r - R 函数错误 `is.positive.definite`

Question

我想测试 R 中的矩阵是否为正定矩阵。我使用了 R 函数 is.positive.definite 但不断收到以下错误消息，尽管我的矩阵作为函数是对称的 isSymmetric 显示。请问这是因为舍入误差吗？

Error in is.positive.definite(S) : argument x is not a symmetric matrix

我的工作代码附在下面。有人可以帮我吗？谢谢。

library(Matrix) # isSymmetric
library(matrixcalc) # is.positive.definite
library(expm) # sqrtm

###################################################################################################

theta0 <- c(0.2, 10)

###################################################################################################

OS.mean <- function(shape, rank, n=10){
  term1 <- factorial(n)/(factorial(rank-1)*factorial(n-rank))
  term2 <- beta(n-rank+1, rank) - beta(n-rank+shape+1, rank)
  term1*term2/shape
}

OS.mean.theta0.10 <- as.matrix(OS.mean(theta0[1], rank=seq(1, 10, by=1)))

###################################################################################################

OSsq.mean <- function(shape, rank, n=10){
  term1 <- factorial(n)/(factorial(rank-1)*factorial(n-rank))
  term2 <- beta(n-rank+1, rank) - 2*beta(n-rank+shape+1, rank) + beta(n-rank+2*shape+1, rank)
  term1*term2/(shape*shape)
}

OSsq.mean.theta0.10 <- as.matrix(OSsq.mean(theta0[1], rank=seq(1, 10, by=1)))

###################################################################################################

OSprod.mean <- function(shape, rank1, rank2, n=10){
  term1 <- factorial(n)/(factorial(rank1-1)*factorial(rank2-rank1-1)*factorial(n-rank2))
  term2 <- beta(n-rank1+1, rank1) - beta(n-rank1+shape+1, rank1)
  term3 <- beta(n-rank2+1, rank2-rank1)
  term4 <- beta(n-rank1+shape+1, rank1) - beta(n-rank1+2*shape+1, rank1)
  term5 <- beta(n-rank2+shape+1, rank2-rank1)
  term1*(term2*term3-term4*term5)/(shape*shape)
}

OS.cov <- function(shape, rank1, rank2){
  OSprod.mean(shape, rank1, rank2) - OS.mean(shape, rank1)*OS.mean(shape, rank2)  
}

###################################################################################################

spacing <- seq(1, 10, by=1)

OS.varcov.10 <- function(shape, n=10){
  V.diag <- diag(c(OSsq.mean.theta0.10 - OS.mean.theta0.10^2))

  V.upper <- matrix(0, nrow=10, ncol=10)
  for(i in 1:9){
    for(j in (i+1):10){
      V.upper[i, j] <- OS.cov(shape, spacing[i], spacing[j])
    }
  }

  V.upper + V.diag + t(V.upper)
}

###################################################################################################

V.theta0.10 <- OS.varcov.10(theta0[1])

kappa(V.theta0.10)

isSymmetric(V.theta0.10)
is.positive.definite(V.theta0.10)

S <- sqrtm(V.theta0.10)
isSymmetric(S)
is.positive.definite(S)

Answer 1

您的 S 矩阵不对称，因为 loss of significance ，但默认的输出小数位数隐藏了它。自己看看:

> options(digits=20)
> S[1,2]
[1] 0.033457660484940172
> S[2,1]
[1] 0.033457660484940213

问题是，matrixcalc 包中的 is.symmetric.matrix 不考虑微小差异(即它只是比较矩阵元素与严格的 == 而不是 all.equal 方法)，而 Matrix 包中的 isSymmetric 可以。如果你四舍五入矩阵，一切都会好起来的:

> S=round(S,10)
> is.symmetric.matrix(S)
[1] TRUE
> is.positive.definite(S)
[1] TRUE