on the variance of fuzzy random variables abstract this paper deals with an expectation and a real valued variance of fuzzy random variables the expectation and the variance of a fuzzy random variable is characterized by frechet s principle in a metric space we study properties of the variance of a fuzzy random variable and compare it with the common variance of real valued random variables using the expectation and the variance of fuzzy random variables we consider a linear regression problem and limit theorems

discriminant analysis of interval data using monte carlo method in assessment of overlap in this paper we show that the method of discriminant analysis da on interval data by data envelopment analysis dea dea discriminant analysis dea da is designed to identify the existence or non existence of an overlap between two groups by separating hyperplane in addition it predicts a new observation to the group which it belongs to data envelopment analysis technique which is developed based on the mathematical programming evaluates the relative efficiency of a set of homogeneous decision making units however there are similarities between dea and da da is a method for separating two sets with previous knowledge meanwhile dea is a technique for separating two sets efficient and inefficient without previous knowledge also goal programming method can be used for both of these methods

the design of granular classifiers a study in the synergy of interval calculus and fuzzy sets in pattern recognition in this study we are concerned with a development of a certain category of granular classifiers referred here to as hyperbox driven classifiers hdc the approach fully capitalizes on the two key technologies of granular computing namely set interval calculus and fuzzy sets in dealing with a description of geometry of patterns data belonging to a certain category we take advantage of the capabilities of sets intervals and their cartesian products when describing a core structure of classes of patterns in the form of some hyperboxes their combinations are referred to as a core structure of the feature space next we refine the geometry of the classifier by bringing forward the concepts of regions of the feature space characterized by fuzzy sets they are sought as a secondary structure the construction of the core structure is realized by means of the supervised version of dbscan one of the popular clustering algorithms of data mining the secondary structure is described by fuzzy sets whose membership functions are a solution to some optimization problem of allocation of degrees of belongingness in the formation of the secondary structure we exploit a concept of the hausdorff distance that determines a distance between some information granule of a well defined geometry and a numeric pattern a point in the highly dimensional feature space the two stage description of the granular classifier is also beneficial in the characterization of the classification error by virtue of the design of the core structure the classification error occurring there assumes very small almost zero values on the other hand the secondary structure coming with a substantial level of overlap between classes is characterized by relatively higher values of the classification error being cognizant of that we suggest a way of forming a synthetic measure built on the membership values and named a separability index that helps quantify the classification error in this setting a series of numeric examples are used to demonstrate the effectiveness of the proposed granular classifiers

a comparison of rotation free triangular shell elements for unstructured meshes many engineering applications require accurate and rapidly computed thin shell elements rotation free rf shell elements include the bending behaviour of thin shells without introducing any additional degrees of freedom compared to a membrane element instead constant curvatures are approximated from the out of plane displacements of a patch of usually four triangular elements a consequence of this is that the accuracy for irregular meshes has been unsatisfactory the aim of this study is to find an rf shell element which is accurate also for unstructured meshes the main difference between existing elements is whether they assume two dimensional constant curvatures over the patch or use superposition of one dimensional constant curvatures for the three pairs of triangles the first assumption fulfils constant curvatures for a kirchhoff plate exactly whereas the second and most common assumption only approximates constant curvatures the first assumption is significantly more resistant to element shape distortions whereas the second assumption is slightly faster to compute and more appropriate on boundaries where one or more elements are missing or several neighbouring elements share a side the combination is significantly more accurate for irregular meshes than other comparable rf elements for linear benchmark tests

image representation by a new optimal non uniform morphological sampling abstract this paper presents a new non uniform subsampling strategy based on mathematical morphology this sampling scheme is determinist and iterative at each iteration a sample position is determined in order to minimize the overall absolute reconstruction error we show that satisfying this criterion is equivalent to maximizing the volume of reconstructed image the sample selection strategy is repeated until reaching either a previously determined number of samples or a maximal admissible error the correspondent reconstruction procedure is also described the algorithm is first presented using only one structuring element then extended to the case of a family of structuring elements experiments on the choice of structuring elements in the family allow us to choose the optimal description set finally we present an application of this novel image representation method on image compression a comparison of results with jpeg standard shows the good performance of our approach especially for high compression ratios

self organizing maps for imprecise data self organizing maps soms consist of a set of neurons arranged in such a way that there are neighbourhood relationships among neurons following an unsupervised learning procedure the input space is divided into regions with common nearest neuron vector quantization allowing clustering of the input vectors in this paper we propose an extension of the soms for data imprecisely observed self organizing maps for imprecise data soms id the learning algorithm is based on two distances for imprecise data in order to illustrate the main features and to compare the performances of the proposed method we provide a simulation study and different substantive applications

evolution equations for continuous scale morphological filtering multiscale signal analysis has emerged as a useful framework for many computer vision and signal processing tasks morphological filters can be used to develop nonlinear multiscale operations that have certain advantages over linear multiscale approaches in that they preserve important signal features such as edges the authors discuss several nonlinear partial differential equations that model the scale evolution associated with continuous space multiscale morphological erosions dilations openings and closings these equations relate the rate of change of the multiscale signal ensemble as scale increases to a nonlinear operator acting on the space of signals the nonlinear operator is characterized by the shape and dimensionality of the structuring element used by the morphological operators generally taking the form of a nonlinear function of certain partial differential operators

morphological filters part i their set theoretic analysis and relations to linear shift invariant filters this paper examines the set theoretic interpretation of morphological filters in the framework of mathematical morphology and introduces the representation of classical linear filters in terms of morphological correlations which involve supremum infimum operations and additions binary signals are classified as sets and multilevel signals as functions two set theoretic representations of signals are reviewed filters are classified as set processing sp or function processing fp conditions are provided for certain fp filters that pass binary signals to commute with signal thresholding because then they can be analyzed and implemented as sp filters the basic morphological operations of set erosion dilation opening and closing are related to minkowski set operations and are used to construct fp morphological filters emphasis is then given to analytically and geometrically quantifying the similarities and differences between morphological filtering of signals by sets and functions the latter case allows the definition of morphological convolutions and correlations toward this goal various properties of fp morphological filters are also examined linear shift invariant filters due to their translation invariance are uniquely characterized by their kernel which is a special collection of input signals increasing linear filters are represented as the supremum of erosions by their kernel functions if the filters are also discrete and have a finite extent impulse response they can be represented as the supremum of erosions only by their minimal with respect to a signal ordering kernel functions stable linear filters can be represented as the sum of at most two weighted suprema of erosions these results demonstrate the power of mathematical morphology as a unifying approach to both linear and nonlinear signal shaping strategies

change point analysis of imprecise time series abstract in this paper we describe how to conduct a change point analysis when dealing with time series imprecisely or vaguely observed i e time ordered observations whose values are not known exactly such as interval or ordinal time series imprecise time series in order to treat such time series we propose to employ a fuzzy approach i e data are parameterized in the form of fuzzy variables then to detect the number and location of change points we employ a deviation measure for fuzzy variables in the framework of atheoretical regression trees art we present simulation results pertaining to the behavior of the proposed approach as well as two empirical applications to real imprecise time series

scale space using mathematical morphology in this paper we prove that the scale space of a one dimensional gray scale signal based on morphological filterings satisfies causality no new feature points are created as scale gets larger for this we refine the standard definition of zero crossing so as to allow signals with certain singularity and use them to define feature points this new definition of zero crossing agrees with the standard one in the case of functions with second order derivative in particular the scale space based on the gaussian kernel g does not need this concept because a filtered signal g f is always infinitely differentiable using this generalized concept of zero crossing we show that a morphological filtering based on opening and hence also closing by duality satisfies causality we note that some previous works have mistakes which are corrected in this paper our causality results do not apply to more general two dimensional gray scale images causality results on alternating sequential filter obtained as byproduct are also included

nonlinear scale space filtering and multiresolution system we derive and demonstrate a nonlinear scale space filter and its application in generating a nonlinear multiresolution system for each datum in a signal a neighborhood of weighted data is used for clustering the cluster center becomes the filter output the filter is governed by a single scale parameter that dictates the spatial extent of nearby data used for clustering this together with the local characteristic of the signal determines the scale parameter in the output space which dictates the influences of these data on the output this filter is thus adaptive and data driven it provides a mechanism for a removing impulsive noise b improved smoothing of nonimpulsive noise and c preserving edges comparisons with gaussian scale space filtering and median filters are made using real images using the architecture of the laplacian pyramid and this nonlinear filter for interpolation we construct a nonlinear multiresolution system that has two features 1 edges are well preserved at low resolutions and 2 difference signals are small and spatially localized this filter implicitly presents a new mechanism for detecting discontinuities differing from techniques based on local gradients and line processes this work shows that scale space filtering nonlinear filtering and scale space clustering are closely related and provides a framework within which further image processing image coding and computer vision problems can be investigated

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

a rotation free thin shell quadrilateral in this paper a four node quadrilateral finite element for the analysis of smooth thin shells is presented the main feature of the element an extension of previous developments in triangles is that the translational displacements of the middle surface are the only degrees of freedom the membrane behavior results from a standard bilinear interpolation of the geometry within the element with the aim of an efficient element in codes with explicit time integration one point quadrature is used in the element area to avoid spurious deformation modes hourglass modes membrane forces resulting from a perturbation stabilization technique are included for the computation of the curvature tensor a patch of five elements the element and the four adjacent elements is defined the curvature field assumed constant within the element is expressed in terms of the deformation gradient at the element boundary and is dependent on the position of the twelve nodes included in the patch in some problems a bending deformed configuration may occur without associated energy a cost effective perturbation stabilization scheme is used to control it general boundary conditions are shown to be easily implemented the element denoted bsq for basic shell quadrilateral is based on a total lagrangian formulation and has been implemented in codes with implicit and explicit integration to assess the element performance and convergence properties a set of numerical examples are presented including geometrically linear and non linear problems with large strain plasticity the results obtained show good convergence properties for several examples different values of the stabilization coefficients have been considered to study the sensitivity of the results to such coefficients in general this sensitivity appears to be low as the mesh is refined and the results are obtained with a fixed set of coefficients

how biased is the apparent error rate of a prediction rule abstract a regression model is fitted to an observed set of data how accurate is the model for predicting future observations the apparent error rate tends to underestimate the true error rate because the data have been used twice both to fit the model and to check its accuracy we provide simple estimates for the downward bias of the apparent error rate the theory applies to general exponential family linear models and general measures of prediction error special attention is given to the case of logistic regression on binary data with error rates measured by the proportion of misclassified cases several connected ideas are compared mallows s cp cross validation generalized cross validation the bootstrap and akaike s information criterion

rotation free finite element for the non linear analysis of beams and axisymmetric shells abstract in this paper a finite element for the non linear analysis of two dimensional beams and axisymmetric shells is presented the element uses classical thin shell assumptions no transverse shear strains the main feature of the element is that it has no rotational degrees of freedom curvatures are computed using geometrical information from the patch of three elements formed by the main element and the two neighbor adjacent elements special attention is devoted to non smooth geometries and branching shells an elastic plastic material law is considered large strains are treated using a logarithmic strain measure and a through the thickness numerical integration of the constitutive equations several examples are presented including linear problems to study convergence properties and non linear problems for both elastic and elastic plastic materials and large strains

m band nonlinear subband decompositions with perfect reconstruction we investigate nonlinear multirate filterbanks with maximal decimation and perfect reconstruction definitions of the desired properties of such structures are given for general nonlinear filterbanks we then consider a triangular representation of linear filterbanks and see that it may be easily extended to the nonlinear case furthermore general nonlinear filterbanks are presented for which perfect reconstruction is either inherently guaranteed or ensured subject to an easily verified condition extensions to bidimensional filters are also discussed and an application for nonlinear multiresolution schemes to feature sieves is shown

a generalized strong law of large numbers strong laws of large numbers have been stated in the literature for measurable functions taking on values on different spaces in this paper a strong law of large numbers which generalizes some previous ones like those for real valued random variables and compact random sets is established this law is an example of a strong law of large numbers for borel measurable nonseparably valued elements of a metric space

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

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

the digital morphological sampling theorem morphological sampling reduces processing time and cost and yet produces results sufficiently close to the result of full processing a morphological sampling theorem is described which states 1 how a digital image must be morphologically filtered before sampling in order to preserve the relevant information after sampling 2 to what precision an appropriate morphologically filtered image can be reconstructed after sampling and 3 the relationship between morphologically operating before sampling and the more computationally efficient scheme of morphologically operating on the sampled image with a sampled structuring element the digital sampling theorem is developed first for the case of binary morphology and then it is extended to gray scale morphology through the use of the umbra homomorphism theorems

map signal estimation in noisy sequences of morphologically smooth images sidiropoulos et al 1994 demonstrated that morphological openings and closings can be viewed as maximum a posteriori map estimators of morphologically smooth signals in signal independent i i d noise the present authors extend these results to the m fold independent observation case and show that the aforementioned estimators are strongly consistent we also demonstrate the validity of a thresholding conjecture sidiropoulos et al 1994 by simulation and use it to evaluate estimator performance taken together these results can help determine the least upper bound m on m which guarantees virtually error free reconstruction of morphologically smooth images

morphological pyramids with alternating sequential filters the aim of this paper is to find a relationship between alternating sequential filters asf and the morphological sampling theorem mst developed by haralick et al 1987 the motivation behind this approach is to take advantage of the computational efficiency offered by the mst to implement morphological operations first we show alternative proofs for opening and closing in the sampled and unsampled domain using the basis functions these proofs are important because they show that it possible to obtain any level of a morphological pyramid in one step rather than the traditional two step procedure this decomposition is then used to show the relationship of the open closing in the sampled and unsampled domain an upper and a lower bound for the above relationships are presented under certain circumstances an equivalence is shown for open closing between the sampled and the unsampled domain an extension to more complicated algorithms using a union of openings and an intersection of closings is also proposed using the hausdorff metric it is shown that a morphologically reconstructed image cannot have a better accuracy than twice the radius of the reconstruction structuring element binary and gray scale examples are presented

morphological filters part ii their relations to median order statistic and stack filters this paper extends the theory of median order statistic os and stack filters by using mathematical morphology to analyze them and by relating them to those morphological erosions dilations openings closings and open closings that commute with thresholding the max min representation of os filters is introduced by showing that any median or other os filter is equal to a maximum of erosions moving local minima and also to a minimum of dilations moving local maxima thus os filters can be computed by a closed formula that involves a max min on prespecified sets of numbers and no sorting stack filters are established as the class of filters that are composed exactly of a finite number of max min operations the kernels of median os and stack filters are collections of input signals that uniquely represent these filters due to their translation invariance the max min functional definitions of these nonlinear iliters is shown to be equivalent to a maximum of erosions by minimal with respect to a signal ordering kernel elements and also to a minimum of dilations by minimal kernel elements of dual filters the representation of stack filters based on their minimal kernel elements is proven to be equivalent to their representation based on irreducible sum of products expressions of boolean functions it is also shown that median filtering and its iterations of any signal by convex 1 d windows is bounded below by openings and above by closings a signal is a root fixed point of the median iff it is a root of both an opening and a closing the open closing and clos opening yield median roots in one pass suppress impulse noise similarly to the median can discriminate between positive and negative noise impulses and are computationally less complex than the median some similar results are obtained for 2 d median filtering

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

optimal filtering of digital binary images corrupted by union intersection noise we model digital binary image data as realizations of a uniformly bounded discrete random set or discrete random set for short which is a mathematical object that can be directly defined on a finite lattice we consider the problem of estimating realizations of discrete random sets distorted by a degradation process that can be described by a union intersection noise model two distinct optimal filtering approaches are pursued the first involves a class of mask filters which arises quite naturally from the set theoretic analysis of optimal filters the second approach involves a class of morphological filters we prove that under i i d noise morphological openings closings unions of openings and intersections of closings can be viewed as map estimators of morphologically smooth signals then we show that by using an appropriate under a given degradation model expansion of the optimal filter we can obtain universal characterizations of optimality that do not rely on strong assumptions regarding the spatial interaction of geometrical primitives of the signal and the noise the results generalize to gray level images in a fairly straightforward manner

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

segmentation of overlapping elliptical objects in silhouette images segmentation of partially overlapping objects with a known shape is needed in an increasing amount of various machine vision applications this paper presents a method for segmentation of clustered partially overlapping objects with a shape that can be approximated using an ellipse the method utilizes silhouette images which means that it requires only that the foreground objects and background can be distinguished from each other the method starts with seedpoint extraction using bounded erosion and fast radial symmetry transform extracted seedpoints are then utilized to associate edge points to objects in order to create contour evidence finally contours of the objects are estimated by fitting ellipses to the contour evidence the experiments on one synthetic and two different real data sets showed that the proposed method outperforms two current state of art approaches in overlapping objects segmentation

nonlinear multiresolution signal decomposition schemes ii morphological wavelets interest in multiresolution techniques for signal processing and analysis is increasing steadily an important instance of such a technique is the so called pyramid decomposition scheme this paper presents a general theory for constructing linear as well as nonlinear pyramid decomposition schemes for signal analysis and synthesis the proposed theory is based on the following ingredients 1 the pyramid consists of a finite or infinite number of levels such that the information content decreases toward higher levels and 2 each step toward a higher level is implemented by an information reducing analysis operator whereas each step toward a lower level is implemented by an information preserving synthesis operator one basic assumption is necessary synthesis followed by analysis yields the identity operator meaning that no information is lost by these two consecutive steps several examples of pyramid decomposition schemes are shown to be instances of the proposed theory a particular class of linear pyramids morphological skeleton decompositions the morphological haar pyramid median pyramids etc furthermore the paper makes a distinction between single scale and multiscale decomposition schemes i e schemes without or with sample reduction finally the proposed theory provides the foundation of a general approach to constructing nonlinear wavelet decomposition schemes and filter banks