Topological Data Analysis on Magnetic Resonance Image Biomarkers Big data provides us the source to extract new insights in various disciplines. In healthcare, big data may help discover the pathology of a disease and new effective treatments. However, the era of big data taxes the ability of many researchers to analyze and interact with data in biomedical research. Traditional statistical methods have limitations to infer relevant features for possibly complex data. Topological Data Analysis (TDA) provides a set of new topological and geometric tools to discover the hidden relations or key features from complex data. It may help identify the key risk factors related to a given disease and reduce the noise impact from other factors. In this paper, we use TDA to analyze the multidimensional data from MRI of knee osteoarthritis patients.

Topological Data Analysis in Investment Decisions Abstract   This article explores the applications of Topological Data Analysis (TDA) in the finance field, especially addressing the primordial problem of asset allocation. Firstly, we build a rationale on why TDA can be a better alternative to traditional risk indicators such as standard deviation using real data sets. We apply Takens embedding theorem to reconstruct the time series of returns in a high dimensional space. We adopt the sliding window approach to draw the timedependent point cloud data sets and associate a topological space with them. We then apply the persistent homology to discover the topological patterns that appear in the multidimensional time series. The temporal changes in the persistence landscapes, which are the realvalued functions that encode the persistence of topological patterns, are captured via Lp norm. The time series of the Lp norms shows that it is better at measuring the dynamics of returns than the standard deviation.  Inspired by our findings, we explore an application of TDA in Enhanced Indexing (EI) that aims to build a portfolio of fewer assets than that in the index to outperform the latter. We propose a twostep procedure to accomplish this task. In step one, we utilize the Lp norms of the assets to propose a filtration technique of selecting a few assets from a larger pool of assets. In step two, we propose an optimization model to construct an optimal portfolio from the class of filtered assets for EI. To test the efficiency of this enhanced algorithm, experiments are carried out on ten data sets from financial markets across the globe. Our extensive empirical analysis exhibits that the proposed strategy delivers superior performance on several measures, including excess mean returns from the benchmark index and tail rewardrisk ratios than some of the existing models of EI in the literature. The proposed filtering strategy is also noted to be beneficial for both riskseeking and riskaverse investors.