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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How does the variance-covariance matrix change when I create a linear combination of two variables? [duplicate]

Suppose I have four normal r.v (X,Y,W,Z) and the variance-covariance matrix is know. If I create a new r.v J=aX+bY (a and b are scalar), what is the new variance-covariance matrix? Thank you
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Not all positive-definite matrices are valid covariance matrices for lognormal variables

A simple method to generate correlated lognormal variables $X_i$ that obey a covariance matrix $C_{\mathrm{ln}}$ with elements $c_{\mathrm{ln}}^{ij}$ is to first compute the covariance matrix ...
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How does unbiased cov.wt compute its value?

What is exactly the calculation used when calling cov.wt((faithful,c(1,2,3,4))$cov)? (Say the data faithful has 4 rows and 2 columns, coordinates x,y) I.e.: What ...
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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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37 views

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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32 views

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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26 views

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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114 views

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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48 views

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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18 views

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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90 views

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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34 views

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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19 views

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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78 views

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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Matlab FACTORAN error on line 162: a covariance matrix is not positive definite [closed]

I have a data set called Z2 that consists of 717 observations (rows) which are described by 33 variables (columns). The data is standardized by using ZSCORES. Additionally, there is no case for which ...
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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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125 views

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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21 views

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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27 views

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 ...
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58 views

Decomposition of inverse covariance matrix

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 ...
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Why is pure sample covariance a bad metric to understand the degree of correlation between two variables?

Covariance helps you understand how variables are linearly related. Would it be possible to have two pairs of variables in a deterministic relationship (i.e. linearly correlated variables) that have ...
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How to obtain the covariance matrix for the parameters estimated by maximizing this log-likelihood of logit/probit

I have this linear latent model : $y^*=\beta_0+\beta_1 X+\epsilon,$ $y^*$ is latent (non observed) variable and $\epsilon\sim iid (0,\sigma^2 I)$ Consider : $y=1$ if $\beta_0+\beta_1 X>0$ and ...
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52 views

Variance / Covariance Matrix - mean of squared errors

I'm trying to build a stats library. I'm following along with the tutorial on multiple regression analysis here: http://reliawiki.org/index.php/Multiple_Linear_Regression_Analysis I have the ...
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129 views

Estimating VAR by GLS versus OLS: efficiency

Suppose I have a VAR model with different regressors in different equations (this could be due to restricting some coefficients of a full VAR($p$) model to zero or having some different exogenous ...
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Covariance of natural logarithms — how to estimate $\sigma (\mathbf X, \mathbf Y)$ from $\sigma (\ln{\mathbf {(X)}},\ln{\mathbf {(Y)}})$?

I have $$\operatorname{Cov} (f{\mathbf {(X)}},f{\mathbf {(Y)}})$$ where $\operatorname{Cov}$ denotes the covariance and $f(\mathbf X)$ is a nonlinear function, i.e. $f(\mathbf X) = \ln(\mathbf X)$. ...
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Variance estimation in a one-factor linear model

I was given a dataset (a mat file) of $100\: 000$ observations, each with $50$ dimensions (coordinates). Denote matrix $X$ a $50\times 100\:000$ matrix in which each column was generated according to: ...
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115 views

Can covariance be derived from means and variances?

In treatment studies it is common to report multiple outcome measures from the same subjects. The treatment effects on these outcomes are typically correlated so this should be taken into account ...
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113 views

Identification of Difference in Weighted Mean and its Standard Error in Clustered Data Using Stata

I already asked this question on the statalist but have not received any comments on this issue yet. So I am going to post it here too. I am concerned with the following problem. Suppose my data ...
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110 views

Best OLS estimators

Hi i am stuck on this one, the question is related to Gauss-Markov theorem: Consider a general alternative to the OLS estimator that is also a linear unbiased estimator, say ${\tilde \beta}$. ...
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Generate Symmetric Positive Definite Matrix with a pre-specified Sparsity

I am trying to generate a correlation matrix $p\times p$ (symmetric p.s.d) with a pre-specified sparsity structure (specified by a graph on $p$ nodes). The nodes that are connected in the graph have ...
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LS vs MLE for Gaussian Conditional Random Field estimation

Is there such a thing as Least Squares estimation for the conditional mean and covariance of a conditional gaussian random field? I'm looking at this paper by Wytock and Kolter 2013, in which they ...
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Is every correlation matrix positive semi-definite?

I am generating correlation matrix by some new algorithm. Generated matrix is non positive semi definite matrix. I'm getting few negative eigenvalues. Rest of eigenvalues are quite equal to ideal ...
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Multivariate normal with singular covariance

I'm an undergraduate student. I read about multivariate normal distribution in hogg and craig. And i wonder why the covariance is allowed to be positive SEMI-definite. I read this Normal distribution ...
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How to generate a large full-rank random correlation matrix with some strong correlations present?

I would like to generate a random correlation matrix $\mathbf C$ of $n \times n$ size such that there are some moderately strong correlations present: square real symmetric matrix of $n \times n$ ...
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Can the law of total covariance apply to variables from different sample spaces?

Wikipedia says this about the law of total covariance (http://en.wikipedia.org/wiki/Law_of_total_covariance): In probability theory, the law of total covariance,[1] covariance decomposition ...
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597 views

Confused about the visual explanation of eigenvectors: how can visually different datasets have the same eigenvectors?

A lot of statistics textbooks provide an intuitive illustration of what the eigenvectors of a covariance matrix are: The vectors u and z form the eigenvectors (well, eigenaxes). This makes sense. ...