Toroidal PCA via density ridges

Principal Component Analysis (PCA) is a well-known linear dimension-reduction technique designed for Euclidean data. In a wide spectrum of applied fields, however, it is common to observe multivariate circular data (also known as toroidal data), rendering spurious the use of PCA on it due to the periodicity of its support. This paper introduces Toroidal Ridge PCA (TR-PCA), a novel construction of PCA for bivariate circular data that leverages the concept of density ridges as a flexible first principal component analog. Two reference bivariate circular distributions, the bivariate sine von Mises and the bivariate wrapped Cauchy, are employed as the parametric distributional basis of TR-PCA. Efficient algorithms are presented to compute density ridges for these two distribution models. A complete PCA methodology adapted to toroidal data (including scores, variance decomposition, and resolution of edge cases) is introduced and implemented in the companion R package ridgetorus. The usefulness of TR-PCA is showcased with a novel case study involving the analysis of ocean currents on the coast of Santa Barbara.Comment: 20 pages, 8 figures, 1 tabl

W′W' and Z′Z' bosons search at s=13\sqrt{s}=13 TeV with the ATLAS detector at the LHC

Invariant and transverse mass distributions of Beyond the Standard Model signals of $Z’$ and $W’$ bosons have been studied using 1 lepton and 2 lepton final states in ATLAS data, resulting in exclusion with more than 95% confidence for 2 and 3 TeV $Z’$ and $W’$, as well as setting exclusion limits of 3.3 TeV on the $Z’$ and 3.7 TeV on the $W’$. Also, data-driven hadronic jet background estimation and high-mass background fitting have been studied and implemented