Monitoring the shape of weather, soundscapes, and dynamical systems: a new statistic for dimension-driven data analysis on large data sets

Abstract: Dimensionality-reduction methods are a fundamental tool in the analysis of large data sets. These algorithms work on the assumption that the "intrinsic dimension" of the data is generally much smaller than the ambient dimension in which it is collected. Alongside their usual purpose of mapping data into a smaller dimension with minimal information loss, dimensionality-reduction techniques implicitly or explicitly provide information about the dimension of the data set. In this paper, we propose a new statistic that we call the $\kappa$-profile for analysis of large data sets. The $\kappa$-profile arises from a dimensionality-reduction optimization problem: namely that of finding a projection into $k$-dimensions that optimally preserves the secants between points in the data set. From this optimal projection we extract $\kappa,$ the norm of the shortest projected secant from among the set of all normalized secants. This $\kappa$ can be computed for any $k$; thus the tuple of $\kappa$ values (indexed by dimension) becomes a $\kappa$-profile. Algorithms such as the Secant-Avoidance Projection algorithm and the Hierarchical Secant-Avoidance Projection algorithm, provide a computationally feasible means of estimating the $\kappa$-profile for large data sets, and thus a method of understanding and monitoring their behavior. As we demonstrate in this paper, the $\kappa$-profile serves as a useful statistic in several representative settings: weather data, soundscape data, and dynamical systems data.