Second-Order Comparison of Gaussian Random Functions and the Geometry of DNA Minicircles

Given two samples of continuous zero-mean iid Gaussian processes on [0,1], we consider the problem of testing whether they share the same covariance structure. Our study is motivated by the problem of determining whether the mechanical properties of short strands of DNA are significantly affected by their base-pair sequence; though expected to be true, had so far not been observed in three-dimensional electron microscopy data. The testing problem is seen to involve aspects of ill-posed inverse problems and a test based on a Karhunen–Loève approximation of the Hilbert–Schmidt distance of the empirical covariance operators is proposed and investigated. When applied to a dataset of DNA minicircles obtained through the electron microscope, our test seems to suggest potential sequence effects on DNA shape.