A Study of Isomap Extensions and Topological Data Analysis with Applications to Financial Data

Abstract: This thesis focuses mainly on nonlinear dimensionality reduction. A comparison of a number of different extensions of the Isomap-algorithm found in the literature is presented as well as an introduction to the underlying theory. Further, two novel extensions of the algorithm are introduced. Residual variance is used as a means of quantifying performance and optimizing parameters. One of the novel approaches, NNL-Isomap, is applied to financial data sets such as the S&P 500, the results are compared with those from PCA and, when appropriate, well-known academic models. The thesis further introduces a selection of concepts from topological data analysis together with corresponding short explanatory examples as well as applies these concepts to some simple financial examples.