Windowed persistent homology: A topological signal processing algorithm applied to clinical obesity data

Overweight and obesity are highly prevalent in the population of the United States, affecting roughly 2/3 of Americans. These diseases, along with their associated conditions, are a major burden on the healthcare industry in terms of both dollars spent and effort expended. Volitional weight loss is attempted by many, but weight regain is common. The ability to predict which patients will lose weight and successfully maintain the loss versus those prone to regain weight would help ease this burden by allowing clinicians the ability to skip treatments likely to be ineffective. In this paper we introduce a new windowed approach to the persistent homology signal processing algorithm that, when paired with a modified, semimetric version of the Hausdorff distance, can differentiate the two groups where other commonly used methods fail. The novel approach is tested on accelerometer data gathered from an ongoing study at the University of Michigan. While most standard approaches to signal processing show no difference between the two groups, windowed persistent homology and the modified Hausdorff semimetric show a clear separation. This has significant implications for clinical decision making and patient care.