Dr. Ranadeep Daw

The criterion for aggregation error (CAGE) is an important metric that aims to measure errors that arise in multiscale (or multi-resolution) spatial data,
referred to as the modifiable areal unit problem and the ecological fallacy. Specifically, CAGE is a measure of between scale variance of eigenvectors in a
Karhunen-Lo\'{e}ve expansion (KLE), motivated by a theoretical result, referred to as the ``null-MAUP-theorem,'' that states that the MAUP/ecological fallacy are not present when this variance is zero. CAGE was originally developed for univariate spatial data, but its use has been applied to multivariate spatial data without the development of a null-MAUP-theorem in the multivariate spatial setting. To fill this gap, we provide theoretical justification for a
multivariate CAGE (MVCAGE), which includes multiscale multivariate extensions of the KLE, Mercer's theorem, and the-null-MAUP theorem. Additionally, we provide technical results that demonstrate that the MVCAGE is preferable to spatial-only CAGE, and extend commonly used basis functions used to compute CAGE to the multivariate spatial setting. Empirical results are provided to demonstrate the use of MVCAGE for uncertainty quantification and
regionalization.

Informative sampling designs can impact spatial prediction, or kriging, in two important ways. First, the sampling design can bias spatial covariance
parameter estimation, which in turn can bias spatial kriging estimates. Second, even with unbiased estimates of the spatial covariance parameters, since the kriging variance is a function of the observation locations, these estimates will vary based on the sample and overestimate the population-based estimates. In this work, we develop a weighted composite likelihood approach to improve spatial covariance parameter estimation under informative sampling designs. Then, given these parameter estimates, we propose three approaches to quantify the effects of the sampling design on the variance estimates in spatial prediction. These results can be used to make informed decisions for population-based inference. We illustrate our approaches using a comprehensive simulation study. Then, we apply our methods to perform spatial prediction on nitrate concentration in wells located throughout central California.

Catastrophe theory was originally proposed to study dynamical systems that exhibit sudden shifts in behavior arising from small changes in input. These
models can generate reasonable explanation behind abrupt jumps in nonlinear dynamic models. Among the different catastrophe models, the Cusp Catastrophe model attracted the most attention due to it's relatively simpler dynamics and rich domain of application. Due to the complex behavior of the response, the parameter space becomes highly non-convex and hence it becomes very hard to optimize to figure out the generating parameters. Instead of solving for these generating parameters, we demonstrated how a Machine learning model can be trained to learn the dynamics of the Cusp catastrophe models, without ever really solving for the generating model parameters. Simulation studies and application on a few famous datasets are used to validate our approach. To our knowledge, this is the first paper of such kind where a neural network based approach has been applied in Cusp Catastrophe model.