Topological Data Analysis is a quickly expanding field but one particular subfield, multidimensional persistence, has hit a dead end. Although TDA is a very active field, it has been proven that the one-dimensional persistence used in persistent homology cannot be generalized to higher dimensions. With this in mind, progress can still be made in the accuracy of approximating it. The central challenge lies in the multiple persistence parameters. Using more than one parameter at a time creates a multi-filtration of the data which cannot be totally ordered in the way that a single filtration can.
The goal of this thesis is to contribute to the development of persistence heat maps by replacing the persistent betti number function (PBN) defined by Xia and Wei in 2015 with a new persistence summary function, the accumulated persistence function (APF) defined by Biscio and Moller in 2016. The PBN function fails to capture persistence in most cases and thus their heat maps lack important information. The APF, on the other hand, does capture persistence that can be seen in their heat maps.
A heat map is a way to visually describe three dimensions with two spatial dimensions and color. In two-dimensional persistence heat maps, the two chosen parameters lie on the x- and y- axes. These persistence parameters define a complex on the data, and its topology is represented by the color. We use the method of heat maps introduced by Xia and Wei. We acquired an R script from Matthew Pietrosanu to generate our own heat maps with the second parameter being curvature threshold. We also use the accumulated persistence function introduced by Biscio and Moller, who provided an R script to compute the APF on a data set. We then wrote new code, building from the existing codes, to create a modified heat map. In all the examples in this thesis, we show both the old PBN and the new APF heat maps to illustrate their differences and similarities. We study the two-dimensional heat maps with respect to curvature applied to two types of parameterized knots, Lissajous knots and torus knots. We also show how both heat maps can be used to compare and contrast data sets.
This research is important because the persistence heat map acts as a guide for finding topologically significant features as the data changes with respect to two parameters. Improving the accuracy of the heat map ultimately improves the efficiency of data analysis. Two-dimensional persistence has practical applications in analyses of data coming from proteins and DNA. The unfolding of proteins offers a second parameter of configuration over time, while tangled DNA may have a second parameter of curvature.
The concluding argument of this thesis is that using the accumulated persistence function in conjunction with the persistent betti number function provides a more accurate representation of two-dimensional persistence than the PBN heat map alone.