| dc.description.abstract | Parameter estimation has wide applications in such fields as finance, biological science, weather prediction, oil deposit detection, etc. Researchers are particularly interested in reconstructing some unknown parameters from the observed data set which may be sparse and noisy. This is a typical inverse problem which tends to be ill-conditioned in many cases. A plethora of literature has been devoted to this area and there has been a concrete progress recently in the design of more efficient estimation techniques. Depending on the model (usually an equation or a system of equations) we choose, we divide the estimation into two categories: stochastic and deterministic parameter estimations. The former involves a stochastic system, usually a stochastic differential equation (SDE) or system of SDEs where unknown parameters are present. A standard estimator to use for stochastic parameter estimation problems is the maximum likelihood estimator (MLE), since it, in many situations, enjoys desirable properties such as consistency and asymptotic normality. One major obstacle in obtaining the MLE is that the transition density of SDE, which is essential for deriving MLE, is often not available. To address this issue, various approximations of transition density was introduced in the past decades. In chapter 1, we will present several popular density approximation schemes including Euler-Maruyama methods and Hermite expansions. We will also introduce the parametrix approximation in which we derive a point-wise approximation of the transition density that is uniform in the parameter. As a consequence, the approximated MLE from parametrix method will eventually converge to the true MLE so that those desired properties of MLE can be preserved. We will see some applications of the parametrix approximation and some necessary preliminaries regarding the ergodicity of SDEs and the consistency of the estimators will also be presented. The deterministic parameter estimation involves a partial differential equation (PDE) or a system of PDEs. A key feature for this type of estimation is the high level of uncertainty for recovering the parameters, i.e. different choices of parameters may all yield reasonable explanation of the data. This is a typical feature for many ill-conditioned inverse problems. The Bayesian inference formulation provides a systematic way to characterize this uncertainty. It incorporates a prior, which is from the historical data before any experiment is done, and a likelihood, which measures how likely the data will be provided that certain parameter value is chosen, to form a posterior density. It generates a neat solution which takes the form of a posterior probability density. However, how to interpret this posterior density is a non-trivial task since the forward model may be very expensive and the discretized parameter field may result in a high dimensional density. As a consequence, efficient sampling techniques are called for to better characterize the posterior. In chapter 2, we will introduce some traditional sampling methods such as Gaussian approximations, MCMC and importance sampling. We also introduce our implicit sampling methods together with its sequential implementation. We will apply these methods to a seismic wave inversion problem where a detailed comparison among other methods demonstrates a clear superiority of our implicit sampling method. Finally in chapter 3, we will give some concluding remarks and point out possible future work. | |
