Topological Data Analysis and Effective Resistance Preserving Isomorphism

Abstract

This thesis explores three topics. First of all, this thesis shows that types of single-nucleotide polymorphism (SNPs) from diabetes patients are more independent to each other than those from normal people. To demonstrate this, we construct a discrete product space of SNPs for a group of 1,182 type 2 diabetes patients and a pediatric control group of 2,364 people and compute the persistent homology using two dissimilarity functions measuring independence of two SNPs. Next, this thesis introduces the persistent shock surface to identifying shocks in the time series. We show that the number of electronic devices can be restored from the overall electric consumption time-series data using the persistent shock surface. Also, we introduce a concept of observable exogeneity of a shock in the autoregressive process and infer what magnitude of a shock can be regarded as an exogenous shock using numerical experiment. Lastly, as an appendix, this thesis introduces the concept of an effective resistance preserving isomorphism and shows that every subgraph isomorphism of two trees is the effective resistance preserving isomorphism. Moreover, it introduced a binary operation for two finite connected unweighted graphs G and H, called attaching H to G as a wing. This operation induces the canonical subgraph isomorphism between G and the resultant graph from the operation. For the finite connected unweighted graph G and H, we prove that the canonical subgraph isomorphism is the effective resistance preserving isomorphism up to a scalar.