a generalization of the deformed algebra of quantum group su 2 q for hopf algebra a generalization of the deformation of lie algebra of the su 2 group is established for the hopf algebra by modifying the j3 component in all of its defining commutators the modification is carried out in terms of a polynomial f of j3 and the q deformation parameter which contains the known q deformation functionals as its particular cases

regression analysis for a functional response functional responses are encountered when units are observed over time although the whole function itself is not observed a sufficiently large number of evaluations as is common with modern recording equipment are assumed to be available functional regression analysis relates the smooth functional response y t to known covariates x by a linear combination of parameter functions t which are to be estimated the model takes the standard form y t x t t t this approach provides an alternative to standard longitudinal data methods used in the biological sciences where less and noisier data necessitate parametric modeling the methodology is illustrated by an application in ergonomics

combining registration and fitting for functional models a registration method can be defined as a process of aligning features of a sample of curves by monotone transformations of their domain the aligned curves exhibit only amplitude variation and the domain transformations called warping functions capture the phase variation in the original curves in this article we precisely define a new type of registration process in which the warping functions optimize the fit of a principal components decomposition to the aligned curves the principal components are effectively the features that this process aligns we discuss the relationship of registration to closure of a function space under convex operations and define consistency for registration methods we define an explicit decomposition of functional variation into amplitude and phase partitions and develop an algorithm for combining registration with principal components analysis and apply it to simulated and real data

the q deformation of symmetry transformations preserving the radial form of the schrodinger equation proofs are given that the symmetry transformations preserving the radial form of the schrodinger equation lead to matching conditions which are essentially the same as the classical ones power law potentials are treated as illustrative examples in particular the dimensional q deformed coulomb system is converted into a q deformed harmonic oscillator acting again in space dimensions we also found that q deformed 1 n energy formulae are covariant under such transformations to first 1 n order

lipschitzian de morgan triplets of fuzzy connectives after studying lipschitzian strong negations lipschitzian de morgan triplets t s n where t is a triangular norm s a triangular conorm and n a strong negation are investigated the relationships between the best lipschitzian constants of functions t s and n are given several examples are included

synchronizing sample curves nonparametrically more and more often the outcome of a study is not a random variable but a noisy function for each experimental unit resulting in a sample of curves typically the individual curves vary not only in amplitude or intensity but also with respect to the time axis different subjects experience certain events sooner or later analyzing such data involves finding out the time changes or curve registration among curves following our previous work where modified dynamic time warping is applied to align two curves we formulate a global minimization problem to align all curves in a sample and to compute the aligned average curve algorithms for solving the minimization problem are presented and tested with simulated and real data the test results are promising the method which involves kernel smoothing of regression functions estimates the time changes and the average of the aligned curves from noisy data large sample asymptotics is derived

incorporating parametric effects into functional principal components analysis summary the ideas of functional principal component analysis are extended to deal with data that are hybrids of functional and parametric effects the parametric effects may be more general than just the addition of a multiple of a given function a detailed development is given in the case of shifts of the time axis for functions observed on a periodic interval and some remarks are made for the extension to a far more general case given data a procrustes fitting method can be used to estimate the parametric effects several possible ways of treating the estimated parameter values are discussed the methods are illustrated by reference to temperature data at 35 canadian weather stations

on properties of functional principal components analysis summary functional data analysis is intrinsically infinite dimensional functional principal component analysis reduces dimension to a finite level and points to the most significant components of the data however although this technique is often discussed its properties are not as well understood as they might be we show how the properties of functional principal component analysis can be elucidated through stochastic expansions and related results our approach quantifies the errors that arise through statistical approximation in successive terms of orders n 1 2 n 1 n 3 2 where n denotes sample size the expansions show how spacings among eigenvalues impact on statistical performance the term of size n 1 2 illustrates first order properties and leads directly to limit theory which describes the dominant effect of spacings thus for example spacings are seen to have an immediate first order effect on properties of eigenfunction estimators but only a second order effect on eigenvalue estimators our results can be used to explore properties of existing methods and also to suggest new techniques in particular we suggest bootstrap methods for constructing simultaneous confidence regions for an infinite number of eigenvalues and also for individual eigenvalues and eigenvectors

stability of weighted penalty based aggregation functions in many practical applications the need arises to aggregate data of varying dimension following from the self identity property some recent studies have looked at the stability of aggregation operators in terms of their behavior as the dimensionality is increased from n 1 to n we use the penalty based representation of aggregation functions in order to investigate the conditions for weighting vectors associated with some important weighted families extending on the results already established for quasi arithmetic means in particular we obtain results for quasi medians and functions that involve a reordering of the inputs such as the owa and order statistics

computing reproducing kernels with arbitrary boundary constraints smoothing with l splines involves the use of the penalty term x 2 2 int lx 2 t dt where l is an arbitrary linear differential operator of order m it also involves the choice of m co

a bayesian measurement error model for misaligned radiographic data an understanding of the inherent variability in micro computed tomography micro ct data is essential to tasks such as statistical process control and the validation of radiographic simulation tools these data present unique challenges to variability analysis due to the relatively low resolution of radiographs and also due to minor variations from run to run which can result in misalignment or magnification changes between repeated measurements of a sample such positioning changes artificially inflate the variability of the data in ways that mask true physical phenomena we present a novel bayesian nonparametric regression model that incorporates both additive and multiplicative measurement error in addition to heteroscedasticity to address this problem we use this model to assess the effects of sample thickness and sample position on measurement variability for an aluminum specimen supplementary materials for this article are available online

assessing the finite dimensionality of functional data summary if a problem in functional data analysis is low dimensional then the methodology for its solution can often be reduced to relatively conventional techniques in multivariate analysis hence there is intrinsic interest in assessing the finite dimensionality of functional data we show that this problem has several unique features from some viewpoints the problem is trivial in the sense that continuously distributed functional data which are exactly finite dimensional are immediately recognizable as such if the sample size is sufficiently large however in practice functional data are almost always observed with noise for example resulting from rounding or experimental error then the problem is almost insolubly difficult in such cases a part of the average noise variance is confounded with the true signal and is not identifiable however it is possible to define the unconfounded part of the noise variance this represents the best possible lower bound to all potential values of average noise variance and is estimable in low noise settings moreover bootstrap methods can be used to describe the reliability of estimates of unconfounded noise variance under the assumption that the signal is finite dimensional motivated by these ideas we suggest techniques for assessing the finiteness of dimensionality in particular we show how to construct a critical point 9q such that if the distribution of our functional data has fewer than q 1 degrees of freedom then we should be willing to assume that the average variance of the added noise is at least 9q if this level seems too high then we must conclude that the dimension is at least q 1 we show that simpler more conventional techniques based on hypothesis testing are generally not effective

functional convex averaging and synchronization for time warped random curves data that can be best described as a sample of curves are now fairly common in science and engineering when the dynamics of development growth or response over time are at issue subjects or experimental units may experience events at different temporal paces for functional data where trajectories may be individually time transformed it is usually inadequate to use commonly used sample statistics such as the cross sectional mean or median or the cross sectional sample variance if one observes time warped curve data i e random curves or random trajectories that exhibit random transformations of the time scale then the usual l2 norm and metric typically are inadequate one may then consider subjecting each observed curve to a time transformation in an attempt to reverse the warping of the time scale before further statistical analysis dynamic time warping alignment curve registration and landmark based methods have been put forward with the goal of finding adequate empirical time transforma

assessing extrema of empirical principal component functions the difficulties of estimating and representing the distributions of functional data mean that principal component methods play a substantially greater role in functional data analysis than in more conventional finite dimensional settings local maxima and minima in principal component functions are of direct importance they indicate places in the domain of a random function where influence on the function value tends to be relatively strong but of opposite sign we explore statistical properties of the relationship between extrema of empirical principal component functions and their counterparts for the true principal component functions it is shown that empirical principal component funcions have relatively little trouble capturing conventional extrema but can experience difficulty distinguishing a shoulder in a curve from a small bump for example when the true principal component function has a shoulder the probability that the empirical principal component function has instead a bump is approximately equal to 1 2 we suggest and describe the performance of bootstrap methods for assessing the strength of extrema it is shown that the subsample bootstrap is more effective than the standard bootstrap in this regard a bootstrap likelihood is proposed for measuring extremum strength exploratory numerical methods are suggested

convergence and consistency results for self modeling nonlinear regression on propose une methode d estimation simultanee et un algorithme iteratif simplifiant le calcul des estimateurs

lipschitz continuity of copulas w r t lp norms abstract in the framework of the stability analysis of real functions the lipschitz continuity of copulas w r t l p norms is investigated emphasis is put on 1 lipschitz continuity as it is the strongest type possible for copulas after illustrating how to identify 1 lipschitz copulas w r t some l p norm the preservation of this 1 lipschitz continuity by certain construction methods such as ordinal sums and convex sums is demonstrated special attention is given to three important classes of copulas namely archimedean copulas extreme value copulas and archimax copulas

estimating multiple pathways of object growth using nonlongitudinal image data this article presents an infinite bayesian mixture of monotonic regression models for analyzing multiple growth pathways of star shaped objects growing over time by using nonlongitudinal data a motivating example is the analysis of nanocrystal growth processes a radius function representation used for the outlines of star shaped objects allows us to represent an object growth i e expansion of outlines by an increasing sequence of random radius functions we propose a monotonic regression model to fit the increasing sequence to nonlongitudinal data ensuring that the fitted radius functions always monotonically increase in the sequence to model the multiple pathways of object growth we propose the dirichlet infinite location mixture of multiple monotonic regression models and use a block gibbs sampler as a numerical solver we apply the proposed model to nonlongitudinal sets of microscopic image data for inferring multiple growth pathways of nanocrystals comparing the inference result with real nan

registration for exponential family functional data we consider the problem of aligning curves from exponential family distributions the approach is based on the combination of alignment and functional principal components analysis and is facilitated by recent extensions of fpca to non gaussian settings our work is motivated by the study of physical activity using accelerometers wearable devices that provide around the clock monitoring of activity and produce non gaussian measurements we apply the proposed methods to activity counts using a poisson distribution and to a binary active vs inactive indicator using a binomial distribution after alignment the trajectories show clear peaks of activity in the morning and afternoon with a dip in the middle of the day

nonparametric maximum likelihood estimation for shifted curves the analysis of a sample of curves can be done by self modelling regression methods within this framework we follow the ideas of nonparametric maximum likelihood estimation known from event history analysis and the counting process set up we derive an infinite dimensional score equation and from there we suggest an algorithm to estimate the shape function for a simple shape invariant model the nonparametric maximum likelihood estimator that we find turns out to be a nadaraya watson like estimator but unlike in the usual kernel smoothing situation we do not need to select a bandwidth or even a kernel function since the score equation automatically selects the shape and the smoothing parameter for the estimation we apply the method to a sample of electrophoretic spectra to illustrate how it works

an integrated approach to ladder and shift operators for the morse oscillator radial coulomb and radial oscillator potentials the morse oscillator radial coulomb and radial harmonic oscillator problems can be solved exactly using a variety of algebraic methods these problems correspond to different realizations of the so 2 1 algebra and a comparison of the generators of the algebra may be used to identify mappings between each pair of systems the resultant transition operators act as ladder or energy changing operators in the cases of the coulomb and harmonic oscillator potentials whereas they act as shift operators acting at constant energy in the case of the morse potential this is a consequence of the so 2 1 dynamical symmetry whereby the morse hamiltonian is expressible solely in terms of the casimir operator of the algebra an alternative algebraic approach the use of the method of supersymmetric quantum mechanics or factorization produces in each case a set of shift operators relations between the various ladder and shift operators may be identified by means of the appropriate mappings and these results can be generalized so as to relate the one dimensional morse oscillator to the radial coulomb and radial harmonic oscillator potentials involving an arbitrary number of angular dimensions

searching for structure in curve samples abstract the shape of a regression curve can to a large extent be characterized by the succession of structural features like extrema inflection points and so on when analyzing a sample of regression curves it is often important to know at an early stage of data analysis which structural features are occurring consistently in each curve of the sample such a definition is usually not easy due to substantial interindividual variation both in the x and the y axis and due to the influence of noise a method is proposed for identifying typical features without relying on an a priori specified functional model for the curves the approach is based on the frequencies of occurrence of structural features as for example maxima in the curve sample along the x axis important tools are nonparametric regression and differentiation and kernel density estimation apart from a theoretical foundation the usefulness of the method is documented by application to two interesting biomedical areas growth and develop

bayesian hierarchical curve registration functional data often exhibit a common shape but with variations in amplitude and phase across curves the analysis often proceeds by synchronization of the data through curve registration in this article we propose a bayesian hierarchical model for curve registration our hierarchical model provides a formal account of amplitude and phase variability while borrowing strength from the data across curves in the estimation of the model parameters we discuss extensions of the model by using penalized b splines in the representation of the shape and time transformation functions and by allowing temporal misalignment of the curves we discuss applications of our model to simulated data as well as to two data sets in particular we use our model in a nonstandard analysis aimed at investigating regulatory network in time course microarray data

statistical tools to analyze data representing a sample of curves the paper is concerned with data representing a sample of smooth curves which can be considered as independent realizations of an underlying biological chemical ldots process such samples of curves often possess the following features there is a typical structural pattern common to all curves of the sample on the other hand individual realizations of the typical shape show different dynamics and intensity in particular typical peaks are shifted from individual to individual differences in dynamics complicate the analysis of samples of curves for example the cross sectional average usually does not reflect an average pattern due to shifts structure is smeared or might even disappear our approach consists in synchronizing the individual curves before determining the average or any further statistics pointwise averaging of the synchronized curves then leads to an average curve which represents the common structure with average dynamics and average intensity the method requires the introduction of new statistical objects they are defined mathematically their properties are discussed and possible estimators are proposed the asymptotic bias and variance of the estimators are derived an application to visually evoked brain potentials illustrates the approach

alignment of curves by dynamic time warping when studying some process or development in different subjects or units be it biological chemical or physical we usually see a typical pattern common to all curves yet there is variation both in amplitude and dynamics between curves following some ideas of structural analysis introduced by kneip and gasser we study a method dynamic time warping with a proper cost function for estimating the shift or warping function from one curve to another to align the two functions for some models this method can identify the true shift functions if the data are noise free noisy data are smoothed by a nonparametric function estimate such as a kernel estimate it is shown that the proposed estimator is asymptotically normal and converges to the true shift function as the sample size per subject goes to infinity some simulation results are presented to illustrate the performance of this method

some tools for functional data analysis multivariate data analysis permits the study of observations which are finite sets of numbers but modern data collection situations can involve data or the processes giving rise to them which are functions functional data analysis involves infinite dimensional processes and or data the paper shows how the theory of l splines can support generalizations of linear modelling and principal components analysis to samples drawn from random functions spline smoothing rests on a partition of a function space into two orthogonal subspaces one of which contains the obvious or structural components of variation among a set of observed functions and the other of which contains residual components this partitioning is achieved through the use of a linear differential operator and we show how the theory of polynomial splines can be applied more generally with an arbitrary operator and associated boundary constraints these data analysis tools are illustrated by a study of variation in temperature precipitation patterns among some canadian weather stations

q deformed ladder and shift operators for three exactly solvable potentials obeying so 2 1 symmetry the quantum deformation algebra soq 2 1 is studied and applied to derive the q analogues of the ladder and shift operators for the radial coulomb radial harmonic oscillator and morse oscillator potentials the q deformed operators in all three cases are found to act like shift operators called q shift operators their possible similarity with the quasi shift operators arising in supersymmetric quantum mechanics or factorization of the radial harmonic oscillator is also pointed out

distribution on warp maps for alignment of open and closed curves alignment of curve data is an integral part of their statistical analysis and can be achieved using model or optimization based approaches the parameter space is usually the set of monotone con

on the use of aggregation operations in information fusion processes this position paper discusses the role of the existing body of fuzzy set aggregation operations in various kinds of problems where the process of fusion of items coming from several sources is central several kinds of membership functions can be useful according to the nature of the information to be merged numerical vs ordinal inputs preferences vs uncertain data observations vs constraints in each case some aggregation operations look more plausible or feasible than others the aim of this discussion is to suggest directions for putting at work the results of recent mathematical investigations in the structure of aggregation operations

a multiresolution approach to time warping achieved by a bayesian prior posterior transfer fitting strategy summary warping is an approach to the reduction and analysis of phase variability in functional observations by applying a smooth bijection to the function argument we propose a natural representation of warping functions in terms of a new type of elementary functions named warping component functions or warplets which are combined into the warping function by composition the inverse warping function is trivial and explicit to obtain a sequential bayesian estimation strategy is introduced which fits a series of models and transfers the posterior of the previous fit into the prior of the next fit model selection is based on a warping analogue to wavelet thresholding combined with bayesian inference

on the q oscillator and the quantum algebra suq 1 1 the authors discuss some problems of the q deformed harmonic oscillator different q bosonisations of the quantum suq 1 1 algebra are given and the corresponding infinite dimensional representations of discrete series are analysed

resolution adaptive fixed rank kriging abstractthe spatial random effects model is flexible in modeling spatial covariance functions and is computationally efficient for spatial prediction via fixed rank kriging frk however the model depends on a class of basis functions which if not selected properly may result in unstable or undesirable results additionally the maximum likelihood ml estimates of the model parameters are commonly computed using an expectation maximization em algorithm which further limits its applicability when a large number of basis functions are required in this research we propose a class of basis functions extracted from thin plate splines the functions are ordered in terms of their degrees of smoothness with higher order functions corresponding to larger scale features and lower order ones corresponding to smaller scale details leading to a parsimonious representation of a nonstationary spatial covariance function with the number of basis functions playing the role of spatial resolution the proposed

self modelling warping functions the paper introduces a semiparametric model for functional data the warping functions are assumed to be linear combinations ofqcommon components which are estimated from the data hence the name self modelling even small values ofqprovide remarkable model flexibility comparable with nonparametric methods at the same time this approach avoids overfitting because the common components are estimated combining data across individuals as a convenient by product component scores are often interpretable and can be used for statistical inference an example of classification based on scores is given abstract from author

template priors in bayesian curve registration in experiments where observations on each experimental unit are functional in nature it is often the case that in addition to variability along the horizontal axis height or amplitude variabilit

smoothed functional principal components analysis by choice of norm the principal components analysis of functional data is often enhanced by the use of smoothing it is shown that an attractive method of incorporating smoothing is to replace the usual l 2 orthonormality constraint on the principal components by orthonormality with respect to an inner product that takes account of the roughness of the functions the method is easily implemented in practice by making use of appropriate function transforms fourier transforms for periodic data and standard principal components analysis programs several alternative possible interpretations of the smoothed principal components as obtained by the method are presented some theoretical properties of the method are discussed the estimates are shown to be consistent under appropriate conditions and asymptotic expansion techniques are used to investigate their bias and variance properties these indicate that the form of smoothing proposed is advantageous under mild conditions indeed milder than those for existing methods of smoothed functional principal components analysis the choice of smoothing parameter by cross validation is discussed the methodology of the paper is illustrated by an application to a biomechanical data set obtained in the study of the behaviour of the human thumb forefinger system

on lipschitz properties of generated aggregation functions this article discusses lipschitz properties of generated aggregation functions such generated functions include triangular norms and conorms quasi arithmetic means uninorms nullnorms and continuous generated functions with a neutral element the lipschitz property guarantees stability of aggregation operations with respect to input inaccuracies and is important for applications we provide verifiable sufficient conditions to determine when a generated aggregation function holds the k lipschitz property and calculate the lipschitz constants of power means we also establish sufficient conditions which guarantee that a generated aggregation function is not lipschitz we found the only 1 lipschitz generated function with a neutral element email protected 0 1

estimating and depicting the structure of a distribution of random functions summary we suggest a nonparametric approach to making inference about the structure of distributions in a potentially infinite dimensional space for example a function space and displaying information about that structure it is suggested that the simplest way of presenting the structure is through modes and density ascent lines the latter being the projections into the sample space of the curves of steepest ascent up the surface of a functionaldata density modes are always points in the sample space and ascent lines are always one parameter structures even when the sample space is determined by an infinite number of parameters they are therefore relatively easily depicted our methodology is based on a functional form of an iterative data sharpening algorithm