Related papers: Ultrametric and Generalized Ultrametric in Computa…

We begin with pervasive ultrametricity due to high dimensionality and/or spatial sparsity. How extent or degree of ultrametricity can be quantified leads us to the discussion of varied practical cases when ultrametricity can be partially or…

A number of topics involving metrics and measures are discussed, including some of the special structure associated with ultrametrics.

We review the theory and practice of determining what parts of a data set are ultrametric. It is assumed that the data set, to begin with, is endowed with a metric, and we include discussion of how this can be brought about if a…

The increasing needs of clustering massive datasets and the high cost of running clustering algorithms poses difficult problems for users. In this context it is important to determine if a data set is clusterable, that is, it may be…

Generalized metrics, arising from Lawvere's view of metric spaces as enriched categories, have been widely applied in denotational semantics as a way to measure to which extent two programs behave in a similar, although non equivalent, way.…

Topological data analysis asks when balls in a metric space $(X,d)$ intersect. Geometric data analysis asks how much balls have to be enlarged to intersect. We connect this principle to the traditional core geometric concept of curvature.…

An ultrametric topology formalizes the notion of hierarchical structure. An ultrametric embedding, referred to here as ultrametricity, is implied by a hierarchical embedding. Such hierarchical structure can be global in the data set, or…

In a companion paper, Murtagh (2012), we discussed how Matte Blanco's work linked the unrepressed unconscious (in the human) to symmetric logic and thought processes. We showed how ultrametric topology provides a most useful…

Finite metric spaces arise in many different contexts. Enormous bodies of data, scientific, commercial and others can often be viewed as large metric spaces. It turns out that the metric of graphs reveals a lot of interesting information.…

We study two aspects of information semantics: (i) the collection of all relationships, (ii) tracking and spotting anomaly and change. The first is implemented by endowing all relevant information spaces with a Euclidean metric in a common…

An ultrametric topology formalizes the notion of hierarchical structure. An ultrametric embedding, referred to here as ultrametricity, is implied by a natural hierarchical embedding. Such hierarchical structure can be global in the data…

Some examples and basic properties of ultrametric spaces are briefly discussed.

Topological data analysis has emerged as a powerful tool for extracting the metric, geometric and topological features underlying the data as a multi-resolution summary statistic, and has found applications in several areas where data…

We combine continuous and integral logics and found a logical framework for metric measure spaces equipped with a family of continuous relations and operations. We prove the ultraproduct theorem and deduce compactness and other usual…

Since the introduction of network psychometrics, several connections to statistical models in "classical" psychometrics (i.e., IRT, SEM, GLM) as well as to approaches from other research fields have been established. In this paper, these…

Data analysis and data mining are concerned with unsupervised pattern finding and structure determination in data sets. "Structure" can be understood as symmetry and a range of symmetries are expressed by hierarchy. Such symmetries directly…

Data analysis and data mining are concerned with unsupervised pattern finding and structure determination in data sets. The data sets themselves are explicitly linked as a form of representation to an observational or otherwise empirical…

Psychometrics and quantitative psychology rely strongly on statistical models to measure psychological processes. As a branch of mathematics, geometry is inherently connected to measurement and focuses on properties such as distance and…

During the past two decades there has been a lot of interest in developing statistical depth notions that generalize the univariate concept of ranking to multivariate data. The notion of depth has also been extended to regression models and…

Visual insights into a wide variety of statistical methods, for both didactic and data analytic purposes, can often be achieved through geometric diagrams and geometrically based statistical graphs. This paper extols and illustrates the…