Paul Soma Undergraduate Research Blog

For this analysis I examined the persistent homology of trigonometric functions, such as f(t) = sin(t), using the JavaPlex Matlab library (see https://appliedtopology.github.io/javaplex/). I first took sliding window samples of the function  f ( t ) = sin( t ). Define a sliding window as , and define the point cloud of sliding windows . where  T  is a set of so-called "starting points" for each sliding window,  d  + 1 is the number of points sampled from  f  in each sliding window and the dimension of the point cloud of sliding windows, and  tau  is the step between points sampled within each sliding window. After generating the points, I created a Vietoris-Rips filtered simplicial complex from the point cloud using JavaPlex. With parameters  d  = 45,  tau  = 0.2, and  T  as the set of 40 equally distributed reals on [0,8*pi], I produced the following barcode: In this plot, ...

I found a baseball hitting data set that had more detailed statistics, including batting average, slugging percentage, and categorical data about what positions each player played during the 2015 season. The parameters for this analysis with Cedar were: normalized data, 10 cluster bins, lens 1 is eccentricity with exponent of 2 and 12 partitions with 50% overlap, not equalized. Lens 2 is density with a Point-wise Gaussian kernel width of 1, and 12 partitions with 50% overlap, not equalized. The conventional wisdom of baseball would say that pitchers are the least effective hitters, and that first and third basemen and some outfielders are typically higher impact or power hitters. When we color by players who played pitcher during the 2015 season, we see the circled group consists only of pitchers. These pitchers averaged less than one plate appearance throughout the entire season. This matches the conventional wisdom that pitchers are typically low-impact hitters. ...