论文标题
有效地更新协方差矩阵及其LDL分解
Efficiently updating a covariance matrix and its LDL decomposition
论文作者
论文摘要
提出了有效更新或减少大量$ M $维观测值的协方差矩阵的方程式。显示与协方差矩阵以及混合更新/下限的更新和淡淡的日期，被证明是排名-K $修改，其中$ k $是添加的新观测值的数量，加上已删除的旧观测值的数量。结果，更新和下调方程将修改的所需乘法数减少为$θ（（k+1）m^2）$，而不是$θ（（（n+k+1）m^2）$或$θ（（n-k+1）m^2）$，如果$ n是$ n $是初始观察的数量。更新的等级$ k $公式还允许应用许多其他已知的身份，从而提供了一种将更新并直接置于协方差矩阵的逆和分解的方法。为了说明，我们提供了一种有效的算法，用于将等级$ K $更新应用于协方差矩阵的LDL分解。
Equations are presented which efficiently update or downdate the covariance matrix of a large number of $m$-dimensional observations. Updates and downdates to the covariance matrix, as well as mixed updates/downdates, are shown to be rank-$k$ modifications, where $k$ is the number of new observations added plus the number of old observations removed. As a result, the update and downdate equations decrease the required number of multiplications for a modification to $Θ((k+1)m^2)$ instead of $Θ((n+k+1)m^2)$ or $Θ((n-k+1)m^2)$, where $n$ is the number of initial observations. Having the rank-$k$ formulas for the updates also allows a number of other known identities to be applied, providing a way of applying updates and downdates directly to the inverse and decompositions of the covariance matrix. To illustrate, we provide an efficient algorithm for applying the rank-$k$ update to the LDL decomposition of a covariance matrix.
