Questions tagged [cholesky]

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15
votes
5answers
2k views

Generate normally distributed random numbers with non positive-definite covariance matrix

I estimated the sample covariance matrix $C$ of a sample and get a symmetric matrix. With $C$, I would like to create $n$-variate normal distributed r.n. but therefore I need the Cholesky ...
7
votes
2answers
5k views

Can I use the Cholesky-method for generating correlated random variables with given mean?

I want to generate correlated random variables with a given correlation matrix, means, and variances. Does the Cholesky decomposition only work when the initial random variables are iids with the same ...
3
votes
1answer
870 views

How to do SVD instead of Cholesky for $L^{T}L$?

The Cholesky decomposition can be used to obtain $A$ from $X = AA^{T}$ (lower triangular version) but also $B$ from $Y = B^{T}B$ (upper triangular version). The SVD can be used to do something similar ...
13
votes
1answer
317 views

Explain how `eigen` helps inverting a matrix

My question relates to a computation technique exploited in geoR:::.negloglik.GRF or geoR:::solve.geoR. In a linear mixed model ...
4
votes
2answers
2k views

Why use upper triangular Cholesky?

Software packages seem to prefer to work with the upper triangular part of the Cholesky factorization, see for example cholupdate. Why is this? It seems that it is ...
4
votes
1answer
630 views

Mahalanobis distance with LDL decomposition

I've got an extended Kalman filter with innovation covariance defined as $\mathbf{W}=\mathbf{H}\mathbf{P}\mathbf{H}^\textrm{T} + \mathbf{R}$. I want to know the squared Mahalanobis distance $\|z\|^2$ ...
3
votes
1answer
309 views

Compute covariance matrix via rank-1 update to $M^\top M$

I have a large, sparse matrix $M\in\mathbb{R}^{n\times p}$. Centering $M$ to compute the covariance matrix $\Sigma$ would, in general, destroy the "zeros aren't stored" property of sparse matrices. ...
3
votes
1answer
345 views

Why is computing ridge regression with a Cholesky decomposition much quicker than using SVD?

By my understanding, for a matrix with n samples and p features: Ridge regression using Cholesky decomposition takes O(p^3) time Ridge regression using SVD takes O(p^3) time Computing SVD when only ...
5
votes
1answer
338 views

Generate random variables with predefined correlation structure AND fixing some values

I need to generate 4 random variables that show a predefined correlation structure Sigma AND where certain values of Vars 1-4 are fixed. As an illustrative example, consider the variables: ...
4
votes
1answer
492 views

Derivative of $x^T A^Ty$ with respect to $\Sigma$ where $A$ is (an upper triangle matrix and ) Cholesky decomposition of $\Sigma$

I would like to evaluate: $$ \frac{ \partial x^T A^Ty}{\partial \Sigma} $$ where $A$ is a Cholesky decomposition of $\Sigma$ and an upper triangle matrix such that $\Sigma = A^T A$, $x$ and $y$ are a ...
4
votes
1answer
824 views

Cholesky factorization and forward substitution less accurate than inversion?

I recently asked this question asking for an efficient way to compute the Mahalanobis distance (without calculating the inverse). The accepted solution was to use the Cholesky factorization and ...
1
vote
0answers
641 views

Trying to use Cholesky decomposition of covariance matrix to sample error ellipsoid

I'm trying to construct an error ellipsoid from a covariance matrix (which exists for a 3D point) and then sample consistent xyz points in this region. In a previous question when I asked about this (...