Adolfo J. Quiroz

Connectivity of the mutual k-nearest-neighbor graph in clustering and outlier detection

For multivariate data sets, we study the relationship between the connectivity of a mutual k-nearest-neighbor graph, and the presence of clustering structure and outliers in the data. A test for detection of clustering structure and outliers is proposed and its performance is evaluated in simulated data.

A permutation-based algorithm for block clustering

Hartigan (1972) discusses the direct clustering of a matrix of data into homogeneous blocks. He introduces a stepwise divisive method for block clustering within a certain class of block structures which induce clustering trees for both row and column margins. While this class of structures is appealing, the stopping criterion for his method, which is based on asymptotic theory and the assumption that the individual elements of the data matrix are normally distributed, is quite restrictive. In this paper we propose a permutation-based algorithm for block clustering within the same class of block structures. By using permutation arguments to decide where to split and when to stop, our algorithm becomes applicable in a wide variety of cases, including matrices of categorical data and matrices of small-to-moderate size. In addition, our algorithm offers considerable flexibility in how block homogeneity is defined. The algorithm is studied in a series of simulation experiments on matrices of known structure, and illustrated in examples drawn from the fields of taxonomy, political science, and data architecture.

A clustering procedure based on the comparison between the k nearest neighbors graph and the minimal spanning tree

We present a procedure for the identification of clusters in multivariate data sets, based on the comparison between the k nearest neighbors graph, Gk, and the minimal spanning tree, MST. Our key statistic is the random quantity the smallest k such that Gk contains the MST. Under regularity assumptions, we show that for i.i.d. data from a density on with compact support having one connected component, , where n denotes sample size, a bound that seems to be sharp, according to simulations. This leads to a consistent test for the identification of crisp clusters. We illustrate the use of our procedure with an example.

Some new tests for multivariate normality

SummaryA family of statistics is presented that can be used for testing goodness of fit to a parametric family. These statistics include Mardias measure of multivariate kurtosis and Moore and Stubblebines test for multivariate normality. The asymptotic distribution of the statistics is found under mild hypotheses on the parametric family and, in the case of multivariate normality, the distribution is shown to be independent of the “true” parameter. A class of tests for multivariate normality is presented and the performance of two such tests in the bivariate case is found in simulations.

Fast random generation of binary, t-ary and other types of trees

Trees, and particularly binary trees, appear frequently in the classification literature. When studying the properties of the procedures that fit trees to sets of data, direct analysis can be too difficult, and Monte Carlo simulations may be necessary, requiring the implementation of algorithms for the generation of certain families of trees at random. In the present paper we use the properties of Prufers enumeration of the set of completely labeled trees to obtain algorithms for the generation of completely labeled, as well as terminally labeled t-ary (and in particular binary) trees at random, i.e., with uniform distribution. Actually, these algorithms are general in that they can be used to generate random trees from any family that can be characterized in terms of the node degrees. The algorithms presented here are as fast as (in the case of terminally labeled trees) or faster than (in the case of completely labeled trees) any other existing procedure, and the memory requirements are minimal. Another advantage over existing algorithms is that there is no need to store pre-calculated tables.

Using the empirical moment generating function in testing for the Weibull and the type I extreme value distributions

We introduce two families of statistics, functionals of the empirical moment generating function process of the logarithmically transformed data, for testing goodness of fit to the two-parameter Weibull distribution or, equivalently, to the type I extreme value model. We show that when affine invariant estimators are used for the parameters of the extreme value distribution, the distributions of these statistics to not depend on the underlying parameters and one of them has a limiting chi-squared distribution. We estimate, via simulations, some finite sample quantiles for the statistics introduced and evaluate their power against a varied set of alternatives.

A vectorial notion of skewness and its use in testing for multivariate symmetry

By modifying the statistic of Malkovich and Afifi (1973), we introduce and study the properties of a notion of multivariate skewness that provides both a magnitude and an overall direction for the skewness present in multivariate data. This notion leads to a test statistic for the nonparametric null hypothesis of multivariate symmetry. Under mild assumptions, we find the asymptotic distribution of the test statistic and evaluate, by simulation, the convergence of the finite sample size percentiles to their limits. We also present an associated test statistic for multivariate normality.

Spherical harmonics in quadratic forms for testing multivariate normality

We study two statistics for testing goodness of fit to the null hypothesis of multivariate normality, based on averages over the standardized sample of multivariate spherical harmonics, radial functions and their products. These statistics (of which one was studied in the two-dimensional case in Quiroz and Dudley, 1991) have, as limiting distributions, linear combinations of chi-squares. In arbitrary dimension, we obtain closed form expressions for the coefficients that describe the limiting distributions, which allow us to produce Monte Carlo approximate limiting quantiles. We also obtain Monte Carlo approximate finite sample size quantiles and evaluate the power of the statistics presented against several alternatives of interest. A power comparison with other relevant statistics is included. The statistics proposed are easy to compute (with Fortran code available from the authors) and their finite sample quantiles converge relatively rapidly, with increasing sample size, to their limiting values, a behaviour that could be explained by the large number of orthogonal functions used in the quadratic forms involved.

ESTIMATION OF A MULTIVARIATE BOX-COX TRANSFORMATION TO ELLIPTICAL SYMMETRY VIA THE EMPIRICAL CHARACTERISTIC FUNCTION*

Let X=(X1, X2,..., Xd)t be a random vector of positive entries, such that for some λ=(λ1,λ2,...,λd)t, the vector X(λ) defined by % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% GaamiwamaaDaaaleaamiaadMgaaSqaaWGaaiikaiabeU7aSnaaBaaa% baGaamyAaiaacMcaaeqaaaaakiabg2da9iaacIcadaWcgaqaaiaadI% fadaqhaaWcbaadcaWGPbaaleaamiabeU7aSnaaBaaabaGaamyAaaqa% baaaaOGaeyOeI0IaaGymaiaacMcaaeaacqaH7oaBdaWgaaWcbaadca% WGPbGaaiilaaWcbeaakiaadMgacqGH9aqpcaaIXaGaeSOjGSKaaiil% aiaadsgaaaaaaa!53BB!\[X_i^{(\lambda _{i)} } = ({{X_i^{\lambda _i } - 1)} \mathord{\left/ {\vphantom {{X_i^{\lambda _i } - 1)} {\lambda _{i,} i = 1 \ldots ,d}}} \right. \kern-\nulldelimiterspace} {\lambda _{i,} i = 1 \ldots ,d}}\]is elliptically symmetric. We describe a procedure based on the multivariate empirical characteristic function for estimating the λis. Asymptotic results regarding consistency of the estimators are given and we evaluate their performance in simulated data. In a one-dimensional setting, comparisons are made with other available transformations to symmetry.

A fast permutation-based algorithm for block clustering

A stepwise divisive procedure for the clustering of numerical data recorded in matrix form into homogeneous groups is introduced. The methodology relates to those proposed by Hartigan (1972) and Duffy and Quiroz (1991). As the latter, the proposed methodology uses the permutation distribution of the data in a block as the reference distribution to make inferences about the presence of clustering structure. A local (within block) criteria and Bayesian sequential decision methodology are used to evaluate the significance of potential partitions of blocks, resulting in an algorithm which is faster than those considered by Duffy and Quiroz (1991). The class of possible clustering structures that our procedure can discover is also larger than those previously considered in the literature.