Fuzzy order statistics and their application to fuzzy clustering

The median and the median absolute deviation (MAD) are robust statistics based on order statistics. Order statistics are extended to fuzzy sets to define a fuzzy median and a fuzzy MAD. The fuzzy c-means (FCM) clustering algorithm is defined for any p-norm (pFCM), including the l₁-norm (1FCM), The 1FCM clustering algorithm is implemented via the alternating optimization (AO) method and the clustering centers are shown to be the fuzzy median. The resulting AO-1FCM clustering algorithm is called the fuzzy c-medians (FCMED) clustering algorithm. An example illustrates the robustness of the FCMED     

Quantile curves and dependence structure for bivariate distributions

Within the context of a general bivariate distribution an intuitive method is presented in order to study the dependence structure of the two distributions. A set of points—level curve—which accumulate the same probability for a fixed quadrant is considered. This procedure provides four level curves which can be considered as the boundary of a generalization of the real interquantile interval. It is shown that the accumulated probability among the level curves depends on the dependence structure of the distribution function where the dependence structure is given by the notion of copula. Furthermore, the case when the marginal distributions are independent is investigated. This result is used to find out positive or negative dependence properties for the variables. Finally, a nonparametric test for independence with a local dependence meaning is performed and applied to different data sets.     

Applying order statistics to detection problems

This paper introduces the concept of a truncated order statistic. We derive analytical expressions for the mean value and the variance of a truncated order statistic with a threshold determined by a given number of the variational series term. The algorithm of noisy signal detection is developed for detection systems, which employ the method of statistical hypotheses testing and the energy detection criterion. The algorithm utilizes statistical characteristics of order statistics.     

Selection and order statistics from correlated normal random variables

This paper investigates order statistics from correlated normal random variables and its application, namely, ranking and selection. We propose a new approach to estimate the percentage points (quantiles) of the correlated normal distributions. The new approach is flexible and can be used to estimate the critical constants for the problem at hand, even when the correlations are unknown or unequal. An experimental performance evaluation demonstrates the validity and efficiency of the procedures.     

Expressions for Rényi and Shannon entropies for bivariate distributions

Exact forms of Rényi and Shannon entropies are determined for 27 continuous bivariate distributions, including the Kotz type distribution, truncated normal distribution, distributions with normal and centered normal conditionals, natural exponential distribution, Freund's exponential distribution, Marshall and Olkin's exponential distribution, exponential mixture distribution, Arnold and Strauss's exponential distribution, McKay's gamma distribution, distribution with gamma conditionals, gamma exponential distribution, Dirichlet distribution, inverted beta distribution, distribution with beta conditionals, beta stacy distribution, Cuadras and Augé's distribution, Farlie Gumbel Morgenstern distribution, logistic distribution, Pearson type VII distribution, Pearson type II distribution, distribution with Cauchy conditionals, bilateral Pareto distribution, Muliere and Scarsini's Pareto distribution, distribution with Pareto conditionals and the distribution with Gumbel conditionals. We believe that the results presented here will serve as an important reference for scientists and engineers in many areas.     

Relations for moments of progressively Type-II censored order statistics from half-logistic distribution with applications to inference

In this paper, we establish several recurrence relations for the single and product moments of progressively Type-II right censored order statistics from a half-logistic distribution. The use of these relations in a systematic recursive manner would enable one to compute all the means, variances and covariances of progressively Type-II right censored order statistics from the half-logistic distribution for all sample sizes n, effective sample sizes m, and all progressive censoring schemes (R₁,…,R_m). The results established here generalize the corresponding results for the usual order statistics due to Balakrishnan (1985). These moments are then utilized to derive best linear unbiased estimators of the scale and location-scale parameters of the half-logistic distribution. A comparison of these estimators with the maximum likelihood estimates is then made. The best linear unbiased predictors of censored failure times is then discussed briefly. Finally, two numerical examples are presented to illustrate all the inferential methods developed here.     

Plotting and checking the bivariate distributions of multiple Gaussian data

The geostatistical modeling of continuous variables relies heavily on the multivariate Gaussian distribution. It is remarkably tractable. The multivariate Gaussian distribution is adopted for K multiple variables (often K is between 2 and 10) and for N multiple locations (often N is in the tens of millions). Our focus is on the relationship between the K variables. Each variable is transformed to be univariate Gaussian, but the multivariate nature of the data is not necessarily Gaussian after univariate transformation. If multiple data variables are deemed non-Gaussian, then additional steps need to be taken such as linearization by alternating conditional expectation (ACE) or multivariate transformation by the stepwise conditional transformation (SCT). Although all L-variate distributions (1<L≤K) should be checked, the bivariate distributions are practically important; there are relatively few data in practice to investigate higher order distributions. A quantitative measure of departure from the bivariate Gaussian distribution is established based on quadrants and the distribution of differences from the theoretically expected distribution. Although approximate, the measure of departure is useful for comparing different distributions and guiding the geostatistician to look closer at some data variables. A scatnscores program is shown that will plot all K(K–1)/2 bivariate cross plots associated with K variables. The correlation coefficients, number of data, degree of departure from the bivariate Gaussian distribution, and bivariate Gaussian probability contours associated with specified cumulative probabilities are shown. The data ID numbers can also be shown to help identify outlier or problematic data.     

一类单据控件的设计与应用

讨论了在企业管理信息系统的开发实践中单据处理业务的基础性、重要性和相似性。运用面向对象的思想归纳了各类常用单据的共同特性,并设计了一个现实的单据控件。讨论了控件的设计要求、各种常用的属性、方法和事件,以一个具体的应用实例对该控件的使用方法和使用效果做了一个简要总结。     

Using algebraic software to compute the moments of order statistics

Here we examine using the algebraic software Mathematica for computing the moments of the order statistics of several frequently used distribution functions, specifically those for the extreme-value, Weibull, double Weibull, logistic and Cauchy distributions. These are illustrated in obtaining the generalised least-squares estimates of the location and scale parameters of the extreme-value distribution.     

Recurrence relations for rational cubic methods I: The Halley method

In this paper we present a system of a priori error bounds for the Halley method in Banach spaces. Our theorem supplies sufficient conditions on the initial point to ensure the convergence of Halley iterates, by means of a system of “recurrence relations”, analogous to those given for the Newton method by Kantorovich, improving previous results by Döring [4]. The error bounds presented are optimal for second degree polynomials. Other rational cubic methods, as the Chebyshev method, will be treated in a subsequent paper.     

A Gaussian mixture Bernoulli filter for extended target tracking with application to an ultra-wideband localization system

This paper presents a Gaussian mixture (GM) implementation of the Bernoulli filter for extended target tracking, which we call the extended target GM Bernoulli (ET-GM-Ber) filter. Closed form expressions for the ET-GM-Ber filter recursions are obtained. A clustering step is integrated into the measurement update stage in order to have a computationally tractable filter. Performance of the proposed filter is tested both on the simulated data and experimental data collected using an ultra-wideband (UWB) localization system. Simulations and experimental results demonstrate the accurate and effective performance of the proposed filter.     

Statistical inference for the lifetime performance index based on generalised order statistics from exponential distribution

In manufacturing industries, the lifetime of an item is usually characterised by a random variable X and considered to be satisfactory if X exceeds a given lower lifetime limit L. The probability of a satisfactory item is then η_L := P(X ≥ L), called conforming rate. In industrial companies, however, the lifetime performance index, proposed by Montgomery and denoted by C_L, is widely used as a process capability index instead of the conforming rate. Assuming a parametric model for the random variable X, we show that there is a connection between the conforming rate and the lifetime performance index. Consequently, the statistical inferences about η_L and C_L are equivalent. Hence, we restrict ourselves to statistical inference for C_L based on generalised order statistics, which contains several ordered data models such as usual order statistics, progressively Type-II censored data and records. Various point and interval estimators for the parameter C_L are obtained and optimal critical regions for the hypothesis testing problems concerning C_L are proposed. Finally, two real data-sets on the lifetimes of insulating fluid and ball bearings, due to Nelson (1982) Nelson, W. (1982), Applied Life Data Analysis, New York: John Wiley &; Sons.[Crossref] , [Google Scholar] and Caroni (2002) Caroni, C. (2002), ‘The correct “ball bearings” data’, Lifetime Data Analysis, 8, 395–399.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar], respectively, and a simulated sample are analysed.     

Denoising with higher order derivatives of bounded variation and an application to parameter estimation

Regularization with functions of bounded variation has been proven to be effective for denoising signals and images. This nonlinear regularization technique, in contrast with linear regularization techniques like Tikhonov regularization, has the advantage that discontinuities in signals and images can be located very precisely. In this paper bounded variation regularization is generalized to functions with higher order derivatives of bounded variation. This concept is applied to locate discontinuities in derivatives, which has important applications in parameter estimation problems.     

Structured matrices in the application of bivariate interpolation to curve implicitization

A non-standard application of bivariate polynomial interpolation is discussed: the implicitization of a rational algebraic curve given by its parametric equations. Three different approaches using the same interpolation space are considered, and their respective computational complexities are analyzed. Although the techniques employed are usually associated to numerical analysis, in this case all the computations are carried out using exact rational arithmetic. The power of the Kronecker product of matrices in this application is stressed.     

Tolerance limits on order statistics in future samples coming from the Pareto distribution

It is often desirable to have statistical tolerance limits available for the distributions used to describe time-to-failure data in reliability problems. For example, one might wish to know if at least a certain proportion, say β, of a manufactured product will operate at least T hours. This question cannot usually be answered exactly, but it may be possible to determine a lower tolerance limit L(X), based on a random sample X, such that one can say with a certain confidence γ that at least 100β% of the product will operate longer than L(X). Then reliability statements can be made based on L(X), or, decisions can be reached by comparing L(X) to T. Tolerance limits of the type mentioned above are considered in this paper, which presents a new approach to constructing lower and upper tolerance limits on order statistics in future samples. Attention is restricted to invariant families of distributions under parametric uncertainty. The approach used here emphasizes pivotal quantities relevant for obtaining tolerance factors and is applicable whenever the statistical problem is invariant under a group of transformations that acts transitively on the parameter space. It does not require the construction of any tables and is applicable whether the past data are complete or Type II censored. The proposed approach requires a quantile of the F distribution and is conceptually simple and easy to use. For illustration, the Pareto distribution is considered. The discussion is restricted to one-sided tolerance limits. A practical example is given.     

A bivariate frailty model for events with a permanent survivor fraction and non-monotonic hazards; with an application to age at first maternity

For certain life cycle events a non-susceptible fraction of subjects will never undergo the event. In demographic applications, examples are provided by marriage and age at first maternity. A model for survival data allowing a permanent survival fraction, non-monotonic failure rates and unobserved frailty is considered here. Regressions are used to explain both the failure time and permanent survival mechanisms and additive correlated errors are included in the general linear models defining these regressions. A hierarchical Bayesian approach is adopted with likelihood conditional on the random frailty effects and a second stage prior defining the bivariate density of those effects. The gain in model fit, and potential effects on inference, from adding frailty is demonstrated in a case study application to age at first maternity in Germany.     

Electromyography signal analysis using wavelet transform and higher order statistics to determine muscle contraction

Abstract: Electromyography gives an electrical representation of neuromuscular activation associated with a contracting muscle. The electromyography signal acquires noise while travelling though different media. The wavelet transform is employed for removing noise from surface electromyography (SEMG) and higher order statistics are applied for analysing the signal. With the appropriate choice of wavelet, it is possible to remove interference noise (denoise) effectively in order to analyse the SEMG. Daubechies wavelets (db2, db4, db5, db6, db8), symmlet (sym4, sym5) and the orthogonal Meyer (dmey) wavelet can efficiently remove noise from the recorded SEMG signals. However, the most effective wavelet for SEMG denoising is chosen by calculating the root mean square difference and signal-to-noise ratio values. Results for both root mean square difference and signal-to-noise ratio show that wavelet db2 performs denoising best out of the wavelets. Furthermore, the higher order statistics method is applied for SEMG signal analysis because of its unique properties when applied to random time series, such as parameter estimation, testing of Gaussianity and linearity, deterministic and non-deterministic signal detection etc. Gaussianity and linearity tests as part of higher order statistics are conducted to understand changes in muscle contraction and to quantify the effectiveness of the noise removal process. According to the results, the SEMG signal becomes less Gaussian and more linear with increased force.