Rational Covariance Extension, Multivariate Spectral Estimation, and Related Moment Problems: Further Results and Applications

This dissertation concerns the problem of spectral estimation subject to moment constraints. Its scalar counterpart is well-known under the name of rational covariance extension which has been extensively studied in past decades. The classical covariance extension problem can be reformulated as a truncated trigonometric moment problem, which in general admits infinitely many solutions. In order to achieve positivity and rationality, optimization with entropy-like functionals has been exploited in the literature to select one solution with a fixed zero structure. Thus spectral zeros serve as an additional degree of freedom and in this way a complete parametrization of rational solutions with bounded degree can be obtained. New theoretical and numerical results are provided in this problem area of systems and control and are summarized in the following. First, a new algorithm for the scalar covariance extension problem formulated in terms of periodic ARMA models is given and its local convergence is demonstrated. The algorithm is formally extended for vector processes and applied to finite-interval model approximation and smoothing problems. Secondly, a general existence result is established for a multivariate spectral estimation problem formulated in a parametric fashion. Efforts are also made to attack the difficult uniqueness question and some preliminary results are obtained. Moreover, well-posedness in a special case is studied throughly, based on which a numerical continuation solver is developed with a provable convergence property. In addition, it is shown that solution to the spectral estimation problem is generally not unique in another parametric family of rational spectra that is advocated in the literature. Thirdly, the problem of image deblurring is formulated and solved in the framework of the multidimensional moment theory with a quadratic penalty as regularization

Synthetic learner: model-free inference on treatments over time

Understanding of the effect of a particular treatment or a policy pertains to many areas of interest -- ranging from political economics, marketing to health-care and personalized treatment studies. In this paper, we develop a non-parametric, model-free test for detecting the effects of treatment over time that extends widely used Synthetic Control tests. The test is built on counterfactual predictions arising from many learning algorithms. In the Neyman-Rubin potential outcome framework with possible carry-over effects, we show that the proposed test is asymptotically consistent for stationary, beta mixing processes. We do not assume that class of learners captures the correct model necessarily. We also discuss estimates of the average treatment effect, and we provide regret bounds on the predictive performance. To the best of our knowledge, this is the first set of results that allow for example any Random Forest to be useful for provably valid statistical inference in the Synthetic Control setting. In experiments, we show that our Synthetic Learner is substantially more powerful than classical methods based on Synthetic Control or Difference-in-Differences, especially in the presence of non-linear outcome models

Autoregressive Spectral Estimation in Noise with Application to Speech Analysis

Electrical Engineerin