Abstract Sparse Bayesian learning (SBL) has attracted substantial interest in recent years for reliable estimation of sparse parameter vectors of dimension much larger than the number of measurements. However, the theory of online sequential estimation of sparsely changing parameter vectors is much less studied. We present a sequential SBL framework for recursive learning of sparse vectors that also change sparsely between successive sampling time periods. Our method uses a hierarchical Bayesian model to recursively estimate the marginal posterior distribution of the parameter vector for each time period, incorporating the sparseness of both this vector and its temporal changes. Our Bayesian model is built around a linear Gaussian state space model and so many quantities of interest can be calculated by using the recursive Bayesian equations. The fast evidence maximization procedure for SBL is developed for recursive Bayesian analysis and the “noise” parameters are efficiently learned solely from the available data in an efficient manner. Numerical experiments verify that exploiting the sparseness of temporal changes of sparse vectors leads to better performance of sparse Bayesian learning. We also examine two applications of sequential SBL: structural system identification for estimating stiffness losses of sequential damage states and recursive reconstruction of image sequences. These illustrative applications validate the effectiveness and robustness of our method.

中文翻译：

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序列稀疏贝叶斯学习在系统识别中的应用，用于损伤评估和图像序列的递归重建

摘要 近年来，稀疏贝叶斯学习（SBL）因其对维度远大于测量数量的稀疏参数向量的可靠估计而引起了极大的兴趣。然而，关于稀疏变化参数向量的在线序列估计理论研究较少。我们提出了一个顺序 SBL 框架，用于稀疏向量的递归学习，这些向量在连续采样时间段之间也稀疏地变化。我们的方法使用分层贝叶斯模型来递归估计每个时间段参数向量的边际后验分布，并结合该向量的稀疏性及其时间变化。我们的贝叶斯模型是围绕线性高斯状态空间模型构建的，因此可以使用递归贝叶斯方程计算许多感兴趣的量。SBL 的快速证据最大化程序是为递归贝叶斯分析而开发的，并且“噪声”参数是以有效的方式仅从可用数据中有效学习的。数值实验证明，利用稀疏向量随时间变化的稀疏性可以提高稀疏贝叶斯学习的性能。我们还研究了连续 SBL 的两种应用：用于估计连续损伤状态刚度损失的结构系统识别和图像序列的递归重建。这些说明性应用验证了我们方法的有效性和稳健性。数值实验证明，利用稀疏向量随时间变化的稀疏性可以提高稀疏贝叶斯学习的性能。我们还研究了连续 SBL 的两种应用：用于估计连续损伤状态刚度损失的结构系统识别和图像序列的递归重建。这些说明性应用验证了我们方法的有效性和稳健性。数值实验证明，利用稀疏向量随时间变化的稀疏性可以提高稀疏贝叶斯学习的性能。我们还研究了连续 SBL 的两个应用：用于估计连续损伤状态刚度损失的结构系统识别和图像序列的递归重建。这些说明性应用验证了我们方法的有效性和稳健性。