jHoles

A Java High Performance Tool For Topological Data Analysis

From algebraic topology a new set of algorithms have been derived. These algorithms are identified with “computational topology” or often pointed out as Topological Data Analysis (TDA) and are used for investigating high-dimensional data in a quantitative manner. TDA is largely used for the analysis of complex systems; for instance Ibekwe et al. [1] used TDA for reconstructing the relationship structure of E. coli O157 and the authors proved that the non-O157 is in 32 soils (16 organic and 16 conventionally managed soils). TDA was also used by De Silva [2] for the analysis of sensor networks and it was successfully applied to the study of viral evolution in biological complex systems [3]. Petri et al. [4] used a homological approach for studying the characteristics of functional brain networks at the mesoscopic level. Computational topology algorithms can be divided in two families: topological data compression and topological data completion. Algorithms for topological data compression aims to represent a collection of higher dimensional data points through a graph, the main algorithm is known as Mapper [5]. Topological data completion, conversely, completes data to more complex structures, i.e. simplicial complexes, that can be easiest analyzed. This subset of algorithms is based on Persistent Homology [6].

In Topology we established the basics of computational topology. In the following we would like to highlight a set of instances of topological data analysis. The following examples can be used by the reader as signpost in his or her personal quest to perform a convenient topological data analysis.

REFERENCES:

[1] A M IBEKWE, J MA, D E CROWLEY, C-H YANG, A M JOHNSON, T C PETROSSIAN, AND P Y LUM. Topological data analysis of Escherichia coli O157: H7 and non-O157 survival in soils. Frontiers in cellular and infection microbiology, 4, 2014.

[2] V DE SILVA AND R GHRIST. Coverage in sensor networks via persistent homology. Algebraic & Geometric Topology, 7(339-358):24, 2007

[3] J-M CHAN, G CARLSSON, AND R RABADAN. Topology of viral evolution. Proceedings of the National Academy of Sciences, 110(46):18566–18571, 2013.

[4] G PETRI, P EXPERT, F TURKHEIMER, R CARHART-HARRIS, D NUTT, P J HELLYER, AND F VACCARINO. Homological scaffolds of brain functional networks. Journal of The Royal Society Interface, 11(101):20140873, 2014.

[5] F MEMOLI, G SINGH AND G CARLSSON. Topological Methods for the Analysis of High Dimensional Data Sets and 3D Object Recognition. 2007.

[6] H EDELSBRUNNER AND J HARER. Computational topology: an introduction. American Mathematical Soc., 2010.