# Tagged Questions

The variance-covariance matrix, or sometimes just covariance matrix, is the matrix whose $(i,j)$ element is the covariance between the $i^\text{th}$ and $j^\text{th}$ variable (or the $i^\text{th}$ and $j^\text{th}$ parameter).

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### Do I use GLM or GEE. Determine the relationship of a continuous covariate on a 2x2 (within Subjects) interaction?

I have a 2x2 within subjects design with a continuous covariate. This covariate significantly alters the 2x2's interaction. I need to Determine the slope of the covariate's influence on the ...
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### Need advice on unbalanced time-series dataset, for use with CAPM regression

I have 40 years of monthly historical returns of around 3000 mutual funds. The dataset contains both active and inactive funds, so some funds have data for the whole period, whereas others will have ...
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### Generate independent random values from a bivariate normal distribution

I am trying to independently select two sets of numbers (set 1 and set 2) from a bivariate normal distribution. I want the ...
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### Correlation preserving transformation conundrum

I have a problem where I need to generate $n$ random variables $\in$ [0,1] (you can think of them as some sort of probabilities) and the variables have a known correlation structure given by a ...
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### Covariance function with circular similarity property

My aim is to fit functions to covariance matrices. Furthermore I would like to have these functions positive definite. For example the figure below shows a fitted covariance matrix modeled using a ...
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### How to calculate the SE of correlation from the covariance matrix in R?

If $\rho_{_X,_Y}=\frac{Cov(X,Y)}{\sigma_X\sigma_Y}$ is correlation between $X$ and $Y$. What is the Standard Error (SE) of $\rho_{_X,_Y}$? For example if: $\sigma_{_X}$ = 0.88, $\sigma_{_Y}$ = 0.44, ...
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### Distribution of trace of the covariance matrix times the inverse of estimated covariance matrix

I try to implement a two stage procedure for a multidimensional confidence region with a fixed shape regarding to the handbook of sequential analysis (Ghosh 1991, p. 33ff). To do that, I have one ...
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### Fit multiple regression model with pairwise deletion (or on a correlation/covariance matrix) in R

I'm trying to fit a multiple regression model with pairwise deletion in the context of missing data. lm() uses listwise deletion, which I'd prefer not to use in my ...
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### Why do all the PLS components together explain only a part of the variance of the original data?

I have a dataset consisting of 10 variables. I ran partial least squares (PLS) to predict a single response variable by these 10 variables, extracted 10 PLS components, and then computed the variance ...
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### Sample covariance matrix and its inverse

Assume we have the sample covariance matrix $S_1 = XX'/k$ which is not positive definite (in fact it is positive semi-definite) and not well conditioned in very large dimension (large $p$, small $k$). ...
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### What is the problem of singular (non-invertible) covariance-variance matrix?

What exactly is the problem of having non-invertible covariance matrix? Why is getting the inverse of this matrix so important? This problem is often encountered when doing regressiong works on ...
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### Estimation Error when neglecting $\mu$ while computing Covariance-Matrix

Error when neglecting $\mu$ while computing Covariance-Matrix I would like to quantify the estimation error I have to accept when estimating the Covariance matrix based on $T$ observations of ...
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### Effects of covariance structures on mixed effects models

What are generally the effects of using a covariance structure on a mixed effect model ? More specifically, in a mixed model, what should be the expected effect of using an AR(1) covariance ...
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### How can a random vector be nonsingular?

I appreciate the help I have been getting on this site today. I had help on a proof that $\text{Cov}\left(\mathbf{Y}\right)$ is nonnegative definite for any random vector $\mathbf{Y}$. According to ...
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### Using $\text{Cov}(A\mathbf{Y}+b) = A\text{Cov}(\mathbf{Y})A^{\prime}$ to prove that $\text{Cov}(\mathbf{Y})$ is nonnegative definite

Suppose $\mathbf{Y}$ is a $n$-dimensional random vector, $A$ is a fixed $r \times n$ matrix, and $b$ is a fixed vector in $\mathbb{R}^n$. I have proven already that ...
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### In Kalman filter, what is the diagnosis when the variance-covariance matrix of the updated distribution is progressively increasing?

Let $\mathbf{\theta}_t$ be a state vector at time $t$ and $p(\mathbf{\theta}_t | \mathbf{y}_{1:t}) = \mathrm{N}(\mathbf{m}_t, \mathbf{C}_t)$ be its posterior distribution. What can I say of the model ...
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### variance-covariance matrix in R for Weibull survival curve

A simple doubt: Can I use values from vcov(ajust) directly? Or do I need some kind of transformation? I mean, for a Weibull model, does ...
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### how to modify covariance matrix of PC scores after rotation of axes

I have performed principal component analysis on a set of observations, retained four principal components and estimated covariance matrix of their scores. Then I have rotated the axes so that the ...
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### What is the covariance called when it is not divided by N?

I noticed that in signal processing they have this term called cross-covariance. The cross covariance function produces covariances of two functions with different lags. At the center of the vector ...
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### Measuring share contribution of each var/cov term to the standard deviation of a sum of variables

Say, for a simple example, I have a random variable $X = \alpha_1 X_1 + \alpha_2 X_2$, where $X_i$ are random variables and $\alpha_i$ are weights. I then calculate the standard deviation of $X$ as ...
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### Automatically fixing ill-conditioning or collinearity

I'm backtesting a regression model, which entails running it on a bunch of bootstrap samples of a "rewound" version of our data set. Unfortunately, in some of these resamplings, I end up getting some ...
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### Estimating the variance of a sum of predictions

I have $N$ plots that were used to estimate a relationship between three predictor variables, $X_1$, $X_2$, $X_3$, and an outcome, $Y$, using a generalized linear (lognormal) model. The resulting ...
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### Covariance of many simple regressions

Assume we have a true model of $$Y=X\beta+\varepsilon,$$ where $Y$ is some outcome , $X$ is a $1\times p$ vector of covariates which have a (non-diagonal) variance-covariance matrix $\Omega$, ...
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### How to do factor analysis when the covariance matrix is not positive definite?

I have a data set that consists of 717 observations (rows) which are described by 33 variables (columns). The data are standardized by z-scoring all the variables. No two variables are linearly ...
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### Asymptotic covariance matrix of the covariance parameters SAS versus lme

I am trying to obtain the asymptotic covariance matrix of the covariance parameters of a mixed-effects model using SAS and R In SAS, this matrix can easily be obtained by using the 'asycov' option in ...
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### Importance of semi-positive definiteness of covariance matrix [duplicate]

Since Covariance matrix is symmetric it is Hermitian (self adjoint) and always diagonalizable. If the matrix has all non zero eigen-values its is a full rank matrix. But what is the importance of the ...
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### Can someone explain the mechanics of the variance-covariance matrix in OLS?

I have read many of similar posts here already as well as other resources on the topic, but they all generally just show the steps that generate this equation: $\hat{\sigma^2}({X}'X)^{-1}$ What I ...
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### Evaluate the multivariate normal using variance matrices $\boldsymbol{\Lambda}+\alpha_{i}\mathbf{a}\mathbf{a}^{T}+\beta_{j}\mathbf{b}\mathbf{b}^{T}$

I need to calculate a huge amount of inverses and determinants to evaluate the pdf of the multivariate Gaussian. Specifically I need to compute the inverses and determinants of the following ...
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### Why is the covariance matrix symmetric [closed]

I am aware that for a 2-d multivariate variable, the cov(x,y)=cov(y,x). This brings about the symmetry of the covariance matrix, but is it possible to have a non--symmetric covariance matrix?
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### covariance of squared terms

Assuming two variables $X1$ ~ $N(0,1)$, $X2$ ~ $N(0,1)$ with $Cov(X1,X2) = a$. Is it possible to derive analytically what the covariance between $X1^2$ and $X2^2$ would be? Empirically (I tried this ...
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### Finding covariance matrix for MLE of correlated outputs

I generate data using the following model: $\begin{pmatrix}Y_1\\Y_2\end{pmatrix} \sim \mathcal{N}\left( \begin{pmatrix}\mathbf{X}\beta_1\\\mathbf{X}\beta_2\end{pmatrix}, \mathbf{\Sigma} \right)$ I ...
Let $\Sigma$ be a covariance matrix and let $$x^T \Sigma^{-1} x = \|Ax\|_2^2.$$ What is the interpretation of matrix $A$? I tried solving for $A$ with an eigenvalue decomposition of $\Sigma$ as ...