Questions tagged [covariance-matrix]

A $k\times k$ matrix of covariances between all pairs of $k$ random variables. It is also called variance-covariance matrix or simply the covariance matrix.

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System GMM yields identical results for any weighting matrix

I am estimating a system of seemingly unrelated regressions (SUR) in R. Each of the equations has one unique regressor and one common regressor. I am using ...
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Why does `systemfit` yield different results for OLS and WLS under cross-equation restrictions?

I am following up on the question "Why does systemfit yield identical results for OLS and WLS?". It deals with estimating a system of linear equations ...
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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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How to test for equal spread in bivariate normal samples with equal means?

I'm working with samples taken from a bivariate normal distribution, where the differences in means is not relevant since all samples are scaled to mean (0,0) anyway, and I'm trying to remember how to ...
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Is there a way to specify multiple covariance structures in nlme for spatio-temporal longitudinal data?

I am trying to build mixed effects models for longitudinal data with spatio-temporal trends. Is there a way to specify multiple covariance structures in nlme or other packages (e.g. both a corARMA()...
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Heteroscedasticity-consistent (robust) standard errors complemented by i) confidence intervals for beta, ii) Tolerance and iii) VIF values in R?

In order to solve heteroscedasticity in my data, I ran a regression with heteroscedasticity-consistent ("robust") standard errors. I would also like to report i) standardized betas together ...
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Mixed models: equivalence between residual covariance structure and random effects?

Is there a way to specify the covariance structure of a within-subject repeated measures model (MMRM with no random effects) such that the model is mathematically equivalent to a mixed model with ...
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exclude random effects component for a repeated measure

I'm analyzing a dataset on the Nurse Licensure Exam, comprising 3000 participants. (n) These 3000 participants were randomly recruited from 13 Sites across the US. (group level variable) About 40% of ...
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Mapping covariance matrices with the datasets

Can someone please help me with this exercise and explain clearly to me? For me personally, when I look at this exercise, the matrix (2) stand out obviously, as it is a isotropic matrix so, it must be ...
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Joint Distribution Formulation of a Spatial X, Spatial Y, and Spatial Error Model

Introductory Problem: I have $n$ points in 3-D space, where I know their X and Y coordinates (not Z), and therefore the distances between points in those 2 dimensions. Each of the three dimensions has ...
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Distance matrix of actual dataset doesn't obey triangle inequality, leading to non-positive definite covariance matrix

Yesterday I asked a question about why my randomly generated distance matrices were leading to Matérn covariance matrices that were not positive definite. The answer there called my attention to the ...
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Calculating Matérn covariance in R returns matrices that are not positive definite

In R, I am trying to calculate Matérn covariance matrices whose inputs are randomly created distance matrices. However, I often end up getting covariance matrices that are not positive definite, ...
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Bootstrap method with 2 Fisher matrices in order to do the cross-correlations between both

I have 2 Fisher matrices where each colum/row represents the information (in Fisher's sense) of astrophysical parameters. These parameters are in the same order for both matrices. Now, I would like to ...
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GMM derivation for diagonal covariance matrices

I was trying to understand the derivation of M step in the EM algorithm for GMM. All the resources available consider only "full covariance" matrices. I wanted to implement GMM for "...
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Sampling from a very high dimensional Gaussian

I would like to a sample from a Gaussian $N(0,K)$ where $K$ is a kernel gram matrix, so that $K=[K_{ij}]$ with $K_{ij} = k(x_i,x_j)$ for some positive definite function $k$. The first issue is that ...
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How do we interpret the covariance matrices $\textbf{U}$ and $\textbf{V}$ in the Matrix Variate Normal Distribution?

Consider the Matrix Normal Distribution. My first question is: how do we interpret the entries $\textbf{X}_{ij}$ of the random matrix $\textbf{X}(n\times p)$? My second question is: how do we ...
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Understanding the multicollinearity issue in relation to linear regression

There are 2 issues that multicollinearity in linear regression leads to Interpretability goes for a toss Parameter confidence intervals are wide and it is difficult to find a parameter significant I ...
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Is the assumption of a diagonal covariance matrix on the latent space in a variational autoencoder in any way restrictive?

The covariance matrix in an autoencoder is assumed to be diagonal. And, I see it mentioned in good places that this is a fairly restrictive assumption. To quote However, in order to simplify the ...
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Clarification on covariance matrix for multidimensional Gaussian distributions

It is a well known property of Gaussian distributions that if $Y = (Y_1, \ldots, Y_n)$, where each $Y_i$ is a real Gaussian random variable, then the components of $Y$ are independent if and only if ...
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Intuition behind between-group covariance matrix from MANOVA?

Suppose that we have samples from $m$ different $p$-dimensional normal multivariate distributions, where they share a common covariance matrix $\Sigma$ but the mean vectors may be different for each ...
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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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Decompose covariance matrix into uncorrelated and correlated part

I have an almost diagonal covariance matrix, I would like to decompose it in an uncorrelated and correlated part: $$ \Sigma = \Sigma_U + \Sigma_C $$ where all the matrices above are covariance ...
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Prove that modified RBF function satisfies Mercer conditions

Suppose that I have a modified RBF kernel function. $k(\mathbf{x},\mathbf{y}) = \exp{(-||\mathbf{x}-P\mathbf{y}||^2 })$ where $\mathbf{x},\mathbf{y}$ represent $d$ dimensional inputs and $P$ is the ...
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Is this formula the calculation of covariance matrix

I came across this formula in a text that says $S$ is the sample covariance matrix where $$S = \sum_{j=1}^n(\mathbf{X}_j - \bar{\mathbf{X}})(\mathbf{X}_j-\bar{\mathbf{X}})'$$. What I am trying to ...
John Smith's user avatar
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Multivariate sample covariance

I have a set of $X_1,...,X_n$ samples with covariance $\Sigma_1,...,\Sigma_n$. The multivariate sample mean is then $$ \left(\sum_{i=1}^n \Sigma_i^{-1} \right)^{-1} \left(\sum_{i=1}^n \Sigma_i^{-1} ...
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Interpreting eigenvalues of non-normalized covariance matrix of physical system

Cross-posted from physics stackexchange Summary: Eigenvalues of a "non-normalized" covariance matrix of time-series measurements from a linear system have units of Action (energy * time). ...
user3716267's user avatar
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Efficient construction of correlation matrix—serial correlation

Given $\rho$, is there a way to efficiently construct this matrix (i.e., as a product of matrices, rather than using a for loop)? $$ \Sigma = \begin{pmatrix} 1 & \rho & \rho^2 &\cdots &...
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Exponentially Weighted Covariance Matrix with Ledoit Wolf Shrinkage

The Ledoit Wolf paper "Honey, I Shrunk the Sample Covariance Matrix" presents the formulation for the shrinkage intensity parameter estimate in Appendix B. The formula for a weighted ...
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Sample Random Effects from Mixed-effects Model

I have a mixed effects model that I'd like to simulate random effects from using a MVN approximation (sampling from the predictive distribution). My question is what is the advantage of simulating ...
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Origin of the term "spherical" in relation to covariance matrices?

I understand that a covariance matrix with all diagonal elements equal, and all off-diagonal elements also equal (but different to the diagonal elements) is called "spherical". I am curious ...
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Distribution of covariance parameter resulting from sum of known- and unknown-covariance noise processes

Let's say I have a set $X$ of $N$ random $p$-dimensional vectors generated by $\mathbf{x}_i = \boldsymbol{\mu} + \Psi_i^{1/2} \boldsymbol{\xi}_i + \Sigma^{1/2} \boldsymbol{\zeta}_i$, where $\...
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difference between GLM covariance matrix from MLE vs. IRLS for non-canonical link

Someone asked a question on Stack Overflow where they noted a difference between Minitab and R (glm) results for the variance-covariance matrix of the parameters, ...
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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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Conditions of the covariance matrix between discrete and continuous variables

Does the covariance matrix for a discrete variable and a set of continuous variables have extra constraints beyond being positive semi-definite as in the case of a real-valued random vector? ...
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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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Beyond AR(1) as a covariance structure for mixed models with repeated measures

I have been reading about alternatives to assuming an AR(1) covariance structure for mixed models with repeated measures (in time). In particular I have heard about Toeplitz and Ante-dependence ...
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Riemannian alignment has no effect

I am trying to implement a Brain-Computer Interface system that should be able to differentiate between rest and movement trials. I am using motor-related cortical potentials for movement detection. ...
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Merge/ "stack-up" two covariance matrices

I'm doing a kalman filter and I need to sum two covariance matrices (Q1 and Q2) regarding two uncertainties I need to join. I'm tracking the position of an object from a referencial with a changing ...
HenryTran's user avatar
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How to handle the residual covariance structure in a mixed model with repeated measurements?

In clinical trials and other areas of applied statistics, we often need to model longitudinal data. In most introductions to modelling longitudinal data with mixed effects models that I have seen, ...
underflow's user avatar
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MLE of multivariate normal distribution when the VCV matrix is full of equations

Short Version: Given a variance covariance matrix for my multivariate normal distribution where the entries are equations of other parameters, how do I find which of those parameter values maximizes ...
A Friendly Fish's user avatar
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1 answer
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Reparameterization of the variance-covariance matrix (`apVar`) of the random-effect parameters estimated by `lme`

The question is about computing the variance of the random-effect parameters estimated when fitting a linear mixed-effect model when the parameterization of the random-effect parameters changes. This ...
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Is it possible to have a multivariate random distribution with all its random variables (pair-wise) reverse correlated?

Continuing this question, I want to ask if its possible to have a multivariate random distribution having simultaneously all its random variables reverse correlated. I think it is impossible, because ...
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Covariance matrix for data

Assume $n*p$ data matrix $X$, where n is the number of observations and p is the number of features. We are interested in the covariance among features. I have seen notations where covariance matrix ...
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Estimating the noise covariance matrix in linear regression

Given y = x*B + e, where x is a p dimensional vector of inputs, and y is a one dimensional response variable. If the noise term is not necessarily i.i.d., but zero-mean Gaussian, how would I estimate ...
Septimus Boshoff's user avatar
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Incrementally Computing $\Sigma_t v_t$ Without Storing $x_i$, $v_i$, or $\Sigma_t$

Motivation: I have a discussion with friends many days ago, at that time I think this problem is very easy so we directly skip, later I realize I cannot solve it XD. The problem is: Given a sequence ...
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A Trivariate Normal Distribution with a Singular Covariance Matrix

I have a very quick but subtle question. Consider a trivariate normal vector: \begin{align*} [V_1,V_2,V_3]'\sim N\left[0, \Sigma\right]\end{align*} Here, if $V_3$ is a convex combination of $V_1$ and $...
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Estimating different random effects based on a between-subject factor in a mixed model?

Background I have a between-subject factor of GROUP with 2 levels (Control and Active) and a within-subject factor of TIME with 3 levels (Time1, Time2, and Time3). Code below to generate some dummy ...
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Can I estimate the confidence intervals when fixed-effect model matrix is rank deficient?

I have run model selection for linear mixed effect models with random intercepts, built with lme4 package in R. The top model contains one continuous variable (A), 2 categorical variables (B and C) ...
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Generative model for random covariance matrices to fit hierarchical data

I have a multivariate dataset with M groups of data, each consisting of N iid measurements of p variables. Say I take the N measurements from a single widget, and M corresponds to the number of ...
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GAMLSS model can't compute vcov using family BEINF0

I am using gamlss to analyze inherently proportional data for % mildew on leaf area. This is an average value of 10 leaves per replicate. I only have 3 replicates. I used BEINF0 for some datasets that ...
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