# Questions tagged [cholesky-decomposition]

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### How is Cholesky decomposition used in ridge regression?

As far as I learnt, Cholesky decomposition can be used only for symmetrical positive definite matrices, but I can see it is used as solver in Sklearn-Ridge package, can somebody explain how it is used ...
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### What is the distribution of a Cholesky transformed variable?

I have a situation where I have a vector $\textbf{x} \sim N(\mu_x, \sigma^2_x)$ and a vector $\textbf{y} \sim N(\mu_y, \sigma^2_y)$. I want to generate a new $\textbf{y}_2$ that transforms $\textbf{y}$...
1 vote
587 views

### Positive semi definite matrix with negative eigenvalues?

From what I know, for any square real matrix A, a matrix generated with the following should be a positive semidefinite (PSD) matrix: ...
1 vote
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### When should we decompose the precision matrix as opposed to the covariance matrix to generate correlated variables?

We can take a covariance matrix $\Sigma$ and decompose this into a lower and upper triangular matrix $\Sigma = U^T U$ where $U$ is the Cholesky matrix. This matrix can be used to transform ...
131 views

### Can any covariance factorization $LL^\top$ be used for sampling?

I thought that any factorization of the for $LL^\top$ of a covariance matrix could be used for correlating random noise according to the covariance. I tried doing this with the following code and ...
127 views

### Using kernlab::kqr(reduced = TRUE), how is the y argument missing in the call to csi()?

I'm trying to perform a kernelized quantile regression on some data using the function kqr() from the kernlab package in R. The ...
171 views

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### Eigenvalue decomposition of a covariance matrix using a fast Cholesky decomposition

Let $\mathbf{C}$ be a $n \times n$ covariance matrix and assume that the LDL' Cholesky decomposition can be obtained efficiently. Can we take advantage of this to obtain a fast eigenvalue ...
1 vote
118 views

### Cholesky decomposition in control variates method (Monte Carlo variance reduction technique)

The control variates method, used as a variance reduction technique for Monte Carlo simulations, takes one new variable $t$, correlated to the random variable $m$ to estimate (using the same notations ...
44 views

### Conserve correlation with simulate data

let me explain you the process, I have random variables in a matrix $X_1$: $260\times3$. I have my correlation matrix $\rho_1$: $3\times3$ from my matrix $X_1$. Now I use a Cholesky decomposition ...
2k 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 ...
191 views

### Cholesky Decomposition (in lmer from lme4)

When I retrace the implementation of lmer from lme4 I faced a question regarding cholesky decomposition used for solving penalized least squares. Consider a Cholesky decomposition of a matrix M with ...
969 views

### Generalized linear regression with custom variance-covariance matrix in R

I want to compute the estimate of $\beta$ for a linear model $Y = X\beta + \varepsilon$ with $$\varepsilon \sim N_d(0, \sigma^2V),$$ where $V$ is a $d\times d$ definitive posive, symmetric matrix. ...
3k views

### Difference between Cholesky decomposition and log-cholesky Decomposition

Is there any difference between a Cholesky decomposition and a log-cholesky decomposition? If yes, what is the difference? In the paper "An R package for dynamic linear models" by Giovanni Petris ( ...
2k views

### Estimating correlation matrix using numeric likelihood maximization

I'm performing maximum likelihood estimation on jointly distributed data and I'm having some issues estimating the correlation terms. I am using an approach based on the Cholesky decomposition, but I ...
3k views

### Cholesky decomposition of the covariance matrix: not positive definite?

I am implementing a multivariate simulation in R and when applying the Cholesky decomposition to the covariance matrix I get: the leading minor of order one is not positive definite How could the ...
525 views

### A transformation from uniform random variable to Gaussian mixture

I am attempting to describe a prior_transform for a multivariate Gaussian mixture in order to estimate the evidence integral of that prior convolved with another likelihood distribution. This is ...
2k views

### Can someone provide a non-technical explanation of how Cholesky Covariance priors work?

I am looking for an explanation of how Cholesky Covariance priors work in the context of mixed effects regression. In particular, when they are applied to the correlations among random effects. What ...
7k views

### Why does the resulting matrix from Cholesky decomposition of a covariance matrix when multiplied by its transpose not give back the covariance matrix?

I have a covariance matrix, S, which I use Cholesky decomposition to find A. It is stated that ...
5k views

### How to calculate the standard deviation for a Gaussian Process?

I am quite new to Gaussian processes. A Gaussian Process looks like the following: Where the dark blue line denotes the mean, and the filled-area denotes the mean+std and mean-std respectively. ...
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### For symmetric matrices, is the Cholesky decomposition better than the SVD? [closed]

I am inverting a sparse, symmetric, ill-conditioned matrix. I have used both SVD and the LDL decomposition. I find that my results are better with the latter. Why? I understand that LDL ...
I am trying to find an intuition on why we require that kernels are positive semi definite and I have found this: We are given a dataset $X$ of size $n \times d$ where $n$ is the number of samples ... 