In the first half of the talk, we will introduce the generalized probabilistic principal component analysis (GPPCA) to study the latent factor model for multiple correlated outcomes, where each factor is modeled by a Gaussian process. The method generalizes the previous probabilistic formulation of principal component analysis (PPCA) by providing the closed-form maximum marginal likelihood estimator of the factor loadings and other parameters. Based on the explicit expression of the precision matrix in the marginal likelihood that we derived, the number of the computational operations is linear with respect to the number of output variables. In the second half of the talk, we will introduce connections of random factor processes with fixed or estimated basis to a number of real applications in probing dynamics, such as differential dynamic microscopy (DDM), agent-based models of collective motions, and geophysical models for ground deformation. In particular, DDM is a widely used Fourier-based scattering tool, for analyzing particle movements from microscopic videos without tracking individual particles. The connections between physical and statistical models enable more accurate estimation of dynamical information and more scalable computation for statistical models.