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by Fredrik Akesson, John P. Lehoczky

Preliminary version, comments are welcome. This paper finds the eigenvalues and eigenvectors of the covariance matrix associated with multi-dimensional Brownian motion and the OrnsteinUhlenbeck processes. The result is given in closed form for the onedimensional Brownian motion. In the general case it involves some numerical computations, but the overall work is a small fraction of the work required by standard methods to compute eigenvalues and eigenvectors of a covariance matrix. The results have applications in a new QuasiMonte Carlo method [1] for computing the expected value of a function depending on a Brownian Motion. 1 One-Dimensional Brownian Motion Let fW (t); t 0g be a standard Brownian motion. Consider the time points t i = T i n; 1 i n. The covariance matrix CW = (c i;j) associated with (W (t 1); : : : ; W (t n)), has the elements c i;j = min(i; j) n T: (1)
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Jonathon Shlens
Principal component analysis (PCA) is a mainstay of modern data analysis - a black box that is widely used but poorly understood. The goal of this paper is to dispel the magic behind this black box. This tutorial focuses on building a solid intuition for how and why principal component analysis works; furthermore, it crystallizes this knowledge by deriving from simple intuitions, the mathematics behind PCA . This tutorial does not shy away from explaining the ideas informally, nor does it shy away from the mathematics. The hope is that by addressing both aspects, readers of all levels will be able to gain a better understanding of PCA as well as the when, the how and the why of applying this technique.
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