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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Correlation between two normally distributed variables

Let a~$\mathcal{N}(\mu_a,{\sigma_a}^2)$,b~$\mathcal{N}(\mu_b,{\sigma_b}^2)$ and c~$\mathcal{N}(\mu_c,{\sigma_c}^2)$. We construct two normal variables x~$a-b$ and y~$a-c$. Can we find the ...
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Can you combined two sources with difference variance to reduce error? [duplicate]

I have two samples of data each estimates of a position x, y with Gaussian noise. One source has a larger variance than the other. Is this source in any way useful ...
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Inverse covariance matrix, off-diagonal entries

Let $\Sigma$ be a covariance matrix. According to the material in this link, If the elements of $\Sigma$ are all positive, most of the off-diagonal elements in $\Sigma^{-1}$ will be negative ...
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Variance-covariance matrix as the sum of variance covariance matrices

I have a variance-covariance matrix, $\mathrm{V}$. This allows me to take a vector, $x$ of independent random variables drawn from a known distribution, and induce a required variance-covariance ...
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Negative variance from inverse Hessian matrix

I used nlm function in R to do the optimization. When I calculated the correlation between estimated parameters using the inverse of Hessian matrix, I got negative values on the diagonal. My questions ...
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Covariance matrix estimation in presence of missing data

I want to estimate a covariance matrix from data with some missing values. Ideally I'd like an R package but python could be ok. R has some built in ways of doing this. You can use ...
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36 views

What is the best way to simulate data for a linear regression model?

I am concerned with simulating data for a linear regression model. I need to control the means, variances, and correlations (covariances) between the predictors and the criterion variable. In ...
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36 views

Expectation of the product of two log normal variables

I am struggling with a proof, and I am wondering if anyone can help or point me to the right direction. Suppose that we have two variables, $X$ and $Y$, and they follow a multivariate normal ...
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37 views

Comparing two variance matrices

I am looking for bibliographical reference for comparing two variance matrices with he following criterion: $\text{Var}[X] \geq \text{Var}[Y] \quad \text{if} \quad \text{Var}[X]-\text{Var}[Y] \succeq ...
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How to use the Huber/White estimator of covariances in a generalized linear mixed model (glmmPQL) in R?

An analysis was implemented in SPSS 22 that uses the "Generalized Linear Mixed Models" feature of the program. Now I am looking for a way to port this to R. I use the glmmPQL() function of the MASS ...
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Is there a way to use the covariance matrix to find coefficients for multiple regression?

For simple linear regression, the regression coefficient is calculable directly from the variance-covariance matrix $C$, by $$ C_{d, e}\over C_{e,e} $$ where $d$ is the dependent variable's index, ...
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80 views

Variance-covariance matrix for ridge regression with stochastic $\lambda$

In ridge regression with design matrix $X$, outcomes $y$, fixed regularization parameter $\lambda$, and errors $\epsilon\sim\mathcal{N}(0, \sigma^2I)$, the computations for the ridge regression ...
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Whitening Transformation using a Hadamard product Variance Matrix

I want to whiten a vector $X$ by transforming the variance-covariance matrix so the variance-covariance matrix of the transformed series will be the identity matrix $I$. $X$ is a time-series column ...
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Cholesky decomposition and confidence ellipsoid

I'm trying to construct an error ellipsoid from a covariance matrix (which exists for a 3D point) and then sample consistent xyz points in this region. (This question succeeds this one.) What I'm ...
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24 views

Testing the random slope with correlated random effects

I have a mixed/random effects model $$\mathbf{y}_i=\mathbf{X}_i\boldsymbol\beta+\mathbf{Z}_i\mathbf b_i+\boldsymbol\epsilon_i,$$ where random effects $\mathbf b_i$ has variance-covariance matrix ...
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66 views

Trying to use Cholesky decomposition of covariance matrix to sample error ellipsoid

I'm trying to construct an error ellipsoid from a covariance matrix (which exists for a 3D point) and then sample consistent xyz points in this region. In a previous question when I asked about this ...
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24 views

Univariate Normal Converted to Multivariate Normal: Covariance Derivation

I am reading the paper available at this link: https://drive.google.com/file/d/0B2_rKFnvrjMARnU1QjB4anR3RDA/edit?usp=sharing I am having trouble understanding section 5.1 (page 2741). Essentially ...
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Weighted sample covariance

I have read the Wikipedia article, and know that the unbiased weighted sample covariance matrix for the row vector $\mathbf{x}_i$ is $$\Sigma=\frac{1}{\sum_{i=1}^{N}w_i - 1}\sum_{i=1}^N w_i ...
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Implications of doing a confirmatory factor analysis with a correlation matrix as input instead of a variance-covariance matrix?

Is this possible, and if so what are the implications of doing things one way rather than another. Is one approach generally preferable? So far I have only been taught to use a variance-covariance ...
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How to calculate variance - covariance matrix of a matrix?

For example, we have an NxP matrix with N rows and P variables. Then we need to calculate a PxP sample covariance matrix. How do I do that?
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Is it necessary to transform an econometric model in order to have only diagonal elements in the error covariance matrix?

Model and its assumptions I'm working on the methodology of a two-way error component model. Here is the model: $$y_{jis} = x_{jis} \beta + \upsilon_{jis}$$ $j$ refers to school, $i$ refers to ...
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Background subtraction for signal and error analysis

I use a CCD to see the split of a energy level due to Zeeman effect. I have a 1 dimensional CCD of 7926 pixel of 7μm each one. My CCD analyze a region 2 dimensional, and then it steps forward 200 ...
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Parameter covariance matrix for a multivariate (matrix-Y) logit model

I've got a partially-observed unidirectional network. Nodes can be linked (0/1) in one of many ways. For now, lets call them $y_1$ and $y_2$. The unit of analysis is the potential network link ...
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Gelman and Rubin convergence diagnostic, how to generalise to work with vectors?

The Gelman and Rubin diagnostic is used to check the convergence of multiple mcmc chains run in parallel. It compares the within-chain variance to the between-chain variance, the exposition is below: ...
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Mean of covariance matrices

I'm trying to generalise a formula that takes the mean of some variances to it work with vectors. I'm not sure it makes sense to take the variance between a bunch of vectors, rather it is more suited ...
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Problems estimating covariance matrices with small $n$, smaller $p$?

It is well known that estimating large covariance matrices from small samples is problematic. For instance, the $p \times p$ sample covariance matrix $\Sigma_n$, estimated from $n$ samples, is not ...
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Sample covariance matrix

$\newcommand{\X}{\mathbf{X}} \renewcommand{\S}{\mathbf{S}} \newcommand{\I}{\mathbf{I}} \newcommand{\1}{\mathbf{1}} $ I found an article with an unusual (for me) covariance matrix. Let $\X$ denote an ...
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How to determine the confidence interval or significance of a covariance estimate

I was wondering if there is a way to determine the significance of a covariance? So, if have two vectors of returns, and then calculate the covariance, how do I determine if the sample covariance is ...
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Specific variance covariance structure in lmer

I have a dataset with cluster correlated data; multiple measurement on the same subject (not over time). I am trying to create two different mixed models using lmer in R with two specific variance ...
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525 views

Variance-Covariance matrix interpretation

Assume we have a linear model Model1 and vcov(Model1) gives the following matrix: ...
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121 views

Significant box's test equality covariance matrices, How bad it is?

Having equal number of cases across two groups (60 total cases), if one wants to perform a split-plot Anova and finds out that the Box's Test of Equality of Covariance Metrices is less than 0.001 then ...
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135 views

How to draw estimates based on variance covariance matrix?

Suppose I fitted a logistic model and get the estimates as well as their vcov matrix. I would realize this: draw length($\beta_s$) independent $\mathcal N(0,1)$ ...
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Condition number of covariance matrix

I am interested in generating a covariance matrix of dimension say 100. I managed to get a correlation matrix with finite condition number. To construct a covariance matrix I need to have standard ...
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Plotting error ellipsoid from 3x3 covariance matrix in R?

I'm hoping to be able to take a 3x3 covariance matrix and turn this into an error ellipsoid but so far I haven't been able to achieve this. I'm very new to R (in fact turned to it to attempt to solve ...
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225 views

Gradient of Gaussian log-likelihood

I'm trying to find the MAP estimate for a model by gradient descent. My prior is multivariate Gaussian with a known covariance matrix. On a conceptual level, I think I know how to do this, but I was ...
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Alternative to visualize hyper-ellipsoid defined by variance-covariance matrix

Let say at the beginning I have two variables $a_{1}$ and $a_{2}$ of the same type then I would have the variance-covariance matrix defined as: $\sum = \begin{bmatrix} \sigma^{2}_{a_{1}} & ...
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42 views

Regression Puzzle GLS var/covar matrix of B given summary statistics

There is a problem posted back in 2009 that I found that has me puzzled. I believe the estimates are from a generalized linear regression because after some thought it seems $X'X$ is impossible to ...
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32 views

Finding parameter bias under omitted variable, with variance covariance notation

Dear CrossValidated community, Can anyone help me to prove the bias in a given parameter of a regression when there is omitted variable? I know to do it using matrices and matrix algebra. For ...
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71 views

Error Calculating MVN Likelihood of Time Series with AR(1) Errors in R

I'm having trouble calculating the likelihood of a time series with AR(1) errors. I am generating my covariance matrix according to page 2 of (http://cran.r-project.org/doc/contri...regression.pdf), ...
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Regression factors and covariance matrix

I am trying to follow someone else's notes. They have two matrices. One is called comfact (company factors). This is a 580 x 5 matrix. The 580 rows represent 580 ...
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If x = y*y, and you know var(y), var(z), and cov(y,z), do I know cov(x,z)?

If I know that x = y*y, and I know a whole of statistics pertaining to y, such as the variance and its covariance with other variables, can I analytically solve for the variance and covariance of x? ...
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Finding the covariance matrix to find the best linear predictor (AR(1) model)?

I need to find the covariance matrix of two given estimates of an AR(1) model $$X_t = \phi X_{t-1} + Z_t$$ to find its best linear predictor of $X_2$, given $X_1$ and $X_3$. Let W = ($X_1, X_3$)' ...
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Variance-covariance structure for random-effects in glmer

What is the default variance-covariance structure for random-effects in glmer in lme4 package? How does one specify other ...
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103 views

How to constrain covariance parameters in sas proc mixed?

I would like to test whether 3 dependent variables (measured with the same participants) differ in variance. My plan is to fit one model in which the 3 variables have the same variance, and one model ...
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22 views

Distance from bivariate Gaussian mean in terms of variance

Not sure if my question is a valid one but I will just put it out here. Consider a bivariate data set $(x_i, y_i)$ $[i=1,...,n]$ to which a bivariate Gaussian Distribution is fitted. Now, consider ...
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Heteroskedasticly Consistent Estimators for Var-Cov Matrix, Large Sample OLS Regression

I have a cross-sectional data sample of nearly 40,000 observations and tests for heteroskedasticity fail to reject the assumption of homoskedasticity. However, it seems common practice to report ...
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Variance Inequality for Random Vectors

I know that if X and Y are random scalar variables, then: \begin{align*} \mathrm{Var}(X+Y) & = \mathrm{Var}(X) + \mathrm{Var}(Y) + 2\mathrm{Corr}(X,Y)\sqrt{\mathrm{Var}(X)\mathrm{Var}(Y)} \\ ...
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Appropriate measure to find smallest covariance matrix

In the textbook I am reading they use positive definiteness (semi-positive definiteness) to compare two covariance matrices. The idea being that is A-B is pd then B is smaller than A. But I'm ...
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102 views

Proposal distributions for covariance matrices in MCMC implementation of hierarchical models [duplicate]

In a MCMC implementation of hierarchical models, with normal random effects and a Wishart prior for their covariance matrix, Gibbs sampling is typically used. However, if we change the distribution ...
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93 views

Contingency table analysis to rank preferences of birds per feature

I have a dataset that contains observations of objects that female blackbirds carry to their nests. The birds have id tags and the objects are grouped categorically with respect to their ...