Abstract
Covariance matrices are used for a wide range of applications in particle physics, including Kálmán filter for tracking purposes or Primary Component Analysis for dimensionality reduction. Based on a novel decomposition of the covariance matrix, a design that requires only one pass of data for calculating the covariance matrix is presented. Two computation engines are used depending on parallelizability of the necessary computation steps. The design is implemented onto a hybrid FPGA/CPU system and yields speed-up of up to 5 orders of magnitude compared to previous FPGA implementation.
Highlights
The covariance matrix K represents the covariances between all the permutations of the different data sets
The covariance matrix will be a symmetrical m × m matrix, with the diagonal Kii being the variance on data set Xi.: K = ccoovvv(a(XXr(1...1X..,1XX.)m2..))
A single-pass covariance algorithm design based on a novel decomposition of the covariance matrix has been implemented and successfully tested
Summary
Covariance is a measure of variability between two different data sets X and Y defined as: cov(X, Y) = E (X − E[X])(Y − E[Y]) . We will use the understanding of the expected value as the sample mean: E(X) = X 1 n n i=1 xi. The covariance matrix K represents the covariances between all the permutations of the different data sets. We use the convention that m denotes dimensionality, i.e. the number of different data sets, and n the number of samples. Data are represented by an m × n matrix X. The covariance matrix is defined as: Ki j = cov(Xi., X j.). The covariance matrix will be a symmetrical m × m matrix, with the diagonal Kii being the variance on data set Xi.:
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