<p>Classical principal component analysis (PCA) is typically formulated under the assumption of independent observations, although many real datasets exhibit temporal, or spatial dependence. We propose a likelihood-based latent variable model that extends probabilistic PCA by introducing correlation in latent scores, thereby accommodating dependence across observations within a unified PCA framework. The model allows flexible correlation structures, including autoregressive, group-wise, and spatial forms, and parameters are estimated via an EM algorithm with closed-form updates for model parameters. The likelihood-based formulation also enables formal testing of independence across observations through likelihood ratio tests. Simulation studies show favorable finite-sample performance in rank selection, parameter estimation, and test calibration, and demonstrate improved performance over misspecified or independence-based alternatives when dependence is present. An application to temperature data further illustrates that the proposed method yields an interpretable decomposition of variability while capturing realistic temporal and spatial dependence patterns.</p>