Abstract
Testing for independence plays a fundamental role in many statistical techniques. Among the nonparametric approaches, the distance-based methods (such as the distance correlation-based hypotheses testing for independence) have many advantages, compared with many other alternatives. A known limitation of the distance-based method is that its computational complexity can be high. In general, when the sample size is n, the order of computational complexity of a distance-based method, which typically requires computing of all pairwise distances, can be O(n2). Recent advances have discovered that in the univariate cases, a fast method with O(n log n) computational complexity and O(n) memory requirement exists. In this paper, we introduce a test of independence method based on random projection and distance correlation, which achieves nearly the same power as the state-of-the-art distance-based approach, works in the multivariate cases, and enjoys the O(nK log n) computational complexity and O( max{n, K}) memory requirement, where K is the number of random projections. Note that saving is achieved when K < n/ log n. We name our method a Randomly Projected Distance Covariance (RPDC). The statistical theoretical analysis takes advantage of some techniques on the random projection which are rooted in contemporary machine learning. Numerical experiments demonstrate the efficiency of the proposed method, relative to numerous competitors.
Highlights
Test of independence is a fundamental problem in statistics, with many existing work including the maximal information coefficient (MIC) [1], the copula based measures [2,3], the kernel based criterion [4] and the distance correlation [5,6], which motivated our current work
The above procedure is motivated by the observation that the asymptotic distribution of the test statistic nΩn can be approximated by a Gamma distribution, whose parameters can be estimated by Eq 3.2 and Eq 3.3
The break-even sample size decreases as the data dimension increases, which implies that our proposed method is more advantageous than the direct method when random variables are of high dimension
Summary
Test of independence is a fundamental problem in statistics, with many existing work including the maximal information coefficient (MIC) [1], the copula based measures [2,3], the kernel based criterion [4] and the distance correlation [5,6], which motivated our current work. When the random variables are univariate, there exist efficient numerical algorithms of time complexity. For the multivariate random variables, we have not found any efficient algorithms in existing papers after an extensive literature survey. This work will meet the demands for numerically efficient independence tests of multivariate random variables. The newly proposed test of independence between two (potentially multivariate) random variables X and Y works as follows. Both X and Y are randomly projected to onedimensional spaces. We will show (in Theorem 3.1) that the newly proposed algorithm enjoys the O(Kn log n) computational complexity and O( max{n, K}) memory requirement, where K is the number of random projections and n is the sample size.
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