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.

翻译：状态空间模型广泛应用于各学科领域以研究不可观测的动态过程。经典估计技术（如卡尔曼滤波和期望最大化）虽能提供重要见解，但在大规模分析中计算成本高昂。稀疏逆协方差估计器虽可降低计算成本，但会因强制施加稀疏性与增加估计偏差之间的权衡，在低信噪比条件下需谨慎评估。为应对这些挑战，我们提出三重解决方案：1）引入基于数据驱动正则化的多重惩罚状态空间（MPSS）模型；2）发展源自反向传播、梯度下降及交替最小二乘的新算法用于求解MPSS模型；3）提出评估正则化参数的K折交叉验证扩展方法。我们通过不同信噪比条件下的简化和复杂仿真（包括大规模合成脑磁/脑电图数据分析）验证了MPSS正则化框架。进一步，我们将MPSS模型应用于真实事件相关脑磁/脑电图数据，同时求解涵盖皮层表面数千个源的脑源定位与功能连接问题。该方法突破了现有方法局限于小规模及感兴趣区域分析的局限，有望实现对认知脑功能更精确、更详尽的探索。