Selecting the step of the least mean squares (LMS) algorithm is an old problem. This study uses a new approach to address this problem resulting in a new algorithm with excellent system identification performance. The LMS algorithm, with time-varying step, size can be shown to be equivalent to the Kalman filter in some conditions. This is as long as the state noise of the Kalman filter and the step size of the LMS algorithm are chosen carefully. The Kalman filter is the optimum linear estimator (Bayesian) given the state and the measurement noise covariance matrices, but these matrices are not always known. This work considers the case where these matrices are not known, in the special cases that the Kalman filter reduces to the LMS. This results in an algorithm to select the step-size of the LMS algorithm with few priors. The optimum step size can be calculated using estimates of the probability density function (PDF) of the coefficient estimation error variance ( q_w ) and measurement noise variance ( q_v ). The PDFs can be estimated from the data using Bayes’ rule and assuming Gaussian reference and measurement noise signals. The resulting algorithm to determine q_w and q_v is a second small Kalman filter, and the outputs of this filter (means and covariances) are used to determine the expected value of the step.

中文翻译：

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高斯信号的贝叶斯步长最小均方算法

选择最小均方（LMS）算法的步长是一个老问题。这项研究使用一种新方法来解决此问题，从而产生了一种具有出色系统识别性能的新算法。LMS算法具有随时间变化的步长，在某些情况下，其大小可以等效于卡尔曼滤波器。只要仔细选择卡尔曼滤波器的状态噪声和LMS算法的步长即可。给定状态和测量噪声协方差矩阵，卡尔曼滤波器是最佳线性估计器（贝叶斯估计），但是这些矩阵并不总是已知的。这项工作考虑了这些矩阵未知的情况，特别是在卡尔曼滤波器简化为LMS的特殊情况下。这导致算法以很少的先验选择LMS算法的步长。 q _{w ^} ）和测量噪声方差（ q _v ）。可以使用贝叶斯规则并假设高斯参考噪声和测量噪声信号从数据中估计PDF。结果算法确定q _{w ^} 和 q _v 是第二个小型卡尔曼滤波器，该滤波器的输出（均值和协方差）用于确定步长的期望值。