Control Variate Approach for Parameterized Estimation via Monte Carlo Simulation
Committee Members: Advisor: Pirooz Vakili, SE/ME; Christos Cassandras, SE/ECE; Calin A. Belta, SE/ME; Marcel Rindisbacher, SMG; Appointed Chair: Sean Andersson, SE/ME
Abstract: Monte Carlo (MC) simulation forms a very flexible and widely used computational method employed in many areas of science and engineering. The focus of this research is on the variance reduction technique of Control Variates (CV) which is a statistical approach used to improve the efficiency of MC simulation. In this thesis we consider parameterized estimation problems where the quantity of interest depends on some model or decision parameter and the estimation is performed by one or more users at one or several parameter values. A “store and reuse” setting is introduced where at a setup stage some information is gathered computationally and stored. The stored information is then used at the estimation phase by users to help with their estimation problems.
Three problems in this setting are addressed. (i) An analysis of the users’ choices at the estimation phase is provided. The information generated at the setup phase is stored in the form of information about a set of random variables that can be used as control variates. Users need to decide whether, and if so how, to use the stored information. A so-called cost-adjusted mean squared error is used as a measure cost of the available estimators and users’ decision is formulated as a constrained minimization problem. (ii) A recent approach to defining generic control variates in parameterized estimation problems is generalized in two distinct directions: the first involves considering an alternative parameterization of the original problem through a change of probability measure. This parameterization is particularly relevant to sensitivity estimation problems with respect to model and decision parameters. In the second, for problems where the quantities of interest are defined on sample paths of stochastic processes that model the underlying stochastic dynamics, systematic control variate selection based on approximate dynamics is proposed. (iii) When common random inputs are used parametric estimation variables become statistically dependent. This dependence is explicitly modeled as a random field and conditions are derived to imply the effectiveness of estimation variables as control variates. Comparisons with the metamodeling approach of Kriging and recently proposed Stochastic Kriging that use similar inputs data to predict the mean of the estimation variable are provided.