This paper discusses the fitting of linear state space models to given multivariate time series in the presence of constraints imposed on the four main parameter matrices of these models. Constraints arise partly from the assumption that the models have a block-diagonal structure, with each block corresponding to an ARMA process, that allows the reconstruction of independent source components from linear mixtures, and partly from the need to keep models identifiable. The first stage of parameter fitting is performed by the expectation maximisation (EM) algorithm. Due to the identifiability constraint, a subset of the diagonal elements of the dynamical noise covariance matrix needs to be constrained to fixed values (usually unity). For this kind of constraints, so far, no closed-form update rules were available. We present new update rules for this situation, both for updating the dynamical noise covariance matrix directly and for updating a matrix square-root of this matrix. The practical applicability of the proposed algorithm is demonstrated by a low-dimensional simulation example. The behaviour of the EM algorithm, as observed in this example, illustrates the well-known fact that in practical applications, the EM algorithm should be combined with a different algorithm for numerical optimisation, such as a quasi-Newton algorithm.

本文讨论了在存在对这些模型的四个主要参数矩阵施加约束的情况下，将线性状态空间模型拟合到给定的多元时间序列的问题。局限性部分来自于模型具有块对角线结构的假设，每个块对应于一个ARMA流程，从而允许从线性混合物重建独立的源成分，部分原因是需要保持模型可识别。参数拟合的第一阶段由期望最大化（EM）算法执行。由于可识别性的限制，动态噪声协方差矩阵的对角元素的子集需要限制为固定值（通常为1）。到目前为止，对于这种约束，尚无封闭格式的更新规则。我们针对这种情况提出了新的更新规则，既可以直接更新动态噪声协方差矩阵，又可以更新该矩阵的矩阵平方根。通过一个低维仿真实例证明了该算法的实际适用性。在此示例中观察到的EM算法的行为说明了一个众所周知的事实，即在实际应用中，EM算法应与其他用于数值优化的算法相结合，例如拟牛顿算法。