Title :
Multivariate Data Analysis Using Persistence-Based Filtering and Topological Signatures
Author :
Rieck, Bastian ; Mara, Hubert ; Leitte, Heike
Author_Institution :
Interdiscipl. Center for Sci. Comput. (IWR), Heidelberg Univ., Heidelberg, Germany
Abstract :
The extraction of significant structures in arbitrary high-dimensional data sets is a challenging task. Moreover, classifying data points as noise in order to reduce a data set bears special relevance for many application domains. Standard methods such as clustering serve to reduce problem complexity by providing the user with classes of similar entities. However, they usually do not highlight relations between different entities and require a stopping criterion, e.g. the number of clusters to be detected. In this paper, we present a visualization pipeline based on recent advancements in algebraic topology. More precisely, we employ methods from persistent homology that enable topological data analysis on high-dimensional data sets. Our pipeline inherently copes with noisy data and data sets of arbitrary dimensions. It extracts central structures of a data set in a hierarchical manner by using a persistence-based filtering algorithm that is theoretically well-founded. We furthermore introduce persistence rings, a novel visualization technique for a class of topological features-the persistence intervals-of large data sets. Persistence rings provide a unique topological signature of a data set, which helps in recognizing similarities. In addition, we provide interactive visualization techniques that assist the user in evaluating the parameter space of our method in order to extract relevant structures. We describe and evaluate our analysis pipeline by means of two very distinct classes of data sets: First, a class of synthetic data sets containing topological objects is employed to highlight the interaction capabilities of our method. Second, in order to affirm the utility of our technique, we analyse a class of high-dimensional real-world data sets arising from current research in cultural heritage.
Keywords :
data analysis; data structures; data visualisation; history; information filtering; pattern classification; topology; algebraic topology; analysis pipeline; arbitrary high-dimensional data sets; central structures; cultural heritage; data points classification; hierarchical manner; high-dimensional real-world data sets; interaction capability; interactive visualization techniques; large data sets; multivariate data analysis; noisy data; parameter space evaluation; persistence rings; persistence-based filtering algorithm; persistent homology; problem complexity; relevant structures; significant structures extraction; stopping criterion; synthetic data sets; topological data analysis; topological features; topological objects; topological signatures; visualization pipeline; Clustering methods; Multivariate data sets; Network topology; Topological persistence; clustering; multivariate data;
Journal_Title :
Visualization and Computer Graphics, IEEE Transactions on
DOI :
10.1109/TVCG.2012.248