interval versions of statistical techniques with applications to environmental analysis bioinformatics and privacy in statistical databases in many areas of science and engineering it is desirable to estimate statistical characteristics mean variance covariance etc under interval uncertainty for example we may want to use the measured values x t of a pollution level in a lake at different moments of time to estimate the average pollution level however we do not know the exact values x t e g if one of the measurement results is 0 this simply means that the actual unknown value of x t can be anywhere between 0 and the detection limit dl we must therefore modify the existing statistical algorithms to process such interval data such a modification is also necessary to process data from statistical databases where in order to maintain privacy we only keep interval ranges instead of the actual numeric data e g a salary range instead of the actual salary most resulting computational problems are np hard which means crudely speaking that in general no computationally efficient algorithm can solve all particular cases of the corresponding problem in this paper we overview practical situations in which computationally efficient algorithms exist e g situations when measurements are very accurate or when all the measurements are done with one or few instruments as a case study we consider a practical problem from bioinformatics to discover the genetic difference between the cancer cells and the healthy cells we must process the measurements results and find the concentrations c and h of a given gene in cancer and in healthy cells this is a particular case of a general situation in which to estimate states or parameters which are not directly accessible by measurements we must solve a system of equations in which coefficients are only known with interval uncertainty we show that in general this problem is np hard and we describe new efficient algorithms for solving this problem in practically important situations

a class of fuzzy clusterwise regression models in this paper we introduce a class of fuzzy clusterwise regression models with lr fuzzy response variable and numeric explanatory variables which embodies fuzzy clustering into a fuzzy regression framework the model bypasses the heterogeneity problem that could arise in fuzzy regression by subdividing the dataset into homogeneous clusters and performing separate fuzzy regression on each cluster the integration of the clustering model into the regression framework allows us to simultaneously estimate the regression parameters and the membership degree of each observation to each cluster by optimizing a single objective function the class of models proposed here includes as special cases the fuzzy clusterwise linear regression model and the fuzzy clusterwise polynomial regression model we also introduce a set of goodness of fit indices to evaluate the fit of the regression model within each cluster as well as in the whole dataset finally we consider some cluster validity criteria that are useful in identifying the optimal number of clusters several applications are provided in order to illustrate the approach

optimal allocation for the comparison of proportions optimal allocation strategies for the comparison of proportions p1 and p2 in two groups are discussed allocations which i maximize the precision in the estimation of the difference p1 p2 ii maximize the precision in the estimation of the ratio p1 p2 iii maximize the power to detect a group difference and iv minimize the total cost of a study are compared the recommended allocation varies markedly with the desired objective furthermore there is a wide range of allocation schemes which are effectively optimal with regard to each criterion considered a study of congenital malformations occurring in pregnancies of diabetic women is used to illustrate how these considerations apply to the planning of a study

a generalized l1 type metric between fuzzy numbers for an approach to central tendency of fuzzy data abstract in dealing with data generated from a random experiment l2 metrics are suitable for many statistical approaches and developments to analyze fuzzy valued experimental data a generalized l2 metric based on the mid spread representation of fuzzy values has been stated and a related methodology to conduct statistics with fuzzy data has been carried out most of the developed methods concern either explicitly or implicitly the mean values of the involved random mechanisms producing fuzzy data other statistical approaches and studies with experimental data consider l1 metrics especially in dealing with errors or in looking for a more robust solution and intuitive interpretation this paper aims to introduce a generalized l1 metric between fuzzy numbers based on a new characterization for them that will be referred to as the mid leftdev rightdev characterization more precisely the metric will take into account both absolute differences in location and absolute differences in shape imprecision of fuzzy numbers moreover one can choose the weight of the influence of the second differences in contrast to the first one after introducing the new characterization for fuzzy numbers as well as the associated l1 metric we will examine some properties finally as an immediate application the problem of minimizing the mean distance between a fuzzy number and the distribution of a random mechanism producing fuzzy number valued data will be given discussed and illustrated

on tests of equivalence via non unity relative risk for matched pair design matched pair design is often used in clinical trials to increase the efficiency of treatment comparison we consider the problem of equivalence test with a relative risk endpoint in matched pair studies with binary outcomes and develop several score and wald type statistics for testing a hypothesis of non unity relative risk examples from an assessment of hiv screening test and a cross over clinical trial of soft contact lenses are used to illustrate the proposed methods through simulations we compare the empirical performance of these tests with the test proposed by lachenbruch and lynch we show that a score test based on a reparameterized multinomial model by tango performs best in the sense that the test satisfactorily controls the type i error rate and its empirical type i error rates are generally much closer to the prespecified nominal significance level than those of the other tests

multi input single output nonlinear blind separation of binary sources the problem of blindly separating multiple binary sources from a single nonlinear mixture is addressed through a novel clustering approach without the use of any optimization procedure the method is based on the assumption that the source probabilities are asymmetric in which case the output probability distribution can be expressed as a linear mixture of the sources we are then able to solve the problem by using a known linear multiple input single output miso blind separation method the overall procedure is very fast and in theory it works for any number of independent binary sources and for a wide range of nonlinear functions in practice the accuracy of the method depends on the estimation accuracy of the output probabilities and the cluster centers it can be quite sensitive to noise especially as the number of sources increases or the number of data samples is reduced however in our experiments we have been able to demonstrate successful separation of up to four sources

testing the equality of two poisson means using the rate ratio in this article we investigate procedures for comparing two independent poisson variates that are observed over unequal sampling frames i e time intervals populations areas or any combination thereof we consider two statistics with and without the logarithmic transformation for testing the equality of two poisson rates two methods for implementing these statistics are reviewed they are 1 the sample based method and 2 the constrained maximum likelihood estimation cmle method we conduct an empirical study to evaluate the performance of different statistics and methods generally we find that the cmle method works satisfactorily only for the statistic without the logarithmic transformation denoted as w2 while sample based method performs better for the statistic using the logarithmic transformation denoted as w3 it is noteworthy that both statistics perform well for moderate to large poisson rates e g 10 for small poisson rates e g 10 w2 can be liberal e g actual type i error rate nominal level 1 2 while w3 can be conservative e g actual type i error rate nominal level 0 8 the corresponding sample size formulae are provided and valid in the sense that the simulated powers associated with the approximate sample size formulae are generally close to the pre chosen power level we illustrate our methodologies with a real example from a breast cancer study copyright 2004 john wiley sons ltd

kernel based nonlinear blind source separation we propose ktdsep a kernel based algorithm for nonlinear blind source separation bss it combines complementary research fields kernel feature spaces and bss using temporal information this yields an efficient algorithm for nonlinear bss with invertible nonlinearity key assumptions are that the kernel feature space is chosen rich enough to approximate the nonlinearity and that signals of interest contain temporal information both assumptions are fulfilled for a wide set of real world applications the algorithm works as follows first the data are implicitly mapped to a high possibly infinite dimensional kernel feature space in practice however the data form a smaller submanifold in feature space even smaller than the number of training data points a fact that has already been used by for example reduced set techniques for support vector machines we propose to adapt to this effective dimension as a preprocessing step and to construct an orthonormal basis of this submanifold the latter dimension reduction step is essential for making the subsequent application of bss methods computationally and numerically tractable in the reduced space we use a bss algorithm that is based on second order temporal decorrelation finally we propose a selection procedure to obtain the original sources from the extracted nonlinear components automatically experiments demonstrate the excellent performance and efficiency of our ktdsep algorithm for several problems of nonlinear bss and for more than two sources

a nonparametric kernel approach to interval valued data analysis this article concerns datasets in which variables are in the form of intervals which are obtained by aggregating information about variables from a larger dataset we propose to view the observed set of hyper rectangles as an empirical histogram and to use a gaussian kernel type estimator to approximate its underlying distribution in a nonparametric way we apply this idea to both univariate density estimation and regression problems unlike many existing methods used in regression analysis the proposed method can estimate the conditional distribution of the response variable for any given set of predictors even when some of them are not interval valued empirical studies show that the proposed approach has a great flexibility in various scenarios with complex relationships between the location and width of intervals of the response and predictor variables

distance based linear discriminant analysis for interval valued data interval valued observations arise in many real life situations as either the precise representation of the objective entity or the representation of incomplete knowledge thus given p features observed over a sample of objects belonging to one of two possible classes each observation can be perceived as a non empty closed and bounded hyperrectangle on r p the aim of the paper is to suggest a p dimensional classification method for random intervals when two or more classes are considered by the generalization of fisher s procedure for linear discriminant analysis the idea consists of finding a directional vector which maximizes the ratio of the dispersion between the classes and within the classes of the observed hyperrectangles a classification rule for new observations is also provided and some simulations are carried out to compare the behavior of the proposed classification procedure with respect to other methods known from the literature finally the suggested methodology are applied on a real life situation example

nonlinear blind source separation using kernels we derive a new method for solving nonlinear blind source separation bss problems by exploiting second order statistics in a kernel induced feature space this paper extends a new and efficient closed form linear algorithm to the nonlinear domain using the kernel trick originally applied in support vector machines svms this technique could likewise be applied to other linear covariance based source separation algorithms experiments on realistic nonlinear mixtures of speech signals gas multisensor data and visual disparity data illustrate the applicability of our approach

extended support vector interval regression networks for interval input output data in many applications it is natural to use interval data because of uncertainty existence in the measurements variability in defining terms such as the temperature during a given day description for extremely behavior such as the maximum wind speed in a given area etc in order to handle such interval data a novel approach called the interval support vector interval regression networks isvirns is proposed the isvirns is extended from our previous work the support vector interval regression networks svirns it is easy to find that svirns can handle interval output data but for input data they must be crisp in this study the hausdorff distance is employed as the distance measure of interval data and is incorporated into the kernel functions of svirns to determine the initial structure of isvirns because the proposed approach can provide a better initial structure for isvirns it can have a faster convergent speed the experimental results with real data sets show the validity of the proposed isvirns

fast and robust fixed point algorithms for independent component analysis independent component analysis ica is a statistical method for transforming an observed multidimensional random vector into components that are statistically as independent from each other as possible we use a combination of two different approaches for linear ica comon s information theoretic approach and the projection pursuit approach using maximum entropy approximations of differential entropy we introduce a family of new contrast functions for ica these contrast functions enable both the estimation of the whole decomposition by minimizing mutual information and estimation of individual independent components as projection pursuit directions the statistical properties of the estimators based on such contrast functions are analyzed under the assumption of the linear mixture model and it is shown how to choose contrast functions that are robust and or of minimum variance finally we introduce simple fixed point algorithms for practical optimization of the contrast functions

a new family of metrics for compact convex fuzzy sets based on a generalized concept of mid and spread one of the most important aspects of the statistical analysis of imprecise data is the usage of a suitable distance on the family of all compact convex fuzzy sets which is not too hard to calculate and which reflects the intuitive meaning of fuzzy sets on the basis of expressing the metric of bertoluzza et al c bertoluzza n corral a salas on a new class of distances between fuzzy numbers mathware soft comput 2 1995 71 84 in terms of the mid points and spreads of the corresponding intervals we construct new families of metrics on the family of all d dimensional compact convex sets as well as on the family of all d dimensional compact convex fuzzy sets it is shown that these metrics not only fulfill many good properties but also that they are easy to calculate and easy to manage for statistical purposes and therefore useful from the practical point of view

robust fuzzy regression analysis in this paper we propose a robust fuzzy linear regression model based on the least median squares weighted least squares lms wls estimation procedure the proposed model is general enough to deal with data contaminated by outliers due to measurement errors or extracted from highly skewed or heavy tailed distributions we also define suitable goodness of fit indices useful to evaluate the performances of the proposed model the effectiveness of our model in reducing the outliers influence is shown by using applicative examples based both on simulated and real data and by a simulation study

comparative analysis of two rates in this paper we examine comparative analysis of rates with a view to each of the usual comparative parameters rate difference rd rate ratio rr and odds ratio or and with particular reference to first principles for rd and rr we show the prevailing statistical practices to be rather poor we stress the need for restricted estimation of variance in the chi square function underlying interval estimation and also point estimation and hypothesis testing for rr analysis we propose a chi square formulation analogous to that for rd and thus one which obviates the present practice of log transformation and its associated use of taylor series approximation of the variance as for or analysis we emphasize that the chi square function introduced by cornfield for unstratified data and extended by gart to the case of stratified analysis is based on the efficient score and thus embodies its optimality properties we provide simulation results to evince the better performance of the proposed parameter constrained procedures over the traditional ones

a blind source separation technique using second order statistics separation of sources consists of recovering a set of signals of which only instantaneous linear mixtures are observed in many situations no a priori information on the mixing matrix is available the linear mixture should be blindly processed this typically occurs in narrowband array processing applications when the array manifold is unknown or distorted this paper introduces a new source separation technique exploiting the time coherence of the source signals in contrast with other previously reported techniques the proposed approach relies only on stationary second order statistics that are based on a joint diagonalization of a set of covariance matrices asymptotic performance analysis of this method is carried out some numerical simulations are provided to illustrate the effectiveness of the proposed method

interval arithmetic based simple linear regression between interval data discussion and sensitivity analysis on the choice of the metric the prediction of a response random interval valued set from an explanatory one has been examined in previous developments these developments have considered an interval arithmetic based linear model between the random interval valued sets and a least squares regression analysis the least squares approach involves a generalized l 2 metric between interval data this metric weights squared distances between data location mid points centers and squared distances between data imprecision spread radius as a consequence estimators of the parameters in the linear model depend on the choice of the weights in the metric to investigate about a suitable choice of weighting in the generalized mid spread metric a theoretical conclusion is first obtained finally the impact of varying the weights is discussed by considering a monte carlo simulation study

from the statistics of data to the statistics of knowledge symbolic data analysis increasingly datasets are so large they must be summarized in some fashion so that the resulting summary dataset is of a more manageable size while still retaining as much knowledge inherent to the entire dataset as possible one consequence of this situation is that the data may no longer be formatted as single values such as is the case for classical data but rather may be represented by lists intervals distributions and the like these summarized data are examples of symbolic data this article looks at the concept of symbolic data in general and then attempts to review the methods currently available to analyze such data it quickly becomes clear that the range of methodologies available draws analogies with developments before 1900 that formed a foundation for the inferential statistics of the 1900s methods largely limited to small by comparison datasets and classical data formats the scarcity of available methodologies for symbolic data also becomes clear and so draws attention to an enor

performing nonlinear blind source separation with signal invariants given a time series of multicomponent measurements x t the usual objective of nonlinear blind source separation bss is to find a source time series s t comprised of statistically independent combinations of the measured components in this paper the source time series is required to have a density function in s mathdot s space that is equal to the product of density functions of individual components this formulation of the bss problem has a solution that is unique up to permutations and component wise transformations separability is shown to impose constraints on certain locally invariant scalar functions of x which are derived from local higher order correlations of the data s velocity mathdot x the data are separable if and only if they satisfy these constraints and if the constraints are satisfied the sources can be explicitly constructed from the data the method is illustrated by using it to recover the contents of two simultaneous speech like sounds recorded with a single microphone

estimation of a simple linear regression model for fuzzy random variables a generalized simple linear regression statistical probabilistic model in which both input and output data can be fuzzy subsets of r p is dealt with the regression model is based on a fuzzy arithmetic approach and it considers the possibility of fuzzy valued random errors specifically the least squares estimation problem in terms of a versatile metric is addressed the solutions are established in terms of the moments of the involved random elements by employing the concept of support function of a fuzzy set some considerations concerning the applicability of the model are made

test statistics and sample size formulae for comparative binomial trials with null hypothesis of non zero risk difference or non unity relative risk when it is required to establish a materially significant difference between two treatments or alternatively to show that two treatments are equivalent standard test statistics and sample size formulae based on a null hypothesis of no difference no longer apply this paper reviews some of the test statistics and sample size formulae proposed for comparative binomial trials when the null hypothesis is of a specified non zero difference or non unity relative risk methods based on restricted maximum likelihood estimation are recommended and applied to studies of pertussis vaccine

approximate interval estimation of the ratio of binomial parameters a review and corrections for skewness various methods for finding confidence intervals for the ratio of binomial parameters are reviewed and evaluated numerically it is found that the method based on likelihood scores koopman 1984 biometrics 40 513 517 miettinen and nurminen 1985 statistics in medicine 4 213 226 performs best in achieving the nominal confidence coefficient but it may distribute the tail probabilities quite disparately using general theory of bartlett 1953 biometrika 40 306 317 1955 biometrika 42 201 203 we correct this method for asymptotic skewness following gart 1985 biometrika 72 673 677 we extend this correction to the case of estimating the common ratio in a series of two by two tables computing algorithms are given and applied to numerical examples parallel methods for the odds ratio and the ratio of poisson parameters are noted

generic gâteaux differentiability of convex functions on small sets the theory of generic differentiability of convex functions on banach spaces is by now a well explored part of infinite dimensional geometry all the attempts to solve this kind of problem have in common as a working hypothesis one special feature of the finite dimensional case namely convex functions are always considered to be defined on convex sets with nonempty interior but typically a convex set in a banach space does not have interior points even when it is not contained in a closed hyperplane so this raises the problem of expounding a theory of differentiability of convex functions defined on small sets i e sets without interior points in our paper n2 we have made an attempt to solve this problem discussing questions of generic frechet differentiability of convex functions on small sets in the present paper we deal with problems of generic gateauxdifferentiability in this context

least squares fitting of compact set valued data in contexts where set valued data occur in a natural manner it is reasonable to ask whether such data can be directly processed for example to develop numerical techniques for constructing multifunction models which are a best fit to a body of set valued observations this note proposes methods of least squares fitting of such data for the case where each datum is a nonempty compact interval of the real line two models will be considered a interval valued output observed from interval valued input and b interval valued output resulting from exact real input

inference on regressions with interval data on a regressor or outcome this paper examines inference on regressions when interval data are available on one variable the other variables being measured precisely let a population be characterized by a distribution p y x v v 0 v 1 where y r 1 x r k and the real variables v v 0 v 1 satisfy v 0 v v 1 let a random sample be drawn from p and the realizations of y x v 0 v 1 be observed but not those of v the problem of interest may be to infer e y x v or e v x this analysis maintains interval i monotonicity m and mean independence mi assumptions i p v 0 v v 1 1 m e y x v is monotone in v mi e y x v v 0 v 1 e y x v no restrictions are imposed on the distribution of the unobserved values of v within the observed intervals v 0 v 1 it is found that the immi assumptions alone imply simple nonparametric bounds on e y x v and e v x these assumptions invoked when y is binary and combined with a semiparametric binary regression model yield an identification region for the parameters that may be estimated consistently by a modified maximum score mms method the immi assumptions combined with a parametric model for e y x v or e v x yield an identification region that may be estimated consistently by a modified minimum distance mmd method monte carlo methods are used to characterize the finite sample performance of these estimators empirical case studies are performed using interval wealth data in the health and retirement study and interval income data in the current population survey

a set arithmetic based linear regression model for modelling interval valued responses through real valued variables a new linear regression model for an interval valued response and a real valued explanatory variable is presented the approach is based on the interval arithmetic comparisons with previous methods are discussed the new linear model is theoretically analyzed and the regression parameters are estimated some properties of the regression estimators are investigated finally the performance of the procedure is illustrated using both a real life application and simulation studies

confidence intervals for the ratio of two binomial proportions sometimes the ratio of two binomial proportions is the parameter of major interest for instance as the risk ratio in a two group cohort study fleiss 1973 or as the likelihood ratio for a diagnostic test procedure mcneill keeler and adelstein 1975 let x and y be independent binomial variates based on sample sizes m and n and parameters p and p2 respectively let 0 pm p2 we consider the construction of confidence intervals for 0 santner and snell 1980 have proposed a nonanalytical procedure which guarantees a lower bound on the confidence coefficient of the 1 a interval for 0 their algorithm is usable for small m and n but may produce conservative intervals in 2 an analytical method for obtaining approximate confidence intervals for 0 is proposed and compared to the one recommended by katz et al 1978

diagrammatic representation for interval arithmetic abstract the paper presents a diagrammatic representation of a standard interval space the so called mr diagram and shows how to represent and perform interval arithmetic and derive its various properties using the diagram the representation is an extension and refinement of the is diagram representation devised earlier by the author to represent interval relations first the mr diagram is defined together with appropriate graphical notions and constructions for basic interval relations and operations second diagrammatic constructions for all standard arithmetic operations are presented several examples of the use of these constructions to aid reasoning about various simple though nontrivial properties of interval arithmetic are included in order to show how the representation facilitates both deeper understanding of the subject matter and reasoning about its properties

inclusion degree tests for the aumann expectation of a random interval an extension of the inclusion test of the expected value of a real random variable in an interval to the case of general random intervals is introduced the hypothesis of strict inclusion is relaxed by considering a measure of the degree of inclusion thus partial inclusions are also tested asymptotic and bootstrap techniques are established the performance of the bootstrap test is also analyzed by means of some simulations a case study regarding the blood pressure classification in adults is considered

confidence limits for the ratio of two rates based on likelihood scores non iterative method this paper describes a method for creating a confidence interval for the ratio of rates using the score statistic this non iterative and easy to apply procedure produces confidence intervals that are suitable for use with poisson data and simulation results indicate that it is close to the nominal level for a wide range of scenarios