Dealing with non-metric dissimilarities in fuzzy central clustering algorithms

Clustering is the problem of grouping objects on the basis of a similarity measure among them. Relational clustering methods can be employed when a feature-based representation of the objects is not available, and their description is given in terms of pairwise (dis)similarities. This paper focuses on the relational duals of fuzzy central clustering algorithms, and their application in situations when patterns are represented by means of non-metric pairwise dissimilarities. Symmetrization and shift operations have been proposed to transform the dissimilarities among patterns from non-metric to metric. In this paper, we analyze how four popular fuzzy central clustering algorithms are affected by such transformations. The main contributions include the lack of invariance to shift operations, as well as the invariance to symmetrization. Moreover, we highlight the connections between relational duals of central clustering algorithms and central clustering algorithms in kernel-induced spaces. One among the presented algorithms has never been proposed for non-metric relational clustering, and turns out to be very robust to shift operations. (C) 2008 Elsevier Inc. All rights reserved

The force of dissimilar analogies in bioethics

Although analogical reasoning has long been a popular method of reasoning in bioethics, current literature does not sufficiently grasp its variety. We assert that the main shortcoming is the fact that an analogy's value is often judged on the extent of similarity between the source situation and the target situation, while in (bio)ethics, analogies are often used because of certain dissimilarities rather than in spite of them. We make a clear distinction between dissimilarities that aim to reinforce a similar approach in the source situation and the target situation and dissimilarities that aim to undermine or denounce a similar approach. The former kind of dissimilarity offers the analogy more normative force than if there were no dissimilarities present; this is often overlooked by authors who regard all relevant dissimilarities as detrimental to the analogy's strength. Another observation is that an evaluation of the normative force of an analogy cannot be made independently of moral principles or theories. Without these, one cannot select which elements in an analogy are morally relevant nor determine how they should be interpreted

A fingerprint based metric for measuring similarities of crystalline structures

Measuring similarities/dissimilarities between atomic structures is important for the exploration of potential energy landscapes. However, the cell vectors together with the coordinates of the atoms, which are generally used to describe periodic systems, are quantities not suitable as fingerprints to distinguish structures. Based on a characterization of the local environment of all atoms in a cell we introduce crystal fingerprints that can be calculated easily and allow to define configurational distances between crystalline structures that satisfy the mathematical properties of a metric. This distance between two configurations is a measure of their similarity/dissimilarity and it allows in particular to distinguish structures. The new method is an useful tool within various energy landscape exploration schemes, such as minima hopping, random search, swarm intelligence algorithms and high-throughput screenings

Functional Multidimensional Scaling

Multidimensional scaling is an important component in analyzing proximity (similarity or dissimilarity) between objects and plays a key role in creating low-dimensional visualizations of objects. Regardless of the progress in this area, traditional solutions of multidimensional scaling problems are inapplicable to the proximity which change in time. In this dissertation, we focus on dissimilarity instead of similarity. Motivated by the studies of functional data analysis, we extend the current multidimensional scaling techniques and propose a functional method to obtain lower-dimensional smooth representations in terms of time-varying dissimilarities. This method incorporates the smoothness approach of functional data analysis by using cubic B-spline basis functions. The model is also designed to arrive at optimal representations such that dissimilarities evaluated by estimated representations are almost the same as original dissimilarities of objects in a low dimension which is easier for people to recognize. We verify the feasibility of the model by running simulations, as well as using the closing prices of the S&P 500 stocks as a real case to analyze their dissimilarities. This case study reconstructs the 500 stocks with this functional multidimensional scaling method and provides us a good visualization on a 2D map for the 500 stocks so that we can see how their dissimilarities change smoothly in each month of the year 2018. Following the analysis of all of the 500 stocks, the cluster analysis of the first 15 stocks is displayed based on some conditions so that it helps us see how the stocks move from month to month and offers a new tool to cluster the stocks in the future

The triangle inequality constraint in similarity judgments

Since Tversky's (1977) seminal investigation, the triangle inequality, along with symmetry and minimality, have had a central role in investigations of the fundamental constraints on human similarity judgments. The meaning of minimality and symmetry in similarity judgments has been straightforward, but this is not the case for the triangle inequality. Expressed in terms of dissimilarities, and assuming a simple, linear function between dissimilarities and distances, the triangle inequality constraint implies that human behaviour should be consistent with Dissimilarity (A,B) + Dissimilarity (B,C) ≥ Dissimilarity (A,C), where A, B, and C are any three stimuli. We show how we can translate this constraint into one for similarities, using Shepard's (1987) generalization law, and so derive the multiplicative triangle inequality for similarities, Sim(A,C)≥Sim(A,B)(dot operator)Sim(B,C) where 0≤Sim(x,y)≤1. Can humans violate the multiplicative triangle inequality? An empirical demonstration shows that they can

Ranking and significance of variable-length similarity-based time series motifs

The detection of very similar patterns in a time series, commonly called motifs, has received continuous and increasing attention from diverse scientific communities. In particular, recent approaches for discovering similar motifs of different lengths have been proposed. In this work, we show that such variable-length similarity-based motifs cannot be directly compared, and hence ranked, by their normalized dissimilarities. Specifically, we find that length-normalized motif dissimilarities still have intrinsic dependencies on the motif length, and that lowest dissimilarities are particularly affected by this dependency. Moreover, we find that such dependencies are generally non-linear and change with the considered data set and dissimilarity measure. Based on these findings, we propose a solution to rank those motifs and measure their significance. This solution relies on a compact but accurate model of the dissimilarity space, using a beta distribution with three parameters that depend on the motif length in a non-linear way. We believe the incomparability of variable-length dissimilarities could go beyond the field of time series, and that similar modeling strategies as the one used here could be of help in a more broad context.Comment: 20 pages, 10 figure

On aggregation operators of transitive similarity and dissimilarity relations

Similarity and dissimilarity are widely used concepts. One of the most studied matters is their combination or aggregation. However, transitivity property is often ignored when aggregating despite being a highly important property, studied by many authors but from different points of view. We collect here some results in preserving transitivity when aggregating, intending to clarify the relationship between aggregation and transitivity and making it useful to design aggregation operators that keep transitivity property. Some examples of the utility of the results are also shown.Peer ReviewedPostprint (published version