协方差矩阵的特征值之一在R中为负

Question

我有一个数据集x。而我用cov(x)来计算x的协方差。我想计算 cov(x) 的平方根倒数。但是我得到 cov(x).

的负特征值

这是我的代码

S11=cov(x)
S=eigen(S11,symmetric=TRUE)
R=solve(S$vectors %*% diag(sqrt(S$values)) %*% t(S$vectors))

这是S的特征值。

c(0.897249923338732, 0.814314811717616, 0.437109871173458, 0.334921280373883, 
0.291910583884559, 0.257388456770167, 0.166787180227719, 0.148268784967556, 
0.121401731579852, 0.0588333377333529, 0.0519459283467876, 0.0472867806813002, 
0.0438199555429584, 0.0355421239839632, 0.0325106968911777, 0.0282860419784165, 
0.0222240269478354, 0.0174657163114068, 0.012318267910606, 0.00980611646284724, 
0.00969450391092417, 0.00804912897151307, 0.00788628666010145, 
0.00681419055130702, 0.00664707528670254, 0.00591471779140177, 
0.00581608875646686, 0.0057489828718098, 0.00564645095578336, 
0.00521029715741059, 0.00503304953884416, 0.0048677189522647, 
0.00395692706081966, 0.00391665618240403, 0.00389825739725093, 
0.00383611535401152, 0.00374242176786387, 0.0035160324422885, 
0.00299245160843966, 0.0029501156885799, 0.00289484923017341, 
0.00287327878694529, 0.0028447265712214, 0.00274130080219099, 
0.00273159993035393, 0.00265595612239575, 0.00261856622830277, 
0.0020004125628823, 0.00199834766485368, 0.00199579695856402, 
0.00198945452395265, 0.00197999810684363, 0.00195954105720554, 
0.00195502875017394, 0.00194143254092788, 0.00192530399875842, 
0.00191287435824908, 0.00187418676921454, 0.00184304720875652, 
0.00181132707713659, 0.00167004122321738, 0.00132136106130093, 
0.001001001001001, 0.001001001001001, 0.001001001001001, 0.00100089827907564, 
0.000999613336959707, 0.000999285885989665, 0.000995390174780253, 
0.000990809217795241, 0.000987333916025995, 0.000984260717691378, 
0.000982735942052615, 0.000971684328336702, 0.000964125499180901, 
0.000961900381008093, 0.000947883827257506, 0.000922293473088298, 
0.000862086463606162, 0.000829687294735196, 0.000732694198613695, 
1.95782839335209e-17, 4.13905030077713e-18, 2.02289095736911e-18, 
8.72989281345777e-19, 3.79161425300691e-19, -7.97468731082902e-20)

Answer 1

虽然理论上估计的协方差矩阵必须是正（半）定的，即没有负值，但在实践中浮点误差可能会违反这一点。对我来说，一个 87×87 的矩阵可能有一个很小的负值（大约 -1*10^(-19)）特征值也就不足为奇了。

根据您想要执行的操作，您可以使用 Matrix 包中的 ?nearPD 来强制您的协方差矩阵为正定矩阵：

Compute the nearest positive definite matrix to an approximate one, typically a correlation or variance-covariance matrix.

此外，首先计算矩阵的 Cholesky 分解 (?chol) 然后将其反转可能会更有效（这在原则上很容易 - 我认为您可以使用 backsolve()).

协方差矩阵的特征值之一在R中为负

one of Eigenvalues of covariance matrix is negative in R

r

eigenvalue

matrix-decomposition