Optimal Variable Weighting for Ultrametric and Additive Trees and K-means Partitioning: Methods and Software

K -means partitioning. We also describe some new features and improvements to the algorithm proposed by De Soete. Monte Carlo simulations have been conducted using different error conditions. In all cases (i.e., ultrametric or additive trees, or K-means partitioning), the simulation results indicate that the optimal weighting procedure should be used for analyzing data containing noisy variables that do not contribute relevant information to the classification structure. However, if the data involve error-perturbed variables that are relevant to the classification or outliers, it seems better to cluster or partition the entities by using variables with equal weights. A new computer program, OVW, which is available to researchers as freeware, implements improved algorithms for optimal variable weighting for ultrametric and additive tree clustering, and includes a new algorithm for optimal variable weighting for K-means partitioning.     

OCLUS: An Analytic Method for Generating Clusters with Known Overlap

The primary method for validating cluster analysis techniques is throughMonte Carlo simulations that rely on generating data with known cluster structure (e.g., Milligan 1996). This paper defines two kinds of data generation mechanisms with cluster overlap, marginal and joint; current cluster generation methods are framed within these definitions. An algorithm generating overlapping clusters based on shared densities from several different multivariate distributions is proposed and shown to lead to an easily understandable notion of cluster overlap. Besides outlining the advantages of generating clusters within this framework, a discussion is given of how the proposed data generation technique can be used to augment research into current classification techniques such as finite mixture modeling, classification algorithm robustness, and latent profile analysis.     

Improved Classification for Compositional Data Using the <Emphasis Type="Italic">α</Emphasis>-transformation

In compositional data analysis, an observation is a vector containing nonnegative values, only the relative sizes of which are considered to be of interest. Without loss of generality, a compositional vector can be taken to be a vector of proportions that sum to one. Data of this type arise in many areas including geology, archaeology, biology, economics and political science. In this paper we investigate methods for classification of compositional data. Our approach centers on the idea of using the α-transformation to transform the data and then to classify the transformed data via regularized discriminant analysis and the k-nearest neighbors algorithm. Using the α-transformation generalizes two rival approaches in compositional data analysis, one (when α=1) that treats the data as though they were Euclidean, ignoring the compositional constraint, and another (when α = 0) that employs Aitchison’s centered log-ratio transformation. A numerical study with several real datasets shows that whether using α = 1 or α = 0 gives better classification performance depends on the dataset, and moreover that using an intermediate value of α can sometimes give better performance than using either 1 or 0.     

Degenerating Families of Dendrograms

Dendrograms used in data analysis are ultrametric spaces, hence objects of nonarchimedean geometry. It is known that there exist p-adic representations of dendrograms. Completed by a point at infinity, they can be viewed as subtrees of the Bruhat-Tits tree associated to the p-adic projective line. The implications are that certain moduli spaces known in algebraic geometry are in fact p-adic parameter spaces of dendrograms, and stochastic classification can also be handled within this framework. At the end, we calculate the topology of the hidden part of a dendrogram.     

Whose Agency Now?

I agree with my readers on the necessary alliance of personal agency and collective agency. My point is to prioritize the former. The reasons to prioritize the latter were excellent, and it was undoubtedly a sound decision to start with this scenario: political and institutional improvement of the collectives, enlightened by progressive social sciences. My argument for suggesting a different priority—toward personal microactions and their emergent effects—relies on the opinion that we are stuck in a sustainability crisis due to our current approach. In the question “whose agency now?”, my stress is on “now”.     

Weighted Graphs with Distances in Given Ranges

Let \( \mathcal{G} \) = (G,w) be a weighted simple finite connected graph, that is, let G be a simple finite connected graph endowed with a function w from the set of the edges of G to the set of real numbers. For any subgraph G′ of G, we define w(G′) to be the sum of the weights of the edges of G′. For any i, j vertices of G, we define D _{i,j}(\( \mathcal{G} \)) to be the minimum of the weights of the simple paths of G joining i and j. The D _{i,j}(\( \mathcal{G} \)) are called 2-weights of \( \mathcal{G} \). Weighted graphs and their reconstruction from 2-weights have applications in several disciplines, such as biology and psychology.Let \( {\left\{{m}_I\right\}}_{I\in \left(\frac{\left\{1,\dots, n\right\}}{2}\right)} \) and \( {\left\{{M}_I\right\}}_{I\in \left(\frac{\left\{1,\dots, n\right\}}{2}\right)} \) be two families of positive real numbers parametrized by the 2-subsets of {1, …, n} with m _I ≤ M _I for any I; we study when there exist a positive-weighted graph G and an n-subset {1, …, n} of the set of its vertices such that D _I (\( \mathcal{G} \)) ∈ [m _I ,M _I ] for any \( I\in \left(\frac{\left\{1,\dots, n\right\}}{2}\right) \). Then we study the analogous problem for trees, both in the case of positive weights and in the case of general weights.     

<Emphasis Type="Italic">The Mathematical Intelligencer</Emphasis> Flunks the Olympics

The Mathematical Intelligencer recently published a note by Y. Sergeyev that challenges both mathematics and intelligence. We examine Sergeyev’s claims concerning his purported Infinity computer. We compare his grossone system with the classical Levi-Civita fields and with the hyperreal framework of A. Robinson, and analyze the related algorithmic issues inevitably arising in any genuine computer implementation. We show that Sergeyev’s grossone system is unnecessary and vague, and that whatever consistent subsystem could be salvaged is subsumed entirely within a stronger and clearer system (IST). Lou Kauffman, who published an article on a grossone, places it squarely outside the historical panorama of ideas dealing with infinity and infinitesimals.     

On the Properties of <Emphasis Type="Italic">α</Emphasis>-Unchaining Single Linkage Hierarchical Clustering

In the election of a hierarchical clustering method, theoretic properties may give some insight to determine which method is the most suitable to treat a clustering problem. Herein, we study some basic properties of two hierarchical clustering methods: α-unchaining single linkage or SL(α) and a modified version of this one, SL?(α). We compare the results with the properties satisfied by the classical linkage-based hierarchical clustering methods.     

A Probability for Classification Based on the Dirichlet Process Mixture Model

In this paper we provide an explicit probability distribution for classification purposes when observations are viewed on the real line and classifications are to be based on numerical orderings. The classification model is derived from a Bayesian nonparametric mixture of Dirichlet process model; with some modifications. The resulting approach then more closely resembles a classical hierarchical grouping rule in that it depends on sums of squares of neighboring values. The proposed probability model for classification relies on a numerical procedure based on a reversible Markov chain Monte Carlo (MCMC) algorithm for determining the probabilities. Some numerical illustrations comparing with alternative ideas for classification are provided.     

Bolzano’s Infinite Quantities

In his Foundations of a General Theory of Manifolds, Georg Cantor praised Bernard Bolzano as a clear defender of actual infinity who had the courage to work with infinite numbers. At the same time, he sharply criticized the way Bolzano dealt with them. Cantor’s concept was based on the existence of a one-to-one correspondence, while Bolzano insisted on Euclid’s Axiom of the whole being greater than a part. Cantor’s set theory has eventually prevailed, and became a formal basis of contemporary mathematics, while Bolzano’s approach is generally considered a step in the wrong direction. In the present paper, we demonstrate that a fragment of Bolzano’s theory of infinite quantities retaining the part-whole principle can be extended to a consistent mathematical structure. It can be interpreted in several possible ways. We obtain either a linearly ordered ring of finite and infinitely great quantities, or a partially ordered ring containing infinitely small, finite and infinitely great quantities. These structures can be used as a basis of the infinitesimal calculus similarly as in non-standard analysis, whether in its full version employing ultrafilters due to Abraham Robinson, or in the recent “cheap version” avoiding ultrafilters due to Terence Tao.     

A maximum likelihood methodology for clusterwise linear regression

This paper presents a conditional mixture, maximum likelihood methodology for performing clusterwise linear regression. This new methodology simultaneously estimates separate regression functions and membership inK clusters or groups. A review of related procedures is discussed with an associated critique. The conditional mixture, maximum likelihood methodology is introduced together with the E-M algorithm utilized for parameter estimation. A Monte Carlo analysis is performed via a fractional factorial design to examine the performance of the procedure. Next, a marketing application is presented concerning the evaluations of trade show performance by senior marketing executives. Finally, other potential applications and directions for future research are identified.     

Heidegger on <Emphasis Type="Italic">Verfallenheit</Emphasis>

The question of Heidegger’s reflections on technology is explored in terms of ‘living with’ technology and including the socio-theoretical (Edinburgh) notion of ‘entanglement’ towards a review of Heidegger’s understanding of technology and media, including the entertainment industry and modern digital life. I explore Heidegger’s reflections on Gelassenheit by way of the Japanese aesthetic conception of life and of art as wabi-sabi understood with respect to Heidegger’s Gelassenheit as the art of Verfallenheit.     

Revisiting <Emphasis Type="Italic">Tom Tom:</Emphasis> Performative Anamnesis and Autonomous Vision in Ken Jacobs’ Appropriations of <Emphasis Type="Italic">Tom Tom the Piper’s Son</Emphasis>

In 1969 the American avant-garde filmmaker Ken Jacobs gained wide recognition with a two-hour long interpretation of a 1905 silent short film. Ever since, the artist has kept on revisiting the same material, each time with a different technological approach. Originally hailed as a prime example of structural filmmaking, Jacobs’ more recent variations on the theme of Tom Tom the Piper’s Son beg for a broader understanding of his methods and the meanings implied. To gain a deeper insight in this on-going mise-en-abyme (and an obsession dominating a large part of his career), this essay expands comments by the artist himself with concepts taken from animation, media-archaeology and Warburg’s Mnemosyne atlas. Rereading a filmic text with minute attention, remediating it from an analogue to an electronic format, and reanimating the original action by adding a variety of intervals: all Jacobs’ strategies are aimed at demonstrating the afterlife of Tom Tom in a contemporary cultural context.     

The Unseen Déjà-Vu: From Erkki Huhtamo’s <Emphasis Type="Italic">Topoi</Emphasis> to Ken Jacobs’ Remakes

This commentary on Edwin Carels’ essay “Revisiting Tom Tom: Performative anamnesis and autonomous vision in Ken Jacobs’ appropriations of Tom Tom the Piper’s Son” broadens up the media-archaeological framework in which Carels places his text. Notions such as Huhtamo’s topos and Zielinski’s “deep time” are brought into the discussion in order to point out the difficulty to see what there is to see and to question the position of the viewer in front of experimental films like Tom Tom the Piper’s Son and its remakes.     

Releasement and Nihilism in the Art of Living with Technology

In this contribution the author tries to formulate an approach to the art of living with technology based on Heidegger’s The Principle of Reason, a work often overlooked by contemporary commentators in the philosophy of technology. This approach couples the concept of releasement to insights hailing from Wolfgang Schirmacher concerning Heidegger’s nihilism.     

The Infinite Turn and Speculative Explanations in Cosmology

Infinity, in various guises, has been invoked recently in order to ‘explain’ a number of important questions regarding observable phenomena in science, and in particular in cosmology. Such explanations are by their nature speculative. Here we introduce the notions of relative infinity, closure, and economy of explanation and ask: to what extent explanations involving relative or real constructed infinities can be treated as reasonable?     

The Documentary Surreal: Film and Painting in Luciano Emmer’s <Emphasis Type="Italic">La Leggenda di Sant’Orsola</Emphasis> (1948) and Henri Storck’s <Emphasis Type="Italic">Le Monde de Paul Delvaux</Emphasis> (1946)

This article deals with the aesthetics of the art documentary of the 1940s and 1950s, which can be considered as the Golden Age of the genre. Prior to the breakthrough of television in Europe, which would usurp and standardize the art documentary, cinematic reproductions of artworks resulted in experimental shorts that were highly self-reflexive. These films became visual laboratories to investigate the tensions between movement and stasis, the two- and three-dimensional, and the real and the artificial—a film on art was self-consciously presented as an art film. Focusing on La Leggenda di Sant’Orsola (1948) by Luciano Emmer and Le Monde de Paul Delvaux (1946) by Henri Storck, this article also investigates how the animation of the static image by the film medium relates to Surrealist practices.     

A Procedure for Estimating the Number of Clusters in Logistic Regression Clustering

This paper studies the problem of estimating the number of clusters in the context of logistic regression clustering. The classification likelihood approach is employed to tackle this problem. A model-selection based criterion for selecting the number of logistic curves is proposed and its asymptotic property is also considered. The small sample performance of the proposed criterion is studied by Monto Carlo simulation. In addition, a real data example is presented. The authors would like to thank the editor, Prof. Willem J. Heiser, and the anonymous referees for the valuable comments and suggestions, which have led to the improvement of this paper.