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Search Results: 1 - 10 of 118 matches for " Fionn Murtagh "
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Hilbert Space Becomes Ultrametric in the High Dimensional Limit: Application to Very High Frequency Data Analysis
Fionn Murtagh
Physics , 2007,
Abstract: 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 set, or local. By quantifying extent or degree of ultrametricity in a data set, we show that ultrametricity becomes pervasive as dimensionality and/or spatial sparsity increases. This leads us to assert that very high dimensional data are of simple structure. We exemplify this finding through a range of simulated data cases. We discuss also application to very high frequency time series segmentation and modeling.
Ultrametric embedding: application to data fingerprinting and to fast data clustering
Fionn Murtagh
Mathematics , 2006,
Abstract: 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 locally present in data. We show how the ultrametricity can be assessed in text or document collections, and in time series signals. An aspect of importance here is that to draw benefit from this perspective the data may need to be recoded. Such data recoding can also be powerful in proximity searching, as we will show, where the data is embedded globally and not locally in an ultrametric space.
The Remarkable Simplicity of Very High Dimensional Data: Application of Model-Based Clustering
Fionn Murtagh
Mathematics , 2008, DOI: 10.1007/s00357-009-9037-9
Abstract: 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 local. By quantifying extent or degree of ultrametricity in a data set, we show that ultrametricity becomes pervasive as dimensionality and/or spatial sparsity increases. This leads us to assert that very high dimensional data are of simple structure. We exemplify this finding through a range of simulated data cases. We discuss also application to very high frequency time series segmentation and modeling.
Symmetry in Data Mining and Analysis: A Unifying View based on Hierarchy
Fionn Murtagh
Mathematics , 2008, DOI: 10.1134/S0081543809020175
Abstract: 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 domain of interest. "Structure" has long been understood as symmetry which can take many forms with respect to any transformation, including point, translational, rotational, and many others. Beginning with the role of number theory in expressing data, we show how we can naturally proceed to hierarchical structures. We show how this both encapsulates traditional paradigms in data analysis, and also opens up new perspectives towards issues that are on the order of the day, including data mining of massive, high dimensional, heterogeneous data sets. Linkages with other fields are also discussed including computational logic and symbolic dynamics. The structures in data surveyed here are based on hierarchy, represented as p-adic numbers or an ultrametric topology.
The Haar Wavelet Transform of a Dendrogram
Fionn Murtagh
Computer Science , 2006, DOI: 10.1007/s00357-007-0007-9
Abstract: We describe a new wavelet transform, for use on hierarchies or binary rooted trees. The theoretical framework of this approach to data analysis is described. Case studies are used to further exemplify this approach. A first set of application studies deals with data array smoothing, or filtering. A second set of application studies relates to hierarchical tree condensation. Finally, a third study explores the wavelet decomposition, and the reproducibility of data sets such as text, including a new perspective on the generation or computability of such data objects.
Big Data Scaling through Metric Mapping: Exploiting the Remarkable Simplicity of Very High Dimensional Spaces using Correspondence Analysis
Fionn Murtagh
Computer Science , 2015,
Abstract: We present new findings in regard to data analysis in very high dimensional spaces. We use dimensionalities up to around one million. A particular benefit of Correspondence Analysis is its suitability for carrying out an orthonormal mapping, or scaling, of power law distributed data. Power law distributed data are found in many domains. Correspondence factor analysis provides a latent semantic or principal axes mapping. Our experiments use data from digital chemistry and finance, and other statistically generated data.
The Haar Wavelet Transform of a Dendrogram: Additional Notes
Fionn Murtagh
Computer Science , 2007,
Abstract: We consider the wavelet transform of a finite, rooted, node-ranked, $p$-way tree, focusing on the case of binary ($p = 2$) trees. We study a Haar wavelet transform on this tree. Wavelet transforms allow for multiresolution analysis through translation and dilation of a wavelet function. We explore how this works in our tree context.
Ultrametric and Generalized Ultrametric in Computational Logic and in Data Analysis
Fionn Murtagh
Computer Science , 2010,
Abstract: Following a review of metric, ultrametric and generalized ultrametric, we review their application in data analysis. We show how they allow us to explore both geometry and topology of information, starting with measured data. Some themes are then developed based on the use of metric, ultrametric and generalized ultrametric in logic. In particular we study approximation chains in an ultrametric or generalized ultrametric context. Our aim in this work is to extend the scope of data analysis by facilitating reasoning based on the data analysis; and to show how quantitative and qualitative data analysis can be incorporated into logic programming.
Between the Information Economy and Student Recruitment: Present Conjuncture and Future Prospects
Fionn Murtagh
Computer Science , 2008,
Abstract: In university programs and curricula, in general we react to the need to meet market needs. We respond to market stimulus, or at least try to do so. Consider now an inverted view. Consider our data and perspectives in university programs as reflecting and indeed presaging economic trends. In this article I pursue this line of thinking. I show how various past events fit very well into this new view. I provide explanation for why some technology trends happened as they did, and why some current developments are important now.
Ultrametric Model of Mind, II: Application to Text Content Analysis
Fionn Murtagh
Computer Science , 2012,
Abstract: 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 representational and computational framework for this. Now we look at the extent to which we can find ultrametricity in text. We use coherent and meaningful collections of nearly 1000 texts to show how we can measure inherent ultrametricity. On the basis of our findings we hypothesize that inherent ultrametricty is a basis for further exploring unconscious thought processes.
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