Essays in simulation and stochastic processes

Abstract

This thesis is concerned with two topics rooted in the analysis of time-series.

In the first, we improve the estimation of rare-event probabilities by stochastic simulation. The proposed method, quasi-monotone splitting uses generalized splitting to estimate integrals with respect to intractable target distributions by instead estimating them with respect to the terminal state of a certain Markov chain, allowing us to use time series methods to study them. We employ two innovations to this end: Problem constraints are exploited to derive a simple, efficient estimation strategy automatically for a tractable problem class, and The performance of the estimator is improved through the use of survival analysis and extreme value theory, in which near-optimal parameters can be derived with minimal intervention. We demonstrate applications of this algorithm to a variety of wireless reliability problems. The performance of the resulting algorithms are competitive with specialized Monte Carlo estimators for specific problems, and provide novel estimators for problems previously lacking known, efficient estimators. Some of the methods in this section were developed for a paper with several co-authors which has now been published.

The second topic is audio signal analysis. An important task here is style transfer, which attempts to synthesize a new signal from two others, a source and a target. The new synthetic signal should possess the microscopic “stylistic” statistics of the source, and the macroscopic “semantic” statistics of the target. We solve this problem using mosaicing style transfer, which decomposes the source signal into microscopic sub-samples, superimposing them to produce the new synthetic signal whose macroscopic statistics approximate the target. In such models, one chooses parameters by minimising some loss function which ideally approximates acoustic similarity as perceived by a human listener. We leverage the insight that human pitch perception is related to the local autocorrelogram of a signal to construct a novel loss function based on a difference between autocorrelograms. This, in combination with a signal approximation method based on orthogonal matching pursuits, results in a novel synthesis algorithm called autocorrelogram mosaicing. This algorithm is the only one we know of with public code that can mosaic with arbitrary pitch transposition of source audio, enabling style transfer between differently tuned instruments while maintaining musical consonance. The strength and weakness of this algorithm for various source materials is demonstrated.