We show that, under certain smoothness conditions, a Brownian martingale, when evaluated at a fixed time, can be represented via an exponential formula at a later time. The time-dependent generator of this exponential operator only depends on the second order Malliavin derivative operator evaluated along a "frozen path". The exponential operator can be expanded explicitly to a series representation, which resembles the Dyson series of quantum mechanics. Our continuous-time martingale representation result can be proven independently by two different methods. In the first method, one constructs a time-evolution equation, by passage to the limit of a special case of a backward Taylor expansion of an approximating discrete time martingale. The exponential formula is a solution of the time-evolution equation, but we emphasize in our article that the time-evolution equation is a separate result of independent interest. In the second method, we use the property of denseness of exponential functions. We provide several applications of the exponential formula, and briefly highlight numerical applications of the backward Taylor expansion.

We introduce a new unsupervised learning problem: clustering widesense stationary ergodic stochastic processes. A covariance-based dissimilarity measure together with asymptotically consistent algorithms is designed for clustering offline and online datasets, respectively. We also suggest a formal criterion on the efficiency of dissimilarity measures, and discuss of some approach to improve the efficiency of our clustering algorithms, when they are applied to cluster particular type of processes, such as self-similar processes with wide-sense stationary ergodic increments. Clustering synthetic data and real-world data are provided as examples of applications.Keywords: cluster analysis¨wide-sense stationary ergodic processesc ovariance-based dissimilarity measure¨self-similar processes MCS (2010): 62-07¨60G10¨62M10

The study of disease-relevant gene modules is one of the main methods to discover disease pathway and potential drug targets. Recent studies have found that most disease proteins tend to form many separate connected components and scatter across the protein-protein interaction network. However, most of the research on discovering disease modules are biased toward well-studied seed genes, which tend to extend seed genes into a single connected subnetwork. In this paper, we propose N2V-HC, an algorithm framework aiming to unbiasedly discover the scattered disease modules based on deep representation learning of integrated multi-layer biological networks. Our method first predicts disease associated genes based on summary data of Genome-wide Association Studies (GWAS) and expression Quantitative Trait Loci (eQTL) studies, and generates an integrated network on the basis of human interactome. The features of nodes in the network are then extracted by deep representation learning. Hierarchical clustering with dynamic tree cut methods are applied to discover the modules that are enriched with disease associated genes. The evaluation on real networks and simulated networks show that N2V-HC performs better than existing methods in network module discovery. Case studies on Parkinson's disease and Alzheimer's disease, show that N2V-HC can be used to discover biological meaningful modules related to the pathways underlying complex diseases.

Expression quantitative trait locus (eQTL) analyses are critical in understanding the complex functional regulatory natures of genetic variation and have been widely used in the interpretation of disease-associated variants identified by genome-wide association studies (GWAS). Emerging evidence has shown that trans-eQTL effects on remote gene expression could be mediated by local transcripts, which is known as the mediation effects. To discover the genome-wide eQTL mediation effects combing genomic and transcriptomic profiles, it is necessary to develop novel computational methods to rapidly scan large number of candidate associations while controlling for multiple testing appropriately. Here, we present eQTLMAPT, an R package aiming to perform eQTL mediation analysis with implementation of efficient permutation procedures in multiple testing correction. eQTLMAPT is advantageous in threefold. First, it accelerates mediation analysis by effectively pruning the permutation process through adaptive permutation scheme. Second, it can efficiently and accurately estimate the significance level of mediation effects by modeling the null distribution with generalized Pareto distribution (GPD) trained from a few permutation statistics. Third, eQTLMAPT provides flexible interfaces for users to combine various permutation schemes with different confounding adjustment methods. Experiments on real eQTL dataset demonstrate that eQTLMAPT provides higher resolution of estimated significance of mediation effects and is an order of magnitude faster than compared methods with similar accuracy.

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We propose a wavelet-based approach to construct consistent estimators of the pointwise Hölder exponent of a multifractional Brownian motion, in the case where this underlying process is not directly observed. The relative merits of our estimator are discussed, and we introduce an application to the problem of estimating the functional parameter of a nonlinear model.

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