Research on random permutations of long-range dependent sequences and on drug target prediction

The dissertation consists of results from research in two areas: probability (long-range dependence) and bioinformatics. Chapter 1 is an introduction. Chapter 2 and Chapter 3 are in the area of probability. We show that a complete random permutation does not destroy the covariances of a sequence of random variables, but merely equalizes them. For stationary sequences with long-range dependence, the common value of the equalized covariances depends on two parameters N and H. N is the length of the original sequence and H is its Hurst parameter which characterizes the asymptotic decay of its covariances. Using the periodogram method, we explain why one is led to think, mistakenly, that the resulting randomized sequence yields an "estimated H close to 1/2". We also analyze the effect of three other types of random permutations (external, internal, two-level) on a long-range dependent finite variance stationary sequence both in the time domain and in the frequency domain. Chapter 4 is in the area of bioinformatics. We consider the problem of identifying drug targets from DNA microarray data. Our approach is to cast the problem as one of identifying outliers in a large, sparse system of simultaneous equations, describing the influence of both the underlying gene regulatory structure and the external effects of the potential drug candidate. We present a variety of empirical results demonstrating the capabilities of our method.