How to use the Kohonen algorithm to simultaneously analyse individuals in a survey

The Kohonen algorithm (SOM, Kohonen,1984, 1995) is a very powerful tool for data analysis. It was originally designed to model organized connections between some biological neural networks. It was also immediately considered as a very good algorithm to realize vectorial quantization, and at the same time pertinent classification, with nice properties for visualization. If the individuals are described by quantitative variables (ratios, frequencies, measurements, amounts, etc.), the straightforward application of the original algorithm leads to build code vectors and to associate to each of them the class of all the individuals which are more similar to this code-vector than to the others. But, in case of individuals described by categorical (qualitative) variables having a finite number of modalities (like in a survey), it is necessary to define a specific algorithm. In this paper, we present a new algorithm inspired by the SOM algorithm, which provides a simultaneous classification of the individuals and of their modalities.Comment: Special issue ESANN 0

Flag Structures on Seifert Manifolds

We consider faithful projective actions of a cocompact lattice of SL(2,R) on the projective plane, with the following property: there is a common fixed point, which is a saddle fixed point for every element of infinite order of the the group. Typical examples of such an action are linear actions, ie, when the action arises from a morphism of the group into GL(2,R), viewed as the group of linear transformations of a copy of the affine plane in RP^{2}. We prove that in the general situation, such an action is always topologically linearisable, and that the linearisation is Lipschitz if and only if it is projective. This result is obtained through the study of a certain family of flag structures on Seifert manifolds. As a corollary, we deduce some dynamical properties of the transversely affine flows obtained by deformations of horocyclic flows. In particular, these flows are not minimal.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol5/paper7.abs.htm

The activity of French Research Ethics Committees and characteristics of biomedical research protocols involving humans: a retrospective cohort study

BACKGROUND: Clinical trials throughout the world must be evaluated by research ethics committees. No one has yet attempted to clearly quantify at the national level the activity of ethics committees and describe the characteristics of the protocols submitted. The objectives of this study were to describe 1) the workload and the activity of Research Ethics Committees in France, and 2) the characteristics of protocols approved on a nation-wide basis. METHODS: Retrospective cohort of 976 protocols approved by a representative sample of 25/48 of French Research Ethics Committees in 1994. Protocols characteristics (design, study size, investigator), number of revisions requested by the ethics committee before approval, time to approval and number of amendments after approval were collected for each protocol by trained research assistant using the committee's files and archives. RESULTS: Thirty-one percent of protocols were approved with no modifications requested in 16 days (95% CI: 14–17). The number of revisions requested by the committee, and amendments submitted by the investigator was on average respectively 39 (95% CI: 25–53) and 37 (95% CI: 27–46), per committee and per year. When revisions were requested, the main reasons were related to information to the patient (28%) and consent modalities (18%). Drugs were the object of research in 68% of the protocols examined. The majority of the research was national (80%) with a predominance of single-centre studies. Workload per protocol has been estimated at twelve and half hours on average for administrative support and at eleven and half hours for expertise. CONCLUSION: The estimated workload justifies specific and independent administrative and financial support for Research Ethics Committees

Direct reading algorithm for hierarchical clustering

Reading the clusters from a data set such that the overall computational complexity is linear in both data dimensionality and in the number of data elements has been carried out through filtering the data in wavelet transform space. This objective is also carried out after an initial transforming of the data to a canonical order. Including high dimensional, high cardinality data, such a canonical order is provided by row and column permutations of the data matrix. In our recent work, we induce a hierarchical clustering from seriation through unidimensional representation of our observations. This linear time hierarchical classification is directly derived from the use of the Baire metric, which is simultaneously an ultrametric. In our previous work, the linear time construction of a hierarchical clustering is studied from the following viewpoint: representing the hierarchy initially in an m-adic, m =10, tree representation, followed by decreasing m to smaller valued representations that include p-adic representations, where p is prime and m is a non-prime positive integer. This has the advantage of facilitating a more direct visualization and hence interpretation of the hierarchy. In this work we present further case studies and examples of how this approach is very advantageous for such an ultrametric topological data mapping

Ward's Hierarchical Clustering Method: Clustering Criterion and Agglomerative Algorithm

The Ward error sum of squares hierarchical clustering method has been very widely used since its first description by Ward in a 1963 publication. It has also been generalized in various ways. However there are different interpretations in the literature and there are different implementations of the Ward agglomerative algorithm in commonly used software systems, including differing expressions of the agglomerative criterion. Our survey work and case studies will be useful for all those involved in developing software for data analysis using Ward's hierarchical clustering method.Comment: 20 pages, 21 citations, 4 figure

Projective deformations of weakly orderable hyperbolic Coxeter orbifolds

A Coxeter $n$-orbifold is an $n$-dimensional orbifold based on a polytope with silvered boundary facets. Each pair of adjacent facets meet on a ridge of some order $m$, whose neighborhood is locally modeled on ${\mathbb R}^n$ modulo the dihedral group of order $2m$ generated by two reflections. For $n \geq 3$, we study the deformation space of real projective structures on a compact Coxeter $n$-orbifold $Q$ admitting a hyperbolic structure. Let $e_+(Q)$ be the number of ridges of order $\geq 3$. A neighborhood of the hyperbolic structure in the deformation space is a cell of dimension $e_+(Q) - n$ if $n=3$ and $Q$ is weakly orderable, i.e., the faces of $Q$ can be ordered so that each face contains at most $3$ edges of order $2$ in faces of higher indices, or $Q$ is based on a truncation polytope.Comment: 43 pages with 7 figures, to appear in Geometry & Topolog

On the Schoenberg Transformations in Data Analysis: Theory and Illustrations

The class of Schoenberg transformations, embedding Euclidean distances into higher dimensional Euclidean spaces, is presented, and derived from theorems on positive definite and conditionally negative definite matrices. Original results on the arc lengths, angles and curvature of the transformations are proposed, and visualized on artificial data sets by classical multidimensional scaling. A simple distance-based discriminant algorithm illustrates the theory, intimately connected to the Gaussian kernels of Machine Learning

On the equivalence between hierarchical segmentations and ultrametric watersheds

We study hierarchical segmentation in the framework of edge-weighted graphs. We define ultrametric watersheds as topological watersheds null on the minima. We prove that there exists a bijection between the set of ultrametric watersheds and the set of hierarchical segmentations. We end this paper by showing how to use the proposed framework in practice in the example of constrained connectivity; in particular it allows to compute such a hierarchy following a classical watershed-based morphological scheme, which provides an efficient algorithm to compute the whole hierarchy.Comment: 19 pages, double-colum

The infinitesimal projective rigidity under Dehn filling

To a hyperbolic manifold one can associate a canonical projective structure and ask whether it can be deformed or not. In a cusped manifold, one can ask about the existence of deformations that are trivial on the boundary. We prove that if the canonical projective structure of a cusped manifold is infinitesimally projectively rigid relative to the boundary, then infinitely many Dehn fillings are projectively rigid. We analyze in more detail the figure eight knot and the Withehead link exteriors, for which we can give explicit infinite families of slopes with projectively rigid Dehn fillings.Comment: Accepted for publication at G