# Questions tagged [cholesky-decomposition]

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### Simulate Gaussian copula with negative pair-wise correlations

I am now trying to simulate a multidimensional (let's say 4 dimensions) Gaussian copula. Given software restrictions, I can only use Excel for this simulation. That is why I am trying to implement the ...
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### Constrained Cholesky Decomposition

Suppose that I have an $(n\times 1)$ vector of random variables, $\varepsilon$. However, I know that $k$ linear combinations of $\varepsilon$ are 0. Specifically, I know that for a $(k\times n)$ ...
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### Covariance inversion for Gaussian process

Background Let $x=f(u_x)\in\mathbb{R}$ and let $y=[f(u_y^1)\cdots f(u_y^{N})]\in\mathbb{R}^N$ for some function $f:u \in \mathbb{R}\mapsto \mathbb{R}$. Given $y$, $u_x$, $u_{y}^1,\dots, u_{y}^{N}$, I ...
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### Sampling with specified covariance matrix and distribution

Given a positive semi-definite $n\times n$ matrix $C$ I would like to construct $n$ random variables $X_1,\dots,X_n$ drawn from $n$ fixed distributions such that $\mathrm{corr}(X_i,X_j) = C_{ij}$. I ...
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### Passing a cholesky decomposition for a matrix with constrained variances to an objective function

I am trying to optimize an objective function $L(\theta)$ in which some parameters that I aim to recover belong to a covariance matrix, $\Sigma$. $\Sigma$ has a unique structure, which includes ones ...
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### Fast Cholesky decomposition of a Toepllitz matrix via embedding in a circulant & fft

As I understand it, the Cholesky decomposition of a Toeplitz matrix can be computed more efficiently by first embedding it in a circulant matrix then using FFT, but I'm having trouble finding any ...
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### Efficient way to encode a set of large covariance matrices

I have a computational model that involves having a set of $K$ covariance matrices, $\{\Sigma_1, ..., \Sigma_K\}$ with each $\Sigma_i \in R^{n \times n}$. Storing all these full covariance matrices is ...
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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 $\textbf{y}$...
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### 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 ...
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### 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 ...
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### 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 ...
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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 ...
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1 vote
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ( ...
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