smallest singular value of random matrices and geometry of random polytopes abstract we study the behaviour of the smallest singular value of a rectangular random matrix i e matrix whose entries are independent random variables satisfying some additional conditions we prove a deviation inequality and show that such a matrix is a good isomorphism on its image then we obtain asymptotically sharp estimates for volumes and other geometric parameters of random polytopes absolutely convex hulls of rows of random matrices all our results hold with high probability that is with probability exponentially in dimension close to 1

a probabilistic approach to the geometry of the lᵨⁿ ball this article investigates by probabilistic methods various geometric questions on b n p the unit ball of ln p we propose realizations in terms of independent random variables of several distributions on b n p including the normalized volume measure these representations allow us to unify and extend the known results of the sub independence of coordinate slabs in b n p as another application we compute moments of linear functionals on b n p which gives sharp constants in khinchine s inequalities on b n p and determines the 2 constant of all directions on b n p we also study the extremal values of several gaussian averages on sections of b n p including mean width and l norm and derive several monotonicity results as p varies applications to balancing vectors in l 2 and to covering numbers of polyhedra complete the exposition

asymptotic shape of a random polytope in a convex body abstract let k be an isotropic convex body in r n and let z q k be the l q centroid body of k for every n n consider the random polytope k n conv x 1 x n where x 1 x n are independent random points uniformly distributed in k we prove that a random k n is asymptotically equivalent to z ln n n k in the following sense there exist absolute constants 1 2 0 such that for all 0 1 2 and all n n n one has i k n c z q k for every q 1 ln n n with probability greater than 1 c 1 exp c 2 n 1 n ii for every q 2 ln n n the expected mean width e w k n of k n is bounded by c 3 w z q k as an application we show that the volume radius k n 1 n of a random k n satisfies the bounds c 4 ln 2 n n n k n 1 n c 5 l k ln 2 n n n for all n exp n

characterization of fuzzy measures constructed by means of triangular norms abstract following the ideas presented by the author e p klement j math anal appl 85 1982 543 565 finite t fuzzy measures are introduced t being a measurable triangular norm we show that a t fuzzy measure is always a fuzzy measure as considered earlier e p klement j math anal appl 25 1980 330 339 then we study the relation to the integral with respect to some classical measure finally for some special triangular norms t we give precise characterizations of the corresponding classes of t fuzzy measures

monotone spline transformations for dimension reduction consider a set of data consisting of measurements ofn objects with respect top variables displayed in ann p matrix a monotone transformation of the values in each column represented as a linear combination of integrated basis splines is assumed determined by a linear combination of a new set of values characterizing each row object two different models are used one an eckart young decomposition model and the other a multivariate normal model examples for artificial and real data are presented the results indicate that both methods are helpful in choosing dimensionality and that the eckart young model is also helpful in displaying the relationships among the objects and the variables also results suggest that the resulting transformations are themselves illuminating

groupwise dimension reduction via envelope method the family of sufficient dimension reduction sdr methods that produce informative combinations of predictors or indices are particularly useful for high dimensional regression analysis in many such analyses it becomes increasingly common that there is available a priori subject knowledge of the predictors for example they belong to different groups while many recent sdr proposals have greatly expanded the scope of the methods applicability how to effectively incorporate the prior predictor structure information remains a challenge in this article we aim at dimension reduction that recovers full regression information while preserving the predictor group structure built upon a new concept of the direct sum envelope we introduce a systematic way to incorporate the group information in most existing sdr estimators as a result the reduction outcomes are much easier to interpret moreover the envelope method provides a principled way to build a variety of prior structures into dimension reduc

redundancy analysis for qualitative variables redundancy analysis also called principal components analysis of instrumental variables is a technique for two sets of variables one set being dependent of the other its aim is maximization of the explained variance of the dependent variables by a linear combination of the explanatory variables the technique is generalized to qualitative variables it then gives implicitly a simultaneous optimal scaling of the dependent qualitative variables examples are taken from the dutch life situation survey 1977 using satisfaction with life and happiness as dependent variables the analysis leads to one well being scale defined by the explanatory variables marital status schooling income and activity

cross validation in principal component analysis summary this paper describes a form of cross validation in the context of principal component analysis which has a number of useful aspects as regards multivariate data inspection and description topics covered include choice of dimensionality identification of influential observations and selection of important variables the methods are motivated by and illustrated on a well known data set 1 data set and objectives jeffers 1967 described two detailed multivariate case studies one of which concerned 19 variables measured on each of 40 winged aphids alatee adelges that had been caught in a light trap the 19 variables are listed in table 1 principal component analysis pca was used to examine the structure in the data and if possible to answer the following questions i how many dimensions of the individuals are being measured ii how many distinct taxa are present in the habitat iii which variables among the 19 are redundant for distinguishing between taxa and which must be retained in future work of the 19 variables 14 are length or width measurements four are counts and one anal fold is a presence absence variable scored 0 or 1 in view of this disparity in variable type jeffers elected to standardise the data and thus effect the pca by finding the latent roots and vectors of the correlation rather than covariance matrix of the data the elements of each latent vector provide the coefficients of one of 19 linear combinations of the standardised original variables that successively maximise sample variance subject to being orthogonal with each other and the corresponding latent root is the sample variance of that linear combination the 19 observations for each aphid were subjected to each of these 19 linear transformations to form the 19 principal component scores for that aphid the above questions were then answered as follows i the latent roots of the correlation matrix were as given in table 1 the four largest comprise 73 0 12 5 3 9 and 2 6 respectively of the total variance 19 0 of the standardised variables the dimensionality of the data was therefore taken to be 2 ii when the scores of the first two principal components for the 40 aphids were plotted against orthogonal axes the resulting 40 points divided into four groups as shown in figure 1 hence four distinct species were identified for the aphids iii from consideration of the size of coefficients in the first three principal components it was concluded that only the four variables length of tibia number of ovipositor

the effects on convergence of substituting parameter estimates into u statistics and other families of statistics substituting an estimator in a statistic will often affect its limiting distribution sukhatme 1958 randles 1982 and pierce 1982 all consider the changes if any in the statistic s limiting normal distribution this paper gives conditions for a law of the iterated logarithm for u statistics which have a kernel with an estimator substituted into it it also gives conditions for both strong and weak convergence applications of the theory are illustrated by constructing a sequential test for scale differences with power one the theory also produces convergence results for adaptive m estimators and for cross validation assessment statistics in addition it is shown how to extend lil results to a broad class of statistics with estimators substituted into them by use of the differential in particular a law of the iterated logarithm is described for adaptive l statistics and is illustrated by an example of de wet and van wyk 1979

certification of projection based reduced order modelling in computational homogenisation by the constitutive relation error in this paper we propose upper and lower error bounding techniques for reduced order modelling applied to the computational homogenisation of random composites the upper bound relies on the construction of a reduced model for the stress field upon ensuring that the reduced stress satisfies the equilibrium in the finite element sense the desired bounding property is obtained the lower bound is obtained by defining a hierarchical enriched reduced model for the displacement we show that the sharpness of both error estimates can be seamlessly controlled by adapting the parameters of the corresponding reduced order model

texture analysis using graphs generated by deterministic partially self avoiding walks texture is one of the most important visual attributes for image analysis it has been widely used in image analysis and pattern recognition a partially self avoiding deterministic walk has recently been proposed as an approach for texture analysis with promising results this approach uses walkers called tourists to exploit the gray scale image contexts in several levels here we present an approach to generate graphs out of the trajectories produced by the tourist walks the generated graphs embody important characteristics related to tourist transitivity in the image computed from these graphs the statistical position degree mean and dispersion entropy of two vertices with the same degree measures are used as texture descriptors a comparison with traditional texture analysis methods is performed to illustrate the high performance of this novel approach

parametric bootstrap methods for testing multiplicative terms in gge and ammi models the genotype main effects and genotype by environment interaction effects gge model and the additive main effects and multiplicative interaction ammi model are two common models for analysis of genotype by environment data these models are frequently used by agronomists plant breeders geneticists and statisticians for analysis of multi environment trials in such trials a set of genotypes e g crop cultivars are compared across a range of environments e g locations the gge and ammi models use singular value decomposition to partition genotype by environment interaction into an ordered sum of multiplicative terms this article deals with the problem of testing the significance of these multiplicative terms in order to decide how many terms to retain in the final model we propose parametric bootstrap methods for this problem models with fixed main effects fixed multiplicative terms and random normally distributed errors are considered two methods are derived a full and a simple parametric bootstrap method these are compared with the alternatives of using approximate f tests and cross validation in a simulation study based on four multi environment trials both bootstrap methods performed well with regard to type i error rate and power the simple parametric bootstrap method is particularly easy to use since it only involves repeated sampling of standard normally distributed values this method is recommended for selecting the number of multiplicative terms in gge and ammi models the proposed methods can also be used for testing components in principal component analysis

principal component analysis with external information on both subjects and variables a method for structural analysis of multivariate data is proposed that combines features of regression analysis and principal component analysis in this method the original data are first decomposed into several components according to external information the components are then subjected to principal component analysis to explore structures within the components it is shown that this requires the generalized singular value decomposition of a matrix with certain metric matrices the numerical method based on the qr decomposition is described which simplifies the computation considerably the proposed method includes a number of interesting special cases whose relations to existing methods are discussed examples are given to demonstrate practical uses of the method

small ball probability estimates for log concave measures we establish a small ball probability inequality for isotropic log concave probability measures there exist absolute constants c1 c2 0 such that if x is an isotropic log concave random vector in r with 2 constant bounded by b and if a is a non zero n n matrix then for every e 0 c1 and y r p ax y 2 6 e a hs 6 e c2 b a hs a op 2 where c1 c2 0 are absolute constants

structure identification of fuzzy model abstract the problems of structure identification of a fuzzy model are formulated a criterion for the verification of a structure is discussed using the criterion an algorithm for identifying a structure is suggested further a successive identification algorithm of the parameters is suggested the proposed methods are applied to an example

on tests applied to residuals abstract the null distribution of a test statistic computed on residuals can be affected by the dependence among the residuals it might be expected however that the dependence decreases as the sample size increases so that the limiting null distribution is the same as if the test were based on independent observations the author investigates whether this large sample equivalence holds for tests of correlation scale difference regression and location difference particular emphasis is placed on showing when rank tests based on residuals have a strongly asymptotically distribution free property u statistic theorems are used to investigate these issues

on the geometry of random convex sets between polytopes and zonotopes abstract in this work we study a class of random convex sets that interpolate between polytopes and zonotopes these sets arise from considering a q t h moment q 1 of an average of order statistics of 1 dimensional marginals of a sequence of n n independent random vectors in r n we consider the random model of isotropic log concave distributions as well as the uniform distribution on an l p n sphere 1 p with respect to the cone probability measure and study the geometry of these sets in terms of the support function and mean width we provide asymptotic formulas for the expectation of these geometric functionals which are sharp up to absolute constants our model includes and generalizes the standard one for random polytopes

a multivariate reduced rank growth curve model with unbalanced data a multivariate reduced rank growth curve model is proposed that extends the univariate reducedrank growth curve model to the multivariate case in which several response variables are measured over multiple time points the proposed model allows us to investigate the relationships among a number of response variables in a more parsimonious way than the traditional growth curve model in addition the method is more flexible than the traditional growth curve model for example response variables do not have to be measured at the same time points nor the same number of time points it is also possible to apply various kinds of basis function matrices with different ranks across response variables it is not necessary to specify an entire set of basis functions in advance examples are given for illustration

canonical analysis of contingency tables with linear constraints a generalized least squares approach is presented for incorporating linear constraints on the standardized row and column scores obtained from a canonical analysis of a contingency table the method is easy to implement and may simplify considerably the interpretation of a data matrix the approach is compared to a restricted maximum likelihood procedure

affine quermassintegrals of random polytopes abstract a question related to some conjectures of lutwak about the affine quermassintegrals of a convex body k in r n asks whether for every convex body k in r n and all 1 k n k k vol n k 1 n g n k vol k p f k n d n k f 1 k n c n k where c 0 is an absolute constant we provide an affirmative answer for some broad classes of random polytopes we also discuss upper bounds for k k when k b 1 n the unit ball of l 1 n and explain how this special instance has implications for the case of a general unconditional convex body k

multiplicative interaction in generalized linear models bilinear oi biadditive multiplicative models for interaction in two way tables provide the major means for studyiing g eniotype by cnvlironllllcmlt initer actioni pioblems in applicatioins the typical accompanying assumptions are those of a noirnally clistributecl error and an identity link these assumptioins are uninecessarily restrictive introduction of multiplictive terms foi i nteraction in generalized linear models removes these restrictions parameter estimates can be obtained by ain iterative pr ocess of altcrinatinlg generalized irow anid columin iregr essions withlini a quasi likclihood set up the best known examples of this class of generalized additive main effects and multiplicative interaction effects gammi models are the ammi models gauch 1988 biomiietrics 44 705 715 and goodmain s rc association imiodels goodman 1981 joil z lr7 of thle amnericain stcatistical associaition 76 320 334 the multiplicative inteiactioin part of gammi models canl be visualized througlh biplots two applications of gammi models are presented for data coming from planlt breeding experimenits the first illustration deals with a log bilinear model for couint data with extra poisson variation the second ililustiatioin concerniis a logit bilinear model for disease incidence data with a special type of variance function anl extensioin of a model presented by wedderbunii 1974 biomiietrikal 61 439 447

principal components analysis of sampled functions this paper describes a technique for principal components analysis of data consisting ofn functions each observed atp argument values this problem arises particularly in the analysis of longitudinal data in which some behavior of a number of subjects is measured at a number of points in time in such cases information about the behavior of one or more derivatives of the function being sampled can often be very useful as for example in the analysis of growth or learning curves it is shown that the use of derivative information is equivalent to a change of metric for the row space in classical principal components analysis the reproducing kernel for the hilbert space of functions plays a central role and defines the best interpolating functions which are generalized spline functions an example is offered of how sensitivity to derivative information can reveal interesting aspects of the data

lp moments of random vectors via majorizing measures abstract for a random vector x in r n we obtain bounds on the size of a sample for which the empirical pth moments of linear functionals are close to the exact ones uniformly on a convex body k r n we prove an estimate for a general random vector and apply it to several problems arising in geometric functional analysis in particular we find a short lewis type decomposition for any finite dimensional subspace of l p we also prove that for an isotropic log concave random vector we only need n p 2 log n sample points so that the empirical pth moments of the linear functionals are almost isometrically the same as the exact ones we obtain a concentration estimate for the empirical moments the main ingredient of the proof is the construction of an appropriate majorizing measure to bound a certain gaussian process

generalized bilinear models generalized bilinear models are presented for the statistical analysis of two way arrays these models combine bilinear models and generalized linear modeling and yield a family of models that includes many existing models as well as suggest other potentially useful ones this approach both unifies and extends models for two way arrays including the ability to treat response and explanatory variables differently in the models and the incorporation of external information about the variables directly into the analysis a unifying framework for the generalized bilinear models is provided by considering four particular cases which have been proposed and used in the existing statistical literature a three step procedure is proposed to analyze data sets by generalized bilinear models two data sets of different nature are analyzed

functional multiple set canonical correlation analysis we propose functional multiple set canonical correlation analysis for exploring associations among multiple sets of functions the proposed method includes functional canonical correlation analysis as a special case when only two sets of functions are considered as in classical multiple set canonical correlation analysis computationally the method solves a matrix eigen analysis problem through the adoption of a basis expansion approach to approximating data and weight functions we apply the proposed method to functional magnetic resonance imaging fmri data to identify networks of neural activity that are commonly activated across subjects while carrying out a working memory task

fitting a regression model for genotype by environment data on heading dates in grasses by methods for nonlinear mixed models summary the analysis of agricultural crop variety trials is usually complicated by the presence of genotype by environment interaction a number of methods and models have been proposed to tackle this problem one of the most common methods is the regression approach due to yates and cochran 1938 journal of agricultural science28 556 580 in which performances of genotypes in the environments are regressed onto environmental means the underlying regression model contains a multiplicative term with two unknown parameters one for genotypes and one for environments in the present paper the model is modified by exchanging the role of genotypes and environments various diagnostic plots show that this modified model is adequate for a data set on heading dates in the grass species dactylis glomerata if environments are considered as a random factor while genotypes are taken as fixed the model falls into the class of nonlinear mixed models recently a number of procedures have been suggested for this class of models which are based on first order taylor series expansion alternatively the model can be estimated by maximum likelihood this paper discusses the application of these methods for estimating parameters of the model

dimensionality estimation for optimal detection of functional networks in bold fmri data estimation of the intrinsic dimensionality of fmri data is an important part of data analysis that helps to separate the signal of interest from noise we have studied multiple methods of dimensionality estimation proposed in the literature and used these estimates to select a subset of principal components that was subsequently processed by linear discriminant analysis lda using simulated multivariate gaussian data we show that the dimensionality that optimizes signal detection in terms of the receiver operating characteristic roc metric goes through a transition from many dimensions to a single dimension as a function of the signal to noise ratio this transition happens when the loci of activation are organized into a spatial network and the variance of the networked task related signals is high enough for the signal to be easily detected in the data we show that reproducibility of activation maps is a metric that captures this switch in intrinsic dimensionality except for reproducibility all of the methods of dimensionality estimation we considered failed to capture this transition optimization of bayesian evidence minimum description length supervised and unsupervised lda prediction and stein s unbiased risk estimator this failure results in sub optimal roc performance of lda in the presence of a spatially distributed network and may have caused lda to underperform in many of the reported comparisons in the literature using real fmri data sets including multi subject group and within subject longitudinal analysis we demonstrate the existence of these dimensionality transitions in real data

a robust approach to independent component analysis of signals with high level noise measurements we propose a robust approach for independent component analysis ica of signals where observations are contaminated with high level additive noise and or outliers the source signals may contain mixtures of both sub gaussian and super gaussian components and the number of sources is unknown our robust approach includes two procedures in the first procedure a robust prewhitening technique is used to reduce the power of additive noise the dimensionality and the correlation among sources a cross validation technique is introduced to estimate the number of sources in this first procedure in the second procedure a nonlinear function is derived using the parameterized t distribution density model this nonlinear function is robust against the undue influence of outliers fundamentally moreover the stability of the proposed algorithm and the robust property of misestimating the parameters kurtosis have been studied by combining the t distribution model with a family of light tailed distributions sub gaussian model we can separate the mixture of sub gaussian and super gaussian source components through the analysis of artificially synthesized data and real world magnetoencephalographic meg data we illustrate the efficacy of this robust approach

dimension estimation in noisy pca with sure and random matrix theory principal component analysis pca is one of the best known methods for dimensionality reduction perhaps the most important problem in using pca is to determine the number of principal components pcs or equivalently choose the rank of the loading matrix many methods have been proposed to deal with this problem but almost all of them fail in the important practical case when the number of observation is comparable to the number of variables i e the realm of random matrix theory rmt in this paper we propose to use stein s unbiased risk estimator sure to estimate with some assistance from rmt the number of principal components the method is applied both on simulated and real functional magnetic resonance imaging fmri data and compared to bic and the laplace method

relationships among several methods of linearly constrained correspondence analysis this paper shows essential equivalences among several methods of linearly constrained correspondence analysis they include fisher s method of additive scoring hayashi s second type of quantification method ter braak s canonical correspondence analysis nishisato s type of quantification method ter braak s canonical correspondence analysis nishisato s anova of categorical data correspondence analysis of manipulated contingency tables bockenholt and bockenholt s least squares canonical analysis with linear constraints and van der heijden and meijerink s zero average restrictions these methods fall into one of two classes of methods corresponding to two alternative ways of imposing linear constraints the reparametrization method and the null space method a connection between the two is established through khatri s lemma

confidence regions for indscal using the jackknife and bootstrap techniques bootstrap and jackknife techniques are used to estimate ellipsoidal confidence regions of group stimulus points derived from indscal the validity of these estimates is assessed through monte carlo analysis asymptotic estimates of confidence regions based on a multiscale solution are also evaluated our findings suggest that the bootstrap and jackknife techniques may be used to provide statements regarding the accuracy of the relative locations of points in space our findings also suggest that multiscale asymptotic estimates of confidence regions based on small samples provide an optimistic view of the actual statistical reliability of the solution

random vectors in the isotropic position abstract letybe a random vector in r n satisfying e y y id letmbe a natural number and lety1 ymbe independent copies ofy we study the question of approximation of the identity operator by finite sums of the tensorsyi yi we prove that for some absolute constantc e 1 m i 1 m y i y i id c log n m e y log m 1 log m provided that the last expression is smaller than 1 we apply this estimate to improve a result of bourgain concerning the number of random points needed to bring a convex body into a nearly isotropic position

analyzing genotype environment data by mixed models with multiplicative terms where yij mean yield of ith genotype in jth environment general mean gi effect of ith genotype ej effect of jth environment ge ij interaction of ith genotype and jth environment eij random error associated with means yij assumed to be distributed as n o ac2 this model is appropriate for the analysis of means from equally replicated data with homoscedastic errors in an analysis of g x e data it may be useful to fit a more specific model describing the interaction the most common of such models is the regression model first suggested by yates and cochran 1938 which was further elaborated by finlay and wilkinson 1963 and eberhart and russell 1966 it may be written as

model selection and validation for yield trials with interaction summary the additive main effects and multiplicative interaction ammi model first applies the additive analysis of variance anova model to two way data and then applies the multiplicative principal components analysis pca model to the residual from the additive model that is to the interaction ammi analysis of yield trial data is a useful extension of the more familiar anova pca and linear regression procedures particularly given a large genotype by environment interaction model selection and validation are considered from both predictive and postdictive perspectives using data splitting and f tests respectively a new york soybean yield trial serves as an example yield trials generate observations of yield ordinarily replicated for a number of genotypes grown in a number of environments site year combinations often the data are rather noisy with a standard deviation for plot yields in excess of 25 of the mean an additional challenging feature of these data is the frequent presence of important and complex genotype by environment ge interactions plant breeders use yield trials to identify promising genotypes and agronomists use them to make recommendations for farmers the level of success in meeting these goals depends critically on two factors i the accuracy of yield estimates and ii the magnitudes of genotype by site genotype by year and genotype by site by year interactions talbot 1984 in essence these two factors reflect within trial accuracy and between trial predictability this paper addresses only the first concern nothing is said regarding between trial predictability or agrotechnology transfer other than to observe that success with withintrial accuracy is a necessary prelude to success with between trial predictability the within trial accuracy of a statistical model may be assessed by two fundamentally different criteria postdictive success concerns a model s fit to its own data whereas predictive success concerns the fit between a model constructed using part of the data and validation data not used in modelling in either case the statistical setting is that of an incompletely specified model where empirical considerations enter into the decision to include a given potential source in the model or alternatively to relegate it to the model s residual bancroft 1964 whenever the data are noisy postdiction and prediction are different tasks and in general the model chosen by predictive criteria will be different and simpler than the model chosen by postdictive criteria because the several replicates of a yield trial constitute a noisy sample a model chosen by predictive criteria may be expected to provide yield estimates that are closer to the true means and to validation observations than will a model chosen by postdictive criteria this claim is readily subjected to experimental test in what follows the relative performances of statistical models chosen

a note on smooth approximation capabilities of fuzzy systems modeling and prediction in some systems requires the simultaneous approximation of mappings and their derivatives to a certain finite order in this paper universal approximation capabilities of fuzzy systems are extended to this situation by showing the denseness of some general classes of fuzzy models in appropriate function spaces where distance between functions is defined in terms of their derivatives requirements are generally very mild and are often limited to imposing sufficient differentiability conditions on the classes of fuzzy systems to be used the cases covered by this paper include additive fuzzy systems with gaussian membership functions and general additive fuzzy models with realistic membership functions some potential application fields are considered including examples from economics and time series prediction

a note on dual scaling of successive categories data three methods for dual scaling of successive categories data are formulated they are mathematically equivalent and provide an appropriate technique to determine stimulus values and category boundaries on a joint scale a problem that nishisato s method 1980a failed to handle

cross validatory choice of the number of components from a principal component analysis a method is described for choosing the number of components to retain in a principal component analysis when the aim is dimensionality reduction the correspondence between principal component analysis and the singular value decomposition of the data matrix is used the method is based on successively predicting each element in the data matrix after deleting the corresponding row and column of the matrix and makes use of recently published algorithms for updating a singular value decomposition these are very fast which renders the proposed technique a practicable one for routine data analysis

simultaneous solution of fuzzy models an application to economic equilibrium analysis abstract the fuzzy modeling technology which uses a rule based description of the relationship between variables is discussed we then turn to the problem of the simultaneous solution of relationships involving fuzzy models we illustrate this problem by considering the equilibrium problem in economics in this case we have two relationships one between price and supply and the other between price and demand and we desire to find the price for which the supply equals the demand it is shown that if the fuzzy models have antecedent fuzzy sets which are of the trapezoidal triangular type then a solution technique can be found

on the issue of defuzzification and selection based on a fuzzy set abstract we are concerned with the problem of selecting a crisp element based on information provided by a fuzzy set a problem which manifests itself in the defuzzification step in fuzzy logic controllers we provide a unifying approach to this selection process among other characteristics this unification puts the defuzzification methods of mean of maxima and center of gravity in the same framework we show that this selection can be viewed as a three step operation transformation of the decision fuzzy set normalization to probability distribution selection based on the probability distribution a number of different procedures for selection are discussed

multiplicative models for interaction in three way anova with applications to plant breeding summary in plant breeding multiplicative models for two way analysis of variance interaction have become a general means of describing genotype by environment interaction these models offer parsimonious descriptions and facilitate interpretations in biological terms a disadvantage of the prevailing dominance of two way multiplicative models is that data with more complicated environmental structure are often forced to fit the two way framework as a partial solution to this problem three way multiplicative models are presented that can be used in addition to the more familiar two way multiplicative models most importantly a three way generalization is given of the twoway singular value decomposition that can be applied for closer inspection of three way analysis of variance interaction much in the same way as its two way counterpart is used for two way interaction two real data sets are analyzed to illustrate how two and three way multiplicative models for interaction jointly provide parsimonious descriptions of various types of genotype byenvironment interaction these parsimonious descriptions are often open to meaningful biological interpretations as well

nonlinear regression analysis of the joint regression model the joint regression model for two way data assumes a linear relation between a continuous response and column effects standard methods for fitting the model condition on estimates of the column effects but including column effects as covariates in the model results in a nonlinear estimation problem we use methods from nonlinear regression to give estimation and inference for joint regression the method is iterative but computationally requires only multiple regression software inference based on asymptotic results from nonlinear regression is conceptually clear and statistically valid we show how to carry out estimation and inference and illustrate with a real example

influence in principal components analysis summary in linear regression the theoretical influence function and the various sample versions of it have an established place as diagnostic tools these same functions are developed here to provide methods for the detection of influential observations in principal components analysis the perturbation theory of real symmetric matrices unifies this development some interesting points of contrast with the regression case are noted and explained theoretically

a new family of constrained principal component analysis cpca abstract several decompositions of the orthogonal projector p x x x x x are proposed with a prospect of their use in constrained principal component analysis cpca in cpca the main data matrix x is first decomposed into several additive components by the row side and or column side predictor variables g and h the decomposed components are then subjected to singular value decomposition svd to explore structures within the components unlike the previous proposal the current proposal ensures that the decomposed parts are columnwise orthogonal and stay inside the column space of x mathematical properties of the decompositions and their data analytic implications are investigated extensions to regularized pca are also envisaged considering analogous decompositions of ridge operators

a note on the additive and multiplicative models in two way contingency tables summary another look at the mandel type least squares fitting of a mixed additive and multiplicative model is discussed for the analysis of two way contingency tables the method is carried out by a two stage process i fit the additive part of the model then take residuals and ii fit the multiplicative part to the residuals

a maximum likelihood method for fitting the wandering vector model after introducing some extensions of a recently proposed probabilistic vector model for representing paired comparisons choice data an iterative procedure for obtaining maximum likelihood estimates of the model parameters is developed the possibility of testing various hypotheses by means of likelihood ratio tests is discussed finally the algorithm is applied to some existing data sets for illustrative purposes

analyzing rectangular tables by joint and constrained multidimensional scaling abstract this paper selectively discusses multidimensional scaling methods addressing both theory and applications most of the discussion is focussed upon the analysis of rectangular tables the term joint mds is introduced for an analysis which gives a meaningful joint representation of the set of row and the set of column objects of such a table so that the dissimilarities either within sets or between sets are optimally approximated from this perspective it is possible to give an unified account of the relationships between classical multidimensional scaling principal components analysis and correspondence analysis by imposing constraints the general model can be structurally changed in a number of ways for example it can be made discrete that is in principle it subsumes subset models and quadratic assignment as special cases it can also be made more flexible in the sense that external information may be used in order to tune the analysis to specific hypotheses finally an elegant joint mds model called multidimensional unfolding is discussed as a technique unfolding unfortunately suffers from being extremely sensitive to ill conditioned data a new way of coping with this problem is indicated and an application is given in which the bootstrap technique succesfully establishes the stability of the results