Temporally coupled Principal Component Analysis: A Probabilistic autoregression method

Despite the apparent spatio-temporal decomposition given by (Probabilistic) Principal Component Analysis ((P)PCA), there is in fact no temporal coupling built into these models. Here we augment PPCA with a temporal model in the latent space by coupling the latent variables in time with an autoregressive model and show that the new model may be viewed as a generalisation of PPCA. We present an algorithm which utilises both expectation maximisation and a forward-backward algorithm to infer the values of the model parameters and demonstrate that it is able to make good estimates of the parameter values for synthetic data. We show that the additional temporal information is advantageous when imputing values for missing observations when compared with two non-temporal PPCA methods, both against synthetic data and real UK industrial production output data.