The majorization approach to multidimensional scaling for Minkowski distances

The majorization method for multidimensional scaling with Kruskal's STRESS has been limited to Euclidean distances only. Here we extend the majorization algorithm to deal with Minkowski distances with 1≤p≤2 and suggest an algorithm that is partially based on majorization forp outside this range. We give some convergence proofs and extend the zero distance theorem of De Leeuw (1984) to Minkowski distances withp>1.     

The Canonical Analysis of Distance

Canonical Variate Analysis (CVA) is one of the most useful of multivariate methods. It is concerned with separating between and within group variation among N samples from K populations with respect to p measured variables. Mahalanobis distance between the K group means can be represented as points in a (K - 1) dimensional space and approximated in a smaller space, with the variables shown as calibrated biplot axes. Within group variation may also be shown, together with circular confidence regions and other convex prediction regions, which may be used to discriminate new samples. This type of representation extends to what we term Analysis of Distance (AoD), whenever a Euclidean inter-sample distance is defined. Although the N × N distance matrix of the samples, which may be large, is required, eigenvalue calculations are needed only for the much smaller K × K matrix of distances between group centroids. All the ancillary information that is attached to a CVA analysis is available in an AoD analysis. We outline the theory and the R programs we developed to implement AoD by presenting two examples.     

Corrected Zegers-ten Berge Coefficients Are Special Cases of Cohen’s Weighted Kappa

It is shown that if cell weights may be calculated from the data the chance-corrected Zegers-ten Berge coefficients for metric scales are special cases of Cohen’s weighted kappa. The corrected coefficients include Pearson’s product-moment correlation, Spearman’s rank correlation and the intraclass correlation ICC(3, 1).     

On the classification of recall strings using lattice-theoretic measures

Lattice theory is used to develop techniques for classifying groups of subjects on the basis of their recall strategies or multiple recall strategies within individual subjects. Using the ordered tree algorithm to represent sets of recall orders, it is shown how both trees and single recall strings can be represented as points within a nonsemimodular, graded lattice. Distances within the lattice structure are used to construct a dissimilarity measure,S, which can then be used to partition the individual recall strings. The measureS between strings is compared to Kendall's tau in three empirical tests, examining differences between individual subjects, differences between groups of subjects, and differences within a subject. It was shown that onlyS could recover the original differences. Differences between comparing chunks versus comparing orders are discussed.The author would like to thank Henry Rueter, Judith Olson, John Jonides, and James Jaccard for many inspiring comments during several stages of this project, two anonymous reviewers for several important insights, and Malhee Lee for her assistance with data collection. This work was supported by NIMH Grant MH 39912. Portions of this work were presented at the annual meeting of the Classification Society in St. John's, Newfoundland, July 1985, and the annual meeting of the Society for Mathematical Psychology in Boston, MA, August 1986.     

Canonical Analysis: Ranks,Ratios and Fits

Measurements of p variables for n samples are collected into a n×p matrix X, where the samples belong to one of k groups. The group means are separated by Mahalanobis distances. CVA optimally represents the group means of X in an r-dimensional space. This can be done by maximizing a ratio criterion (basically one- dimensional) or, more flexibly, by minimizing a rank-constrained least-squares fitting criterion (which is not confined to being one-dimensional but depends on defining an appropriate Mahalanobis metric). In modern n < p problems, where W is not of full rank, the ratio criterion is shown not to be coherent but the fit criterion, with an attention to associated metrics, readily generalizes. In this context we give a unified generalization of CVA, introducing two metrics, one in the range space of W and the other in the null space of W, that have links with Mahalanobis distance. This generalization is computationally efficient, since it requires only the spectral decomposition of a n×n matrix.     

On the Complexity of Ordinal Clustering

Given a set of pairwise distances on a set of n points, constructing an edgeweighted tree whose leaves are these n points such that the tree distances would mimic the original distances under some criteria is a fundamental problem. One such criterion is to preserve the ordinal relation between the pairwise distances. The ordinal relation can be of the form of total order on the distances or it can be some partial order specified on the pairwise distances. We show that the problem of finding a weighted tree, if it exists, which would preserve the total order on pairwise distances is NP-hard. We also show the NP-hardness of the problem of finding a weighted tree which would preserve a particular kind of partial order called a triangle order, one of the most fundamental partial orders considered in computational biology.     

Leibniz’s Theory of Space

In this paper I offer a fresh interpretation of Leibniz’s theory of space, in which I explain the connection of his relational theory to both his mathematical theory of analysis situs and his theory of substance. I argue that the elements of his mature theory are not bare bodies (as on a standard relationalist view) nor bare points (as on an absolutist view), but situations. Regarded as an accident of an individual body, a situation is the complex of its angles and distances to other co-existing bodies, founded in the representation or state of the substance or substances contained in the body. The complex of all such mutually compatible situations of co-existing bodies constitutes an order of situations, or instantaneous space. Because these relations of situation change from one instant to another, space is an accidental whole that is continuously changing and becoming something different, and therefore a phenomenon. As Leibniz explains to Clarke, it can be represented mathematically by supposing some set of existents hypothetically (and counterfactually) to remain in a fixed mutual relation of situation, and gauging all subsequent situations in terms of transformations with respect to this initial set. Space conceived in terms of such allowable transformations is the subject of Analysis Situs. Finally, insofar as space is conceived in abstraction from any bodies that might individuate the situations, it encompasses all possible relations of situation. This abstract space, the order of all possible situations, is an abstract entity, and therefore ideal.     

Reality,Systems and Impure Systems

Impure systems contain Objects and Subjects: Subjects are human beings. We can distinguish a person as an observer (subjectively outside the system) and that by definition is the Subject himself, and part of the system. In this case he acquires the category of object. Objects (relative beings) are significances, which are the consequence of perceptual beliefs on the part of the Subject about material or energetic objects (absolute beings) with certain characteristics.The IS (Impure System) approach is as follows: Objects are perceptual significances (relative beings) of material or energetic objects (absolute beings). The set of these objects will form an impure set of the first order. The existing relations between these relative objects will be of two classes: transactions of matter and/or energy and inferential relations. Transactions can have alethic modality: necessity, possibility, impossibility and contingency. Ontic existence of possibility entails that inferential relations have Deontic modality: obligation, permission, prohibition, faculty and analogy. We distinguished between theorems (natural laws) and norms (ethical, legislative and customary rules of conduct).     

Distinguishing and Classifying from n-ary Properties

We present a hierarchical classification based on n-ary relations of the entities. Starting from the finest partition that can be obtained from the attributes, we distinguish between entities having the same attributes by using relations between entities. The classification that we get is thus a refinement of this finest partition. It can be computed in O(n + m ²) space and O(n · p · m ^5/2) time, where n is the number of entities, p the number of classes of the resulting hierarchy (p is the size of the output; p < 2n) and m the maximum number of relations an entity can have (usually, m ? n). So we can treat sets with millions of entities.     

On Spatial and Temporal ex mensura Boundaries

In this paper we will propose an empirical analysis of spatial and temporal boundaries. Unlike other proposals, which deal mainly with the commonsense level of the subject, we will ground our explication on well-established scientific practice and language. In this way we show how to reconsider in an innovative way questions such as the distinction between the bona fide boundaries and the fiat boundaries, the thickness and the ownership of the boundaries. At the same time we propose a division between ex mensura boundaries and qui vulgo dicuntur boundaries. What is it, therefore, that divides the atmosphere from the water? [Leonardo da Vinci, Notebooks]      

Constructivism, Cognition, and Science – An Investigation of Its Links and Possible Shortcomings

This paper addresses the questions concerningthe relationship between scientific andcognitive processes. The fact that both,science and cognition, aim at acquiring somekind of knowledge or representationabout the “world” is the key for establishing alink between these two domains. It turns outthat the constructivist frameworkrepresents an adequate epistemologicalfoundation for this undertaking, as its focusof interest is on the (constructive)relationship between the world and itsrepresentation. More specifically, it will beshown how cognitive processes and their primaryconcern to construct a representation of theenvironment and to generate functionallyfitting behavior can act as the basis forembedding the activities and dynamics of theprocess of science in them by making use ofconstructivist concepts, such as functionalfitness, structure determinedness, etc.Cognitive science and artificiallife provide the conceptual framework of representational spaces and their interactionbetween each other and with the environmentenabling us to establish this link betweencognitive processes and thedevelopment/dynamics of scientific theories.The concepts of activation, synaptic weight,and genetic (representational) spaces arepowerful tools which can be used as“explanatory vehicles”for a cognitivefoundation of science, more specifically forthe “context of discovery” (i.e., thedevelopment, construction, and dynamics ofscientific theories and paradigms).Representational spaces do not only offer us abetter understanding of embedding science incognition, but also show, how theconstructivist framework, both, can act as anadequate epistemological foundation for theseprocesses and can be instantiated by theserepresentational concepts from cognitivescience. The final part of this paper addresses somemore fundamental questions concerning thepositivistic and constructivist understandingof science and human cognition. Among otherthings it is asked, whether a purelyfunctionalist and quantitative view of theworld aiming almost exclusively at itsprediction and control is really satisfying forour intellect (having the goal of achieving aprofound understanding of reality).     

Recognition of Robinsonian dissimilarities

We present an O(n ³)-time, O(n ²)-space algorithm to test whether a dissimilarity d on an n-object set X is Robinsonian, i.e., X admits an ordering such that i≤j≤k implies that d(x _i,x_k)≥max {d(x_i,x_j),d(x_j,x_k)}.     

From Doing to Performing Phenomenology: Implications and Possibilities

This commentary on Kurt Vanhoutte and Nele Wynants’s of ‘Performing phenomenology: negotiating presence in intermedial theatre’ focuses on the implications of staging phenomenological research. In my opinion the authors missed an opportunity to stress more what W (Double U), a performance of CREW has to offer postphenomenology and what it actually means to ‘perform’ phenomenology. I will not only argue that W (Double U) because of its performative nature offers a reflection on postphenomenology, but also that the performance must be understood as a specific kind of research, conducted simultaneously from a theoretical and aesthetic orientation, leading to a complex interaction between perception and reflection, and offering a valuable, different perspective on postphenomenological research issues. W (Double U) in this respect functions as a ‘theoretical object’, producing a specific kind of embodied knowledge. Finally I will emphasize the possible radical potential in W (Double U), because I do believe that the performance, although it might not lead explicitly to social change, does have an important social and political relevance that the authors do not really delve into.     

An Optimal Algorithm To Recognize Robinsonian Dissimilarities

A dissimilarity D on a finite set S is said to be Robinsonian if S can be totally ordered in such a way that, for every i < j < k, D (i, j) ≤ D (i, k) and D (j, k) ≤ D (i, k). Intuitively, D is Robinsonian if S can be represented by points on a line. Recognizing Robinsonian dissimilarities has many applications in seriation and classification. In this paper, we present an optimal O (n ²) algorithm to recognize Robinsonian dissimilarities, where n is the cardinal of S. Our result improves the already known algorithms.     

Points of View Beyond Models: Towards a Formal Approach to Points of View as Access to the World

According to Vázquez and Liz (Found Sci 16(4): 383–391, 2011), Points of View (PoV) can be considered in two different ways. On the one hand, they can be explained following the model of propositional attitudes. This model assumes that the internal structure of a PoV is constituted by a subject, a set of contents, and a set of relations between the subject and those contents. On the other hand, we can analyze points of view taking as a model the notions of location and access. If we choose to follow the second approach, instead of the first one, the internal structure of a PoV is not directly addressed, and the emphasized features of PoV are related to the function that PoV are intended to have. That is, PoV are directly identified by their role and they can solely be understood as ways of accessing the world that bring some kind of perspective about it. Having this in mind, we would like to propose a notation that explains how to understand such access as a sort of models (that can allow the creation of concepts), independently of whether the precise PoV under consideration is impersonal or non-impersonal, its kind of content, and its subjective or objective character. First, we will present an account of some previous approaches to the study of points of view. Then, we will analyze what kind of structure the world is assumed to posses and how the access to it is possible. Third, we will develop a notation that explains PoV as qualitative dimensions by means of which it is possible to valuate objects and states of the world.     

Additive Biclustering: A Comparison of One New and Two Existing ALS Algorithms

The additive biclustering model for two-way two-mode object by variable data implies overlapping clusterings of both the objects and the variables together with a weight for each bicluster (i.e., a pair of an object and a variable cluster). In the data analysis, an additive biclustering model is fitted to given data by means of minimizing a least squares loss function. To this end, two alternating least squares algorithms (ALS) may be used: (1) PENCLUS, and (2) Baier’s ALS approach. However, both algorithms suffer from some inherent limitations, which may hamper their performance. As a way out, based on theoretical results regarding optimally designing ALS algorithms, in this paper a new ALS algorithm will be presented. In a simulation study this algorithm will be shown to outperform the existing ALS approaches.     

Recognizing Treelike k-Dissimilarities

A k-dissimilarity D on a finite set X, |X|????k, is a map from the set of size k subsets of X to the real numbers. Such maps naturally arise from edgeweighted trees T with leaf-set X: Given a subset Y of X of size k, D(Y ) is defined to be the total length of the smallest subtree of T with leaf-set Y . In case k?=?2, it is well-known that 2-dissimilarities arising in this way can be characterized by the so-called ??4-point condition??. However, in case k?>?2 Pachter and Speyer (2004) recently posed the following question: Given an arbitrary k-dissimilarity, how do we test whether this map comes from a tree? In this paper, we provide an answer to this question, showing that for k????3 a k-dissimilarity on a set X arises from a tree if and only if its restriction to every 2?k-element subset of X arises from some tree, and that 2?k is the least possible subset size to ensure that this is the case. As a corollary, we show that there exists a polynomial-time algorithm to determine when a k-dissimilarity arises from a tree. We also give a 6-point condition for determining when a 3-dissimilarity arises from a tree, that is similar to the aforementioned 4-point condition.