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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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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28 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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31 views

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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10 views

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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29 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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18 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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42 views

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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91 views

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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51 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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46 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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81 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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69 views

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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34 views

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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2answers
79 views

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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50 views

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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333 views

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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62 views

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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546 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. ...
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52 views

How to select a subsample of fixed size to maximize its total PCA variance?

I would like to use PCA to help design my genomics experiment. I can only afford to perform my experiment on a limited number of genotypes so would like to maximize the variation of the ones I ...
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22 views

Can any one give me inf how I can form X, Z, R, G, and A matrices using dummy variables using the posted info below?

Y=Xb +Zu + e, where y represents a vector of observed (measured) phenotype values, b is vector of unknown parameters for “fixed" effects, while X is corresponding design matrix, u is vector of ...
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197 views

What is the correct formula for between-class scatter matrix in LDA?

At one point in the process of applying linear discriminant analysis (LDA), one has to find the vector $v$ that maximizes the ratio $vBv'/vWv'$, where $B$ is the "between-class scatter" matrix, and ...
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27 views

covariance matrix of residuals from a fitted model to decorrelate residuals

I fit a geeglm model with clustered data and now I would like to decorrelate the residuals of the model in order to run model diagnostics. I read that if I can obtain the covariance matrix of the ...
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79 views

Variance of Beta IV

I'm trying to calculate the variance of the Instrumental Variables (IV) estimator $${\hat \beta _{IV}} = {\left( {{Z^T}X} \right)^{ - 1}}{Z^T}y = \beta + {\left( {{Z^T}X} \right)^{ - 1}}{Z^T}u$$ (or, ...
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55 views

Correlated random draws with graph structured correlation

I have a problem where I have a graph structure, such that some nodes are connected to other nodes i.e. we have an adjacency matrix of size n*n with a 1 corresponding to a connection and 0 otherwise. ...
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68 views

What are the implications of estimating a covariance matrix from a correlated sample?

Given a sample of $n$ independent observations $x_1,...,x_n$ (where $x_i$ are $p$-dimensional column vectors), the $p \times p$ sample covariance matrix is defined as ...
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71 views

Determinant of the covariance matrix in a normal distribution

Suppose a $p \times 1$ vector $x \sim N_p(\boldsymbol 0, \boldsymbol \Sigma_1)$. Now, There is another covariance matrix $\boldsymbol \Sigma_2$. We know that $|\boldsymbol \Sigma_2| < |\boldsymbol ...
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57 views

Generating data with a given sample covariance matrix

Given a covariance matrix $\boldsymbol \Sigma_s$, how to generate data such that it would have the sample covariance matrix $\hat{\boldsymbol \Sigma} = \boldsymbol \Sigma_s$? More generally: we are ...
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11 views

What if the variance-covariance matrix of a sum of two random vectors? [duplicate]

If X is a px1 random vector with mean Mu(x) and variance-covariance matrix sigma(x) and if Y is a qx1 random vector with mean Mu(y) and variance-covariance matrix sigma(y). If p=q, what would be the ...
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30 views

Comparing and Interpreting covariances

I had a discussion about covariance recently and it would be nice to hear your feedback about this. Let's say we have a dataset of $n$ samples with $d$ attributes. For simplicity, let's say 3 of ...
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77 views

Manual calculation of variance-covariance from published confidence intervals

I would like to obtain the variance-covariance matrix from a published set of regression outputs. These outputs provide a mean value and confidence intervals. I will convert the confidence intervals ...
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20 views

Estimating covariance for naive Bayes

I am a beginner in Pattern Recognition and started reading up Bayesian classifiers. I came across the case of naive Bayes with equal covariance in all dimensions. Given sufficient data, how does one ...
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244 views

How to interpret glmer results (variance, correlation and ICC)

I'm a beginner in statistics and I have to run multilevel logistic regressions. I am confused with the results as they differ from logistic regression with just one level. I don't know how to ...
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73 views

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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23 views

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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214 views

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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41 views

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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185 views

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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113 views

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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90 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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2answers
151 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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1answer
57 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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133 views

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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236 views

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, ...
3
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1answer
140 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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92 views

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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49 views

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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42 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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132 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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1answer
38 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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90 views

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 ...