective applications based on the statistical model are discussed in designing new network protocols.

The passive and active sonar equations are reviewed with regard to the definitions of individual terms and their units. It is shown on the one hand that ambiguities exist in conventional definitions that can lead to misinterpretation, and on the other that under special conditions correction terms are needed that are not included in the standard equations. Revised definitions are proposed that remove the need for some of the correction terms and reduce the likelihood of misunderstanding. Particular emphasis is placed on the terms 'propagation loss' and 'source level' for both active and passive sonar. For example, the effect on changes in impedance between the source and receiver of the sound is considered in the case of passive sonar, and between the source, receiver and scatterer of sound in the case of active sonar.

The most applied principles for parameter estimation are maximum likelihood and least square error. This paper presents a new principle with regard to the parameter estimation of the three-parameter Weibull distribution. By transforming the cumulative distribution function, constructed is a mapping from the value of the random variable and its corresponding cumulative distribution probability to the scale parameter. The scale parameter estimated by such a mapping is the random variable value and the corresponding cumulative distribution probability dependent when the shape parameter and/ or location parameter applied in the mapping is subject to error. Given a set of random variable values or a set of sample values, a larger error in the shape/location parameter brings about larger differences between the scale parameter values obtained with the individual random variable values or sample values, respectively. Based on such a causal relationship between the discrepancy and the shape/ location parameter value applied in the mapping relation, a new parameter estimation method is proposed. For the Weibull distribution parameter estimation according to a set of sample values, the right shape parameter and location parameter are those minimizing the discrepancy between the scale parameter values obtained with the individual sample values, respectively. Case studies demonstrate that the proposed method outperforms the maximum likelihood method and the Weibull plot-based least squares method.

In this study, two series of data for extreme rainfall events are generated based on Annual Maximum and Partial Duration Methods, derived from 102 rain-gauge stations in Peninsular from 1982-2012. To determine the optimal threshold for each station, several requirements must be satisfied and Adapted Hill estimator is employed for this purpose. A semi-parametric bootstrap is then used to estimate the mean square error (MSE) of the estimator at each threshold and the optimal threshold is selected based on the smallest MSE. The mean annual frequency is also checked to ensure that it lies in the range of one to five and the resulting data is also de-clustered to ensure independence. The two data series are then fitted to Generalized Extreme Value and Generalized Pareto distributions for annual maximum and partial duration series, respectively. The parameter estimation methods used are the Maximum Likelihood and the L-moment methods. Two goodness of fit tests are then used to evaluate the best-fitted distribution. The results showed that the Partial Duration series with Generalized Pareto distribution and Maximum Likelihood parameter estimation provides the best representation for extreme rainfall events in Peninsular Malaysia for majority of the stations studied. Based on these findings, several return values are also derived and spatial mapping are constructed to identify the distribution characteristic of extreme rainfall in Peninsular Malaysia.

Two classical methods for estimation of Weibull parameters, namely, the moment estimation and the maximum likelihood estimation are reviewed. It is demonstrated that, assuming the location parameter to be zero, the errors made in estimating both the scale and the shape parameters by these methods increase linearly with the true value of the location parameter. An alternative method for Weibull parameter estimation is proposed. This method provides estimates for all three parameters when the shape parameter is close to one, and estimates the scale and the shape parameters more accurately than the other two methods for higher values of the shape parameter. Furthermore, unlike the other methods, this method is capable of separating the constituent components in a multi-component sample and estimating the parameters of each component. This property makes the method suitable for statistical analysis of composite material strength and fatigue life data.

A model is proposed for simulating the group dynamics and acoustic echoes of aggregate, discrete biologics in midfrequency (2–5 kHz) active undersea sonar. Using a Lagrangian description, group behavior is modeled by a set of Newtonian forces and stochastic perturbations on a collection of point masses. Interactions comprise a grouping force, which gives a characteristic separation distance, and an arrayal force, which tends to align the velocities of the group’s constituents. The group dynamics is characterized in terms of their statistical and asymptotic properties and studied as a function of the model parameters. Several models for the acoustic response are considered and studied for both individual and aggregate response. Finally, the resulting synthetic echoes are compared against those of actual biologics measured on a midfrequency active sonar system. The synthetic echoes are found to reproduce well both the qualitative characteristics of measured biologics and the resulting tracking and classification features found in a typical active sonar signal and information processing chain.

The study of extremes has attracted the attention of scientists, engineers, actuaries, policy makers, and statisticians for many years. Extreme value theory (EVT) deals with the extreme deviations from the median of probability distributions and is used to study rare but extreme events. EVT’s main results characterize the distribution of the sample maximum or the distribution of values above a given threshold. In this study, EVT has been used to construct a model on the extreme and rare earthquakes that have happened in the United States from 1700 to 2011.The primary goal of fitting such a model is to estimate the amount of losses due to those extreme events and the probabilities of such events. Several diagnostic methods (for example, QQ plot and Mean Excess Plot) have been used to justify that the data set follows generalized Pareto distribution (GPD). Three estimation techniques have been employed to estimate parameters. The consistency and reliability of estimated parameters have been observed for different threshold values. The purpose of this study is manifold: first, we investigate whether the data set follows GPD, by using graphical interpretation and hypothesis testing. Second, we estimate GPD parameters using three different estimation techniques. Third, we compare consistency and reliability of estimated parameters for different threshold values. Last, we investigate the bias of estimated parameters using a simulation study. The result is particularly useful because it can be used in many applications (for example, disaster management, engineering design, insurance industry, hydrology, ocean engineering, and traffic management) with a minimal set of assumptions about the true underlying distribution of a data set. KEYWORDS: Extreme Value Theory; QQ Plot; Mean Excess Plot; Mean Residual Plot; Peak Over Threshold; Generalized Pareto Distribution; Maximum Likelihood Method; Method of Moments; Probability-Weighted Moments; Shapiro-Wilk test; Anderson- Darling Test

ABSTRACTThe generalized Pareto distribution (GPD) is important in the analysis of extreme values, especially in modeling exceedances over thresholds. Most of the existing methods for estimating the scale and shape parameters of the GPD suffer from theoretical and/or computational problems. A new hybrid estimation method is proposed in this article, which minimizes a goodness-of-fit measure and incorporates some useful likelihood information. Compared with the maximum likelihood method and other leading methods, our new hybrid estimation method retains high efficiency, reduces the estimation bias, and is computation friendly.

Contrary to common belief, Fisher‐Tippett's extreme value (EV) theory does not typically apply to annual rainfall maxima. Similarly, Pickands' extreme excess (EE) theory does not typically apply to rainfall excesses above thresholds on the order of the annual maximum. This is true not just for long‐averaging durationsdbut also for shortdand in the high‐resolution limit asd→ 0. We reach these conclusions by applying large‐deviation theory to multiplicative rainfall models with scale‐invariant structure. We derive several asymptotic results. One is that, asd→ 0, the annual maximum rainfall intensity ind,Iyr,d, has generalized extreme value (GEV) distribution with a shape parameterkthat is significantly higher than that predicted by EV theory and is always in the EV2 range. The value ofkdepends not on the upper tail of the marginal distribution but on regions closer to the body. Under the same conditions, the excesses above levels close to the annual maximum have generalized Pareto distribution with parameterkthat is always higher than that predicted by Pickands' EE theory. For finited, the distribution ofIyr,dis not GEV, but in accordance with empirical evidence is well approximated by a GEV distribution with shape parameterkthat increases asddecreases. We propose a way to estimatekunder preasymptotic conditions from the scaling properties of rainfall and suggest a near‐universalk(d) relationship. The new estimator promises to be more accurate and robust than conventional estimators. These developments represent a significant conceptual change in the way rainfall extremes are viewed and evaluated.

High volatility, thick tails and extreme value theory in value-at-risk estimation

We have analyzed 39 major ammonium nitrate (AN) accidents (1916-2022) that reveal systemic catastrophic risks have not diminished over the past century. This challenges the prevailing narrative of improved process safety of AN on account of iterative guidelines and technology advances. Applying extreme value theory (EVT), we model temporal occurrence as a homogeneous Poisson process (maximum likelihood estimate λ ≈ 0.23 events/year) and casualty severity as heavy-tailed (generalized Pareto distribution, shape parameter ξ ≈ 1.13). These findings indicate that there are persistent systemic patterns. This is shown by the implied "shadow mean" that exceeds the sample mean by 1.5-19-fold, and societal risk profiles occupy the "intolerable" region per frequency-number curve criteria. To address this problem, we establish an EVT-based systemic monitoring framework with quantitative benchmarks, identifying 50-year (∼315 fatalities) and 100-year (∼715 fatalities) return levels as a data-driven baseline for assessing safety performance. This ensemble framework provides a necessary tool for stakeholders to monitor realized systemic risks over mechanism-driven simulations or narrative expectations.

The present investigation introduces an adaptive beamforming scheme that separates the computationally intensive multidimensional beamforming into two simple modules, which are line and circular array beamformers. This approach minimizes the number of degrees of freedom associated with the adaptation process and provides near instantaneous convergence, which is essential for integrated active-passive sonar and medical ultrasound imaging applications. The proposed adaptive processing concept has been implemented in an integrated active-passive real-time sonar deploying a cylindrical array. Preliminary efforts include also the application of a line array adaptive beamformer in medical ultrasound imaging systems. Real data results from the adaptive and conventional beamforming outputs of the cylindrical array sonar system and the line array of the ultrasound system demonstrate the superior performance of the adaptive beamformer in suppressing the reverberation and cluttering effects in active sonar and medical ultrasound applications. Moreover, for passive sonar applications, the adaptive processing provides substantially improved angular resolution performance as compared with that of the conventional beamformer. Both these two performance improvements for a line and a cylindrical array sonar are of particular importance for mine hunting operations.