  State-space models are widely employed across various research disciplines to
study unobserved dynamics. Conventional estimation techniques, such as Kalman
filtering and expectation maximisation, offer valuable insights but incur high
computational costs in large-scale analyses. Sparse inverse covariance
estimators can mitigate these costs, but at the expense of a trade-off between
enforced sparsity and increased estimation bias, necessitating careful
assessment in low signal-to-noise ratio (SNR) situations. To address these
challenges, we propose a three-fold solution: 1) Introducing multiple penalised
state-space (MPSS) models that leverage data-driven regularisation; 2)
Developing novel algorithms derived from backpropagation, gradient descent, and
alternating least squares to solve MPSS models; 3) Presenting a K-fold
cross-validation extension for evaluating regularisation parameters. We
validate this MPSS regularisation framework through lower and more complex
simulations under varying SNR conditions, including a large-scale synthetic
magneto- and electro-encephalography (MEG/EEG) data analysis. In addition, we
apply MPSS models to concurrently solve brain source localisation and
functional connectivity problems for real event-related MEG/EEG data,
encompassing thousands of sources on the cortical surface. The proposed
methodology overcomes the limitations of existing approaches, such as
constraints to small-scale and region-of-interest analyses. Thus, it may enable
a more accurate and detailed exploration of cognitive brain functions.
