Smoothed dynamic factor analysis for identifying trends in multivariate time series

Abstract

Abstract Ecological processes are rarely directly observable, and inference often relies on estimating hidden or latent processes. State‐space models have become widely used for this task because of their ability to simultaneously estimate the multiple sources of variation (natural variability and variance attributed to observation errors). For multivariate time series, a second aim is often dimension reduction, or estimating a number of latent processes that are smaller than the number of observed time series. Dynamic factor analysis (DFA) has been used for performing time‐series dimension reduction, where latent processes are modelled as random walks. Whereas this may be suitable for some situations, random walks may be too flexible for other cases. Here, we introduce a new class of models, where latent processes are modelled as smooth functions (basis splines, penalized splines or Gaussian process models). We implement these models in our bayesdfa r package, which uses the rstan package for fitting. After evaluating model performance with simulated data, we apply conventional models and our smooth trend models to two long‐term datasets from the west coast of the United States: (a) a 35‐year dataset of pelagic juvenile rockfishes and (b) a 39‐year dataset of fisheries catches. Our simulations demonstrate that models matching the underlying trend smoothness make better out‐of‐sample predictions, but this advantage diminishes with increasing levels of observation error. For both case studies, the best smooth trend models had higher predictive accuracy, and yielded more precise predictions, compared to the conventional approach. The smooth trend factor models introduced here offer a new approach for state‐space dimension reduction of multivariate time series. These flexible Bayesian models may be particularly useful for data that are clumped in time, for data with high signal to noise ratios and generally for data where the underlying trend is assumed to be relatively smooth.