Gaussian orthogonal latent factor processes for large incomplete matrices of correlated data

We introduce the Gaussian orthogonal latent factor processes for modeling and predicting large correlated data. To handle the computational challenge, we first decompose the likelihood function of the Gaussian random field with multi-dimensional input domain into a product of densities at the orthogonal components with lower dimensional inputs. The continuous-time Kalman filter is implemented to efficiently compute the likelihood function without making approximation. We also show that the posterior distribution of the factor processes are independent, as a consequence of prior independence of factor processes and orthogonal factor loading matrix. For studies with a large sample size, we propose a flexible way to model the mean in the model and derive the closed-form marginal posterior distribution. Both simulated and real data applications confirm the outstanding performance of this method.