通过透视Cholesky分解生成具有秩不足协方差的多元正态rv

Question

I'm just beating my head against the wall trying to get a Cholesky decomposition to work in order to simulate correlated price movements.

I use the following code:

cormat <- as.matrix(read.csv("http://pastebin.com/raw/qGbkfiyA"))
cormat <- cormat[,2:ncol(cormat)]
rownames(cormat) <- colnames(cormat)
cormat <- apply(cormat,c(1,2),FUN = function(x) as.numeric(x))

chol(cormat)
#Error in chol.default(cormat) : 
#    the leading minor of order 8 is not positive definite

cholmat <- chol(cormat, pivot=TRUE)
#Warning message:
#    In chol.default(cormat, pivot = TRUE) :
#    the matrix is either rank-deficient or indefinite

rands <- array(rnorm(ncol(cholmat)), dim = c(10000,ncol(cholmat)))
V <- t(t(cholmat) %*% t(rands))

#Check for similarity
cor(V) - cormat  ## Not all zeros!

#Check the standard deviations
apply(V,2,sd) ## Not all ones!

I'm not really sure how to properly use the pivot = TRUE statement to generate my correlated movements. The results look totally bogus.

Even if I have a simple matrix and I try out "pivot" then I get bogus results...

cormat <- matrix(c(1,.95,.90,.95,1,.93,.90,.93,1), ncol=3)

cholmat <- chol(cormat)
# No Error

cholmat2 <- chol(cormat, pivot=TRUE)
# No warning... pivot changes column order

rands <- array(rnorm(ncol(cholmat)), dim = c(10000,ncol(cholmat)))
V <- t(t(cholmat2) %*% t(rands))

#Check for similarity
cor(V) - cormat  ## Not all zeros!

#Check the standard deviations
apply(V,2,sd) ## Not all ones!

Answer 1

There are two errors with your code:

1. You did not use pivoting index to revert the pivoting done to the Cholesky factor. Note, pivoted Cholesky factorization for a semi-positive definite matrix A is doing:

    P'AP = R'R 

   where P is a column pivoting matrix, and R is an upper triangular matrix. To recover A from R , we need apply the inverse of P (ie, P' ):

    A = PR'RP' = (RP')'(RP') 

   Multivariate normal with covariance matrix A , is generated by:

    XRP' 

   where X is multivariate normal with zero mean and identity covariance.

2. Your generation of X

    X <- array(rnorm(ncol(R)), dim = c(10000,ncol(R))) 

   is wrong. First, it should not be ncol(R) but nrow(R) , ie, the rank of X , denoted by r . Second, you are recycling rnorm(ncol(R)) along columns, and the resulting matrix is not random at all. Therefore, cor(X) is never close to an identity matrix. The correct code is:

    X <- matrix(rnorm(10000 * r), 10000, r) 

---

As a model implementation of the above theory, consider your toy example:

A <- matrix(c(1,.95,.90,.95,1,.93,.90,.93,1), ncol=3)

We compute the upper triangular factor (suppressing possible rank-deficient warnings) and extract inverse pivoting index and rank:

R <- suppressWarnings(chol(A, pivot = TRUE))
piv <- order(attr(R, "pivot"))  ## reverse pivoting index
r <- attr(R, "rank")  ## numerical rank

Then we generate X . For better result we centre X so that column means are 0.

X <- matrix(rnorm(10000 * r), 10000, r)
## for best effect, we centre `X`
X <- sweep(X, 2L, colMeans(X), "-")

Then we generate target multivariate normal:

## compute `V = RP'`
V <- R[1:r, piv]

## compute `Y = X %*% V`
Y <- X %*% V

We can verify that Y has target covariance A :

cor(Y)
#          [,1]      [,2]      [,3]
#[1,] 1.0000000 0.9509181 0.9009645
#[2,] 0.9509181 1.0000000 0.9299037
#[3,] 0.9009645 0.9299037 1.0000000

A
#     [,1] [,2] [,3]
#[1,] 1.00 0.95 0.90
#[2,] 0.95 1.00 0.93
#[3,] 0.90 0.93 1.00