Questions tagged [cholesky-decomposition]
The cholesky-decomposition tag has no usage guidance.
87
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Distribution of positive semidefinite matrices that are generated by uniformly distributed positive definite matrices
Let $\mathcal{A}=\{ A_1,A_2,\dots,A_n \} \subseteq \mathcal{S}^p_{++}$ be a set of real positive-definite matrices sampled uniformly with a fixed trace (say, using this algorithm).
To convert each $...
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How to sample efficiently from an inverse Wishart distribution?
I am trying to understand the code from pybasicbayes, which defines the following function to sample from an inverse Wishart:
...
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Interpretation of impulse response analysis - Cholesky decomposition output in R
I am doing an impulse response analysis involving 3 time series A, B, and C in R.
Following Lutkepohl approach, I used the log and diff functions to make them stationary.
After creating the VAR model, ...
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General matrix decomposition downgrading algorithm for sampling
I would like to sample from a multivariate Gaussian distribution with covariance matrix $\Sigma - uu^T $, where $u$ is a vector and $\Sigma - uu^T $ is PSD. I have knowledge of a non-Cholesky ...
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Spectral decomposition and quadratic forms
Let $\mathbf{x}\sim N(0,I)$ and $A$ a real-valued square matrix. The spectral decomposition allows us to rewrite a quadratic form $\mathbf{x}^\top A \mathbf{x}$ as a sum of iid chi-squared random ...
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Statistical interpretation of diagonal of Cholesky decomposition?
Given a set of $m$ examples $x$ arranged as rows in $m\times n$ data matrix X, consider Cholesky decomposition of covariance matrix $X'X$.
Is there a statistical interpretation of diagonal entries of ...
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Effect of using Cholesky transpose
I am generating random normal samples Y with covariance C using a well known procedure:
Let L be the Cholesky decomposition of C, such that $C = LL^T$.
Now given a matrix of random numbers $X, x_{ij}...
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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:
...
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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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What is the expectation of the Cholesky factor of a Wishart distributed random matrix?
Let a $d-\text{dimensional}$ Wishart random variable with $\nu$ degrees of freedom $\Sigma$ be distributed according to $\mathcal{W}(\Sigma|\Sigma_0, \nu) \propto |\Sigma|^\frac{\nu-d-1}2\exp{(-\...
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A new method for processing music scores?
I have developed a method and python script:
https://github.com/githubuser1983/algorithmic_python_music
which allows the user to input a midi file and then chose a few numbers as parameters, and the ...
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How do I generate $n$ random variables that follow a correlation matrix with individually log normal distributions?
Short and sweet: I'd like to model $n$ random variables representing price changes of individual assets. Each of these should be distributed as a log normal variable with a median of 1. Is there a way ...
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Does the Cholesky decomposition of a covariance matrix lead to a lower triangular matrix with positive diagonals?
We know that an $N\times N$ covariance matrix $\Sigma$ is symmetric positive definite, and can be factorized using Cholesky decomposition as follows
\begin{equation}
\Sigma=LL'
\end{equation}
where $L$...
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Drawing samples from matrix normal
I have to generate $n \times m$ sample ($A$) from a matrix normal distribution, given two covariance matrices:
$n \times n$ row covariance matrix (matrix $B$) (defines the covariance between the rows ...
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Cholesky decomposition lower triangular in Gaussian process sampling
I am trying to intuitively understand the Cholesky decomposition in gaussian process function sampling. I understand it as as the square root of the covariance matrix being the multivariate ...
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Proof that variance-covariance matrix of var(b|X) - var(b*|X) is positive-semidefinite [closed]
I'm having trouble finding the proof to show that the variance-covariance matrix of var (b|X) - var (b*|X) is positive-semidefinite.
OLS estimator =
GLS estimator =
Hint:
Note that A is the Cholesky ...
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Relationship between Cholesky decomposition and matrix inversion?
I've been reviewing Gaussian Processes and, from what I can tell, there's some debate whether the "covariance matrix" (returned by the kernel), which needs to be inverted, should be done so ...
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Efficient computation of marginalized multivariate normal posterior distribution
In general,if we know that the marginal Gaussian distribution for some variable $\textbf{x}$ and a conditional Gaussian distribution for some $\textbf{y}|\textbf{x}$ of the forms:
$$p(\textbf{x}) = \...
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Moore Penrose Pseudo-Inverse Fast Algorithm in R [closed]
I want to apply Moore Penrose Pseudo-Inverse on my matrix, which is a 20,000 * 20,000 symmetric matrix with rank 19,999. I found ginv() function from the ...
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What is the EVD operation that replaces Cholesky decomposition in lme4qtl R package?
I am working with pedigreemm and lme4qtl R packages.
If I understand it correctly, both are extensions of the lme4 package, which doesn't allow for correlation between random effect clusters, and so ...
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Discrepancy in Cholesky decomposition matrix from variance covariance matrix obtained in Stata and R using regression
This is a question to someone who knows both R and Stata. For a simple linear regression model, I was trying to estimate the Cholesky decomposition matrix from the variance-covariance matrix of the ...
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798
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Cholesky decomposition or alternative for negatively correlated data simulations
I want to generate some signals that have a correlation distribution around a specific pre-defined correlation value (i.e., the distribution of the values of their correlation matrix is around a ...
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Geometric interpretation of Cholesky Decomposition
I understand that a square matrix, say $A$, can be thought of as a linear transformation within the same space. I could be as simple as basis change or some other transformation.
In this way of ...
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How to show that $X = LY$ where $Y\sim N(0,I)$?
Let $X\sim MVN(0,\Sigma)$ denote a random vector having the multivariate normal distribution with mean $0$ and covariance matrix $\Sigma$.
Suppose we want to sample from $X\sim MVN(0,\Sigma)$. ...
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Difference between sparse cholesky and cholesky decomposition
I am confused as to the difference between the regular cholesky decomposition and the sparse cholesky decomposition (i.e. that which is performed by LAPACK's CHOLMOD...
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Showing a useful result for Wisharts and Multivariate Beta random matrices
Let $\mathbf{A} \sim \text{Wishart}_m\left(k_a,\mathbf{V} \right)$ and $\mathbf{B} \sim \text{Wishart}_m\left(k_b,\mathbf{V} \right)$ be two full rank Wishart random matrices. Define
$$
\mathbf{S} = \...
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How to Incorporate Skew Into Simulated Data?
Suppose I have a dataset $\mathbf{X}$ which is a $n \times m$ matrix of $n$ independent realizations of some $m$-dimensional random vector $\mathbf{x}$. I want to generate a new dataset $\mathbf{X}'$ ...
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Betareg error when running bootstrap-like simulations (Error in chol.default(K) : the leading minor of order 16 is not positive definite
A previous post discussed a similar case on non-positive definite covariance matrices resulting when producing half-normal residual plots using the package betareg. However, I would like additional ...
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Getting sets of random correlated variables
For the training of a machine learning model I need to add additional features, and these features are correlated. I need to run the model N times adding these features with random values, and for ...
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Add Noise to a Dataset
I've generated a dataset of 100 elements from a 3-variate Gaussian distribution with parameters $\mu = 0$ and $\Sigma = \begin{pmatrix}1 & \rho_1 & \rho_2 \\ \rho_1 & 1 & \rho_1 \\ \...
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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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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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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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Relation of kernels and Cholesky decomposition
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 ...
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Cholesky Shock - Interpretation of logs in IRF Models
I came across a few articles here and there that conclude:
When the data (say variables X, Y) for an impulse response function are on log level, the y-axis depicts the % response of Y to a 1% shock ...
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