Crisp and fuzzy <Emphasis Type="Italic">k</Emphasis>-means clustering algorithms for multivariate functional data

Functional data analysis, as proposed by Ramsay (Psychometrika 47:379–396, 1982), has recently attracted many researchers. The most popular approach taken in recent studies of functional data has been the extension of statistical methods for the analysis of usual data to that of functional data (e.g., Ramsay and Silverman in Functional data Analysis Springer, Berlin Heidelberg New York, 1997, Applied functional data analysis: methods and case studies. Springer, Berlin Heidelberg New York, 2002; Mizuta in Proceedings of the tenth Japan and Korea Joint Conference of Statistics, pp 77–82, 2000; Shimokawa et al. in Japan J Appl Stat 29:27–39, 2000). In addition, several methods for clustering functional data have been proposed (Abraham et al. in Scand J Stat 30:581–595, 2003; Gareth and Catherine in J Am Stat Assoc 98:397–408, 2003; Tarpey and kinateder in J Classif 20:93–114, 2003; Rossi et al. in Proceedings of European Symposium on Artificial Neural Networks pp 305–312, 2004). Furthermore, Tokushige et al. (J Jpn Soc Comput Stat 15:319–326, 2002) defined several dissimilarities between functions for the case of functional data. In this paper, we extend existing crisp and fuzzy k-means clustering algorithms to the analysis of multivariate functional data. In particular, we consider the dissimilarity between functions as a function. Furthermore, cluster centers and memberships, which are defined as functions, are determined at the minimum of a certain target function by using a calculus-of-variations approach.     

Visualizing hypothesis tests in multivariate linear models: the heplots package for R

Hypothesis-error (or “HE”) plots, introduced by Friendly (J Stat Softw 17(6):1–42, 2006a; J Comput Graph Stat 16:421–444, 2006b), permit the visualization of hypothesis tests in multivariate linear models by representing hypothesis and error matrices of sums of squares and cross-products as ellipses. This paper describes the implementation of these methods in the heplots package for R, as well as their extension, for example from two to three dimensions and by scaling hypothesis ellipses and ellipsoids in a natural manner relative to error. This is a paper for the proceedings of the Directions in Statistical Computing conference.     

Detecting and modeling nonlinearity in the gas furnace data

Using a modification of the Hinich, J Time Ser Anal 3(3):169–176, (1982) bispectrum test for nonlinearity and Gaussianity, the residuals of the Tiao and Box, J Am Stat Assoc 76:802–816, (1981) constrained and unconstrained VAR models for the gas furnace data reject the assumption of Gaussianity and linearity over a grid of bandwidths for estimating the bispectrum. These findings call into question the specification of the linear VAR and VARMA models assumed by Tiao and Box, J Am Stat Assoc 76:802–816, (1981). Utilizing the alternative Hinich J Nonparametr Stat 6:205–221, (1996) nonlinearity test, the residuals of the VAR model were shown to exhibit episodic nonlinearity. The sensitivity of the findings to outliers is investigated by estimating and testing the residuals of L1 and MINIMAX models from 1–6 lags. Building on the linear dynamic specification, a multivariate adaptive regression splines (MARS) model is estimated, using two software implementations, and shown to remove the nonlinearity in the residuals. Leverage plots were used to illustrate the “cost” of imposing a linearity assumption. Out-of-sample forecasting tests from 1–6 periods ahead found that using the sum-of-squared errors criteria, the MARS model out performed ACE, GAM and projection pursuit models.     

A Distributive Lattice Structure Connecting Dyck Paths,Noncrossing Partitions and 312-avoiding Permutations

In [Ferrari, L. and Pinzani, R.: Lattices of lattice paths. J. Stat. Plan. Inference 135 (2005), 77–92] a natural order on Dyck paths of any fixed length inducing a distributive lattice structure is defined. We transfer this order to noncrossing partitions along a well-known bijection [Simion, R.: Noncrossing partitions. Discrete Math. 217 (2000), 367–409], thus showing that noncrossing partitions can be endowed with a distributive lattice structure having some combinatorial relevance. Finally we prove that our lattices are isomorphic to the posets of 312-avoiding permutations with the order induced by the strong Bruhat order of the symmetric group.     

Random Block Matrices and Matrix Orthogonal Polynomials

In this paper we consider random block matrices, which generalize the general beta ensembles recently investigated by Dumitriu and Edelmann (J. Math. Phys. 43:5830–5847, 2002; Ann. Inst. Poincaré Probab. Stat. 41:1083–1099, 2005). We demonstrate that the eigenvalues of these random matrices can be uniformly approximated by roots of matrix orthogonal polynomials which were investigated independently from the random matrix literature. As a consequence, we derive the asymptotic spectral distribution of these matrices. The limit distribution has a density which can be represented as the trace of an integral of densities of matrix measures corresponding to the Chebyshev matrix polynomials of the first kind. Our results establish a new relation between the theory of random block matrices and the field of matrix orthogonal polynomials, which have not been explored so far in the literature.     

Contribution to the bandwidth choice for kernel density estimates

In the present paper we focus on the problem of the bandwidth choice for the kernel density estimates. The problem of finding the optimal bandwidth belongs to the crucial problems of the kernel estimates. As a criterion of quality of the estimates the L ₂ type measure is used. A special iterative method based on a relevant estimation of mean integrated square error given in papers Müller and Wang (Prob Theor Relat Fields 85:523–538, 1990), Jones et al. (Ann Stat 19:1919–1932, 1991) is suggested. Moreover the idea of maximal smoothing principle (Terrell in J Am Stat Assoc 85:470–477, 1990) is extended to the higher order kernels. A simulation study brings a comparison of the proposed method and the cross-validation method. Research supported by the GACR:402/04/1308.     

A shorter proof of Kanter’s Bessel function concentration bound

We give a shorter proof of Kanter’s (J. Multivariate Anal. 6, 222–236, 1976) sharp Bessel function bound for concentrations of sums of independent symmetric random vectors. We provide sharp upper bounds for the sum of modified Bessel functions I₀(x) + I₁(x), which might be of independent interest. Corollaries improve concentration or smoothness bounds for sums of independent random variables due to Čekanavičius & Roos (Lith. Math. J. 46, 54–91, 2006); Roos (Bernoulli, 11, 533–557, 2005), Barbour & Xia (ESAIM Probab. Stat. 3, 131–150, 1999), and Le Cam (Asymptotic Methods in Statistical Decision Theory. Springer, Berlin Heidelberg New York, 1986).      

Stein structures and holomorphic mappings

We prove that every continuous map from a Stein manifold X to a complex manifold Y can be made holomorphic by a homotopic deformation of both the map and the Stein structure on X. In the absence of topological obstructions, the holomorphic map may be chosen to have pointwise maximal rank. The analogous result holds for any compact Hausdorff family of maps, but it fails in general for a noncompact family. Our main results are actually proved for smooth almost complex source manifolds (X,J) with the correct handlebody structure. The paper contains another proof of Eliashberg’s (Int J Math 1:29–46, 1990) homotopy characterization of Stein manifolds and a slightly different explanation of the construction of exotic Stein surfaces due to Gompf (Ann Math 148(2): 619–693, 1998; J Symplectic Geom 3:565–587, 2005).      

N-distance tests for a composite hypothesis of goodness of fit

In this paper, we consider an application of N-distance theory for testing composite hypotheses of goodness of fit. This work is a continuation of our research started in [A. Bakshaev, Goodness of fit and homogeneity tests on the basis of N-distances, J. Stat. Plan. Inference, 139(11):3750–3758, 2009; A. Bakshaev, Nonparametric tests based on N-distances, Lith. Math. J., 48(4):368–379, 2008]. Particular attention is paid to normality and exponentiality tests. A comparative Monte Carlo power study for the proposed criteria is provided. Different alternatives in uni- and bivariate cases are investigated.     

Classification of (0,2)-geometries embedded in AG (3,q)

In De Clerck and Delanote (Des. Codes Cryptogr, 32: 103–110, 2004) it is shown that if a (0,α)-geometry with α ≥  3 is fully embedded in AG (n,q) then it is a linear representation. In De Feyter (J. Combin Theory Ser A, 109(1): 1–23, 2005; Discrete math, 292: 45–54, 2005) the (0,2)-geometries fully embedded in AG(3,q) are classified apart from two open cases. In this paper, we solve these two open cases. This classification for AG(3,q) is used in De Feyter (Adv Geom, 5: 279–292, 2005) to classify the (0,2)-geometries fully embedded in AG(n,q).      

Multi-self-similar Markov processes on ℝ<Subscript>+</Subscript><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript> and their Lamperti representations

 A classical result, due to Lamperti, establishes a one-to-one correspondence between a class of strictly positive Markov processes that are self-similar, and the class of one-dimensional Lévy processes. This correspondence is obtained by suitably time-changing the exponential of the Lévy process. In this paper we generalise Lamperti's result to processes in n dimensions. For the representation we obtain, it is essential that the same time-change be applied to all coordinates of the processes involved. Also for the statement of the main result we need the proper concept of self-similarity in higher dimensions, referred to as multi-self-similarity in the paper. The special case where the Lévy process ξ is standard Brownian motion in n dimensions is studied in detail. There are also specific comments on the case where ξ is an n-dimensional compound Poisson process with drift. Finally, we present some results concerning moment sequences, obtained by studying the multi-self-similar processes that correspond to n-dimensional subordinators. Received: 22 August 2002 / Revised version: 10 February 2003 Published online: 15 April 2003 RID="*" ID="*" MaPhySto – Centre for Mathematical Physics and Stochastics, funded by a grant from the Danish National Research Foundation Mathematics Subject Classification (2000): 60G18, 60G51, 60J25, 60J60, 60J75 Key words or phrases: Lévy process – Self-similarity – Time-change – Exponential functional – Brownian motion – Bessel process – Piecewise deterministic Markov process – Moment sequence     

On testing extreme value conditions

Applications of univariate extreme value theory rely on certain as- sumptions. Recently, two methods for testing these extreme value conditions are derived by [Dietrich, D., de Haan, L., Hüsler, J., Extremes 5: 71–85, (2002)] and [Drees, H., de Haan, L., Li, D., J. Stat. Plan. Inference, 136: 3498–3538, (2006)]. In this paper we compare the two tests by simulations and investigate the effect of a possible weight function by choosing a parameter, the test error and the power of each test. The conclusions are useful for extreme value applications.     

Bootstrap Approximation to Distributions of Finite Population U-statistics

We show that the without replacement bootstrap of Booth, Butler and Hall (J. Am. Stat. Assoc. 89, 1282–1289, 1994) provides second order correct approximation to the distribution function of a Studentized U-statistic based on simple random sample drawn without replacement. In order to achieve similar approximation accuracy for the bootstrap procedure due to Bickel and Freedman (Ann. Stat. 12, 470–482, 1984) and Chao and Lo (Sankhya Ser. A 47, 399–405, 1985) we introduce randomized adjustments to the resampling fraction.      

Determining <Emphasis Type="Italic">p</Emphasis>-values for systems cointegration tests with a prior adjustment for deterministic terms

Following Doornik (J Econ Surv 12:573–593, 1998) I present a procedure to approximate the asymptotic distributions of systems cointegration tests with a prior adjustment for deterministic terms suggested by Lütkepohl (Econometrica 72:647–662, 2004), Saikkonen and Lütkepohl (Econometric Theory 16:373–406, 2000a, J Business Econ Stat 18:451–464, 2000b, Time Series Anal 21:435–456, 2000c) and Saikkonen and Luukkonen (J Econ 81:93–126, 1997). These tests rely upon different assumptions as to the inclusion of deterministic components such as a constant, a linear trend or a level shift. The asymptotic distributions, which are functions of Brownian motions, are approximated by Gamma distributions. Only estimates of the mean and variance of the asymptotic test distributions are needed to fit the Gamma distributions. Such estimates are obtained from response surfaces. The required coefficients to compute the asymptotic moments are presented in this paper. Via the fitted Gamma distributions one can, then, easily derive p-values or arbitrary percentiles.     

Polyhedral surfaces in wedge products

We introduce the wedge product of two polytopes. The wedge product is described in terms of inequality systems, in terms of vertex coordinates as well as purely combinatorially, from the corresponding data of its constituents. The wedge product construction can be described as an iterated “subdirect product” as introduced by McMullen (Discrete Math 14:347–358, 1976); it is dual to the “wreath product” construction of Joswig and Lutz (J Combinatorial Theor A 110:193–216, 2005). One particular instance of the wedge product construction turns out to be especially interesting: The wedge products of polygons with simplices contain certain combinatorially regular polyhedral surfaces as subcomplexes. These generalize known classes of surfaces “of unusually large genus” that first appeared in works by Coxeter (Proc London Math Soc 43:33–62, 1937), Ringel (Abh Math Seminar Univ Hamburg 20:10–19, 1956), and McMullen et al. (Israel J Math 46:127–144, 1983). Via “projections of deformed wedge products” we obtain realizations of some of the surfaces in the boundary complexes of 4-polytopes, and thus in \mathbb R³{{\mathbb R}^3} . As additional benefits our construction also yields polyhedral subdivisions for the interior and the exterior, as well as a great number of local deformations (“moduli”) for the surfaces in \mathbb R³{{\mathbb R}^3} . In order to prove that there are many moduli, we introduce the concept of “affine support sets” in simple polytopes. Finally, we explain how duality theory for 4-dimensional polytopes can be exploited in order to also realize combinatorially dual surfaces in \mathbb R³{{\mathbb R}^3} via dual 4-polytopes.     

A general divergence criterion for prior selection

The paper revisits the problem of selection of priors for regular one-parameter family of distributions. The goal is to find some “objective” or “default” prior by approximate maximization of the distance between the prior and the posterior under a general divergence criterion as introduced by Amari (Ann Stat 10:357–387, 1982) and Cressie and Read (J R Stat Soc Ser B 46:440–464, 1984). The maximization is based on an asymptotic expansion of this distance. The Kullback–Leibler, Bhattacharyya–Hellinger and Chi-square divergence are special cases of this general divergence criterion. It is shown that with the exception of one particular case, namely the Chi-square divergence, the general divergence criterion yields Jeffreys’ prior. For the Chi-square divergence, we obtain a prior different from that of Jeffreys and also from that of Clarke and Sun (Sankhya Ser A 59:215–231, 1997).     

Analysis of a class of nonlinear subdivision schemes and associated multiresolution transforms

This paper is devoted to the convergence and stability analysis of a class of nonlinear subdivision schemes and associated multiresolution transforms. As soon as a nonlinear scheme can be written as a specific perturbation of a linear and convergent subdivision scheme, we show that if some contractivity properties are satisfied, then stability and convergence can be achieved. This approach is applied to various schemes, which give different new results. More precisely, we study uncentered Lagrange interpolatory linear schemes, WENO scheme (Liu et al., J Comput Phys 115:200–212, 1994), PPH and Power-P schemes (Amat and Liandrat, Appl Comput Harmon Anal 18(2):198–206, 2005; Serna and Marquina, J Comput Phys 194:632–658, 2004) and a nonlinear scheme using local spherical coordinates (Aspert et al., Comput Aided Geom Des 20:165–187, 2003). Finally, a stability proof is given for the multiresolution transform associated to a nonlinear scheme of Marinov et al. (2005).     

A direct approach to the weighted admissibility of observation operators for bounded analytic semigroups

In this note we adopt the approach in Bonnit et al. (Czechoslov. Math. J. 60(2):527–539, 2010) to give a direct proof of some recent results in Haak and Le Merdy (Houst. J. Math., 2005) and Haak and Kunstmann (SIAM J. Control Optim. 45:2094–2118, 2007) which characterizes the L ^p -admissibility of type α depending on p of unbounded observation operators for bounded analytic semigroups.     

The Weierstrass—Enneper representation using hodographic coordinates on a minimal surface

In this paper we obtain the general solution to the minimal surface equation, namely its local Weierstrass-Enneper representation, using a system of hodographic coordinates. This is done by using the method of solving the Born-Infeld equations by Whitham. We directly compute conformal coordinates on the minimal surface which give the Weierstrass-Enneper representation. From this we derive the hodographic coordinate ρ∈ D ⊂ ℂ and σ its complex conjugate which enables us to write the Weierstrass-Enneper representation in a new way.