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Persistent Homology of Trigonometric Functions

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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, ...
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Topological Analysis of Baseball Batting Data

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For this analysis I examined baseball hitting data from the 2015 season from the Lahman Database at http://www.seanlahman.com/. My objective was to categorize different types of offensive players.  This following plots were created by using CEDAR with eccentricity and density as the lens functions. The coloring on this first plot is based on number of games played in a season by each player. The blue nodes contain players with a lower number of games played, while red indicates a higher number of games played. As we will see in other plots, the number of games played is correlated with many other statistics. This intuitively makes sense, because players who play in more games have more opportunities to increase their other statistics.  The first identifiable group was "low impact hitters and pitchers". These players are identified by the low number of games they played in, and thus low numbers of other statistics. Pitchers fall into this category because th...
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