Contributions to algebraic pattern recognition and integrated likelihood ratio confidence intervals.

Nonlinear polynomial equations describing positive-dimensional solution sets are called real varieties that in general may be quite complex but locally look like smooth manifolds. In the first chapter, we describe ways to numerically construct useful parameterizations of real varieties using a combination of Monte Carlo strategies to stochastically explore real varieties and tools from deep learning. In the second chapter, in the opposite direction, we describe ways to recognize algebraic patterns when only points with noise are provided by using our proposed RSS model with model selection strategies. In the third and fourth chapters, we propose integrated-likelihood-ratio confidence interval estimations for a Poisson rate parameter with one or two misclassification parameters using double sampling. We also compare the performance of our proposed integrated-likelihood-ratio CI estimations with other CI estimation strategies using both Monte Carlo simulations and real-world examples.