有效更新协方差阵及其LDL分解
Efficiently updating a covariance matrix and its LDL decomposition
摘要：
方程提出了有效地更新或downdate大量的$m$维观测的协方差阵。对协方差阵的更新和下调以及混合更新/下调被显示为秩- $k$修改，其中$k$是添加的新观测值的数量加上删除的旧观测值的数量。因此，更新和降级方程减少了修改$Θ((k+1)m^2)$而不是$Θ((n+k+1)m^2)$或$Θ((n-k+1)m^2)$所需的乘法次数，其中$n$是初始观测值的数量。具有用于更新的秩- $k$公式还允许应用多个其它已知的恒等式，从而提供将更新和向下更新直接应用于协方差阵的逆和分解的方式。为了说明，我们提供了一种有效的算法，用于将秩$k$更新应用于协方差阵的LDL分解。
英文摘要：
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.
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