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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Generate a random covariance matrix with specified eigenspectra and diagonal elements?

I want to generate a random covariance matrix ($c \in \mathcal{R}^{n \times n} $) whose eigenspectra, i.e., $n$ eigenvalues $e_0 \in \mathcal{R}^{n\times 1}$ and diagonal elements $c_{ii} \,\, i=1 \,\,...
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Is the sum of two singular covariance matrices also singular?

I have two sample covariance matrices, computed from $n$ samples, less than $p$ variables: they are singular then. I know that the sum of two covariance matrices is also a covariance matrix. My ...
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Demonstration and Interpretation between a Fisher matrix and its dual space which is covariance matrix

I have a simple (maybe not) issue about the interpretation of the link between Fisher information matrix and its inverse which is the covariance matrix. How to formulate that a line of Covariance ...
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What is the cubic expectation (third-order moment) of a complex gaussian vector (say, E[$aa^{T}a$])?

Note: I also posted this question on MATHEMATICS. For a real gaussian vector, an explicit formula for the cubic expectation can be found in Matrix Reference Manual (search 'Cubic Expectations' in this ...
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The Variance Covariance Matrix of an Estimator Stacking Two OLS Estimators

I am looking for how to derive the variance covariance matrix (henceforth, VCOV) of an estimator stacking two OLS estimators. Suppose that we have two OLS estimators: $$\hat{\alpha}\sim N(\alpha,\;\...
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Apply Shrinkage Coefficient to Exponentially Weighted Moving Covariance Matrices

Title says it all. I'm working on doing volatility forecasting and like the approach of exponentially weighted moving covariance matrices, but I also know that applying shrinkage coefficients further ...
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Estimate the mean vector and the covariance matrix using the simple returns

I would appreciate help with how to to estimate the mean vector and the covariance matrix using the simple returns in R. I have historical (weekly) values of five stocks from a capital market for a ...
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Correlation matrix from pairwise correlations with specified structure

I need to simulate multivariate normal samples with a pre-specified correlation structure. The structure is such that the bigger the (GPS) distance between two points, the smaller the correlation (...
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Estimate 2D covariance from 3D matrix where 3rd column contains probability density values

I have an nx3 matrix where the first 2 columns contain uniformly distributed random (x, y) points and the 3rd column contains pdf values evaluated at each point. The pdf values are computed from a ...
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Variance-covariance structure for random-effects in `lme4` or `nlme` (covariance specification)

I am running a multilevel growth model with multiple random slopes. In the Mplus software, I can specify exactly which random effect covariances are estimated (and which are not). For example, I can ...
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Effect of using Cholesky transpose

I am generating random normal samples Y with covariance C using a well known procedure: Let L be the Cholesky decomposition of C, such that $C = LL^T$. Now given a matrix of random numbers $X, x_{ij}...
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Covariance matrix of beta coefficients for constrained multiple regression

I have a linear least-squares problem with constraints that two of the coefficients must be non-negative. For a typical (unconstrained) least squares estimation, I know that the variance-covariance ...
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Why do some poeple claim, that choice of the working correlation in GEE doesn't affect the marginal coefficients?

I found this discussion: GEE: choosing proper working correlation structure Cite: Correlation structure in GEE, unlike mixed models, does not affect the marginal parameter estimates (which you are ...
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What is the purpose to have the "independent" covariance structure in GEE or GLS?

The methods of estimation like GLS or GEE are especially helpful, when there are clusters of data, like repeated observations, many per cluster=subject. Such observations are naturally correlated in ...
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is it possible to use a bayesian neural network to calculate the covariance matrix among a set of predicted values?

as title says, i have created a little bayesian fcnn with dropout layers trained on some cosmological data that come with a covariance matrix that is almost 0 everywhere. My goal was to use the NN to ...
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Recover linear transformation of Covariance Matrix

Suppose that I can observe $\Sigma_1$ (r x r), and $\Sigma_2$ (r x r)= $A \Sigma_1 A'$, for some arbitrary (r x r) rotation matrix $A$. Is it possible to recover $A$ in any meaningful way?
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Confusion on calculating Mahalanobis distance

I am slightly confused as to how you calculate Mahalanobis distance given a set of data. I have tried asking my tutor for help but he does not seem interested in helping what so ever and I am ...
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Building Covariance Matrix From Samples and not Features

In most of the applications, I see that the covariance matrix is calculated on feature space, by first applying normalization steps on $X$ then the covariance is calculated by $C = X^{T}X$ where $X \...
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Principal Component's Direction for a Matrix

Can anyone give a brief mathematical derivation on how to calculate principal components in PCA for a given covariance matrix let's say - \begin{pmatrix} 5 & 2\\ 2 & 5 \end{pmatrix} ?
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How to derive the covariance matrix from a rotated ellipse?

I'd like to derive the covariance matrix that defines a given ellipse. Information I have: length of major axis $\lambda_1$ length of minor axis $\lambda_2$ angle of rotation of the ellipse is $\...
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Create correlated random variables from uncorrelated variables [duplicate]

Say there are $n$ mean-zero random normal variables $\varepsilon_i,...,\varepsilon_n$ with a $n \times n$ covariance matrix $\Sigma$. I have $n$ mean-zero random normal variables $u_i, ...,u_n$ which ...
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Need help obtaining Cov Parameter Estimates in PROC GLIMMIX

I'm really struggling with something that should be a simple solution but for the life of me, I've been struggling for a while now trying to figure it out. I apologize in advance for my lack of ...
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Problems finding GLS solution from estimated covariance matrix

I am using a mathematical function to estimate the covariance matrix for some process from the variances and then using this covariance matrix in a generalised least squares estimation of the slope ...
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Show that $\frac{1}{n}\sum_{i=1}^n (x_i -\hat{\mu}^\top)\cdot(x_i -\hat{\mu}^\top)^\top=E[(X-\hat{\mu})(X-\hat{\mu})^\top]$ [closed]

In a lecture my lecturer used the fact that $$\frac{1}{n}\sum_{i=1}^n (x_i -\hat{\mu}^\top)\cdot(x_i -\hat{\mu}^\top)^\top=E[(X-\hat{\mu})(X-\hat{\mu})^\top]$$where $X$ is our data matrix and $\hat{\...
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What is the analog of precision matrix for cross-covariance matrices?

For a covariance matrix, I am aware of the precision matrix, the covariance matrix inverse. What's the analog for that for a cross covariance matrix, i.e. $E[XY^{\top}]-E[X]E[Y^{\top}]$ for two random ...
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matrix partition and estimator variance

Consider the GLM $y=X\beta+u$ such that (i) explanatory variables are non stochastic and linear independent (ii) explanatory variables are asymptotically linear independent (iii) the error process are ...
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covariance matrix for three correlated variables

Suppose I have a covariance matrix for three random variable $X1,X2,X3$ $$ \begin{bmatrix} 1&0.5& \rho \\0.5&1&0.5 \\\rho&0.5&1 \end{bmatrix} $$ I know I can solve for valid ...
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Finding the eigenvalues of covariance matrix [duplicate]

Came across this as an interview question that I saw online: given a covariance matrix with diagonals being all 1 and the off-diagonals being c, what are its eigenvalues? Going by the definition $Av = ...
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Estimated covariance matrix and sample covariance matrix of SEM

Normal Covariance I have tried looking high and low for an answer to this question, but I seem to never get a great answer on it. First, I think I'm knowledgeable about what a covariance matrix is. It ...
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How to determine matrix D in the following formula: D = ADA' - R +I

I am trying to isolate the matrix D from the following equation: $$ D = ADA' - R+I $$ Within this equation D is the covariance matrix. There can be assumed that A and R matrices are 3x3 matrices with ...
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When should we decompose the precision matrix as opposed to the covariance matrix to generate correlated variables?

We can take a covariance matrix $\Sigma$ and decompose this into a lower and upper triangular matrix $\Sigma = U^T U$ where $U$ is the Cholesky matrix. This matrix can be used to transform ...
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Environmental variance model: How to obtain slope, RMSE and regression p-values for each level of fixed effects in a linear mixed model made with lmer

Dear everyone, I am trying to obtain linear regression details for my linear mixed model (LMM) created with the lme4 package in R. I have a LMM with Yield as response variable and four levels of ...
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Variance of a random vector (different of the covariance matrix)

Tipically, the variance of a p-dimensional random vector $$X = (X_1,...,X_p)$$ is defined as a the covariance matrix given by: $$E[ (X- EX)^T(X- EX) ]$$ But, in the second page of this paper, the ...
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Multinomial probit: can covariance of coefficients be calculated from predicted probabilities?

When producing a GLM (generalized linear model), one usually wants to have an estimate of the variance-covariance matrix of the fitted coefficients, which happens to have a closed form solution with a ...
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Show that the autocovariance matrix is positive definite

I've been working through the textbook Time Series Analysis and its Applications (R. H. Shumway & D. S. Stoffer 2ed). The topic I'm looking at is forecasting using ARMA models. The below assumes ...
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Methods for comparing structure of multiple covariance matrices

I am familiar with methods to compare the global structure of multiple covariance matrices (eg. Calsbeek, B. and C. J. Goodnight (2009). “Empirical comparison of G matrix test statistics: finding ...
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minimum determinant covariance matrix and covariance

I am trying to understand minimum determinant covariance. I gather from this stack exchange post that it tries to select a subset of data that is tightly distributed to exclude anomalies, and it does ...
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Computing the covariance matrix for the same variable among several groups from a single long column

I have a simple dataset (say a dataframe) that, reduced to its essence, looks like this: ...
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Why is the cholesky decomposition the correct matrix square root to sample from? [duplicate]

I recently discovered that not all matrix square roots are the same because they are basically rotations of one another. Assuming that $X$ is already mean centered, we could use some different ...
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Can any covariance factorization $LL^\top$ be used for sampling?

I thought that any factorization of the for $LL^\top$ of a covariance matrix could be used for correlating random noise according to the covariance. I tried doing this with the following code and ...
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Find information of unknown elements through correlated values

I'm not entirely sure how to word this question, so my apologies outright. How can I estimate the amount of total information in a system by only inspecting specific elements in that system? For ...
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Why are MLE for high dimensional multivariate gausian covariance matrix likely to be ill-conditioned

In a book I'm reading (Probabilistic Machine Learning: An Introduction) the author suggested that in high dimensions, the MLE estimate for the covariance matrix for multivariate gaussian is often ...
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About the calculation of covariance matrix in mahalanobis distance: How $W^TW$ is equal to the covariance matrix? [closed]

I was reading about deep metric learning (from here) and came across the mahalanobis distance. I understood why we can not use euclidean distance if the distribution is not isotropic (the covariance ...
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Convergence of Gaussian random variables

Let $(f_n)$ be a sequence of 0-mean Gaussian densities on $\mathbb{R}^d$ and assume $f$ is limit of $(f_n)$. Question 1 How does one determine the type of convergence by looking at the corresponding ...
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Observed Fisher Information and confidence intervals

I'm trying to put confidence intervals on parameters fitted through MLE through the inversion of the observed Fisher information matrix. More specifically, I define the observed FIM as: $$ J_{n}(\hat{...
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Estimating the sample size of pilot study by using variance covariance as an estimator for structural equation modeling

I am not from an advanced statistical background, therefore feel free to correct and teach me if you found any mistakes that I have made below. Recently, I am interested and learning a Monte Carlo ...
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regularisation: choice of noise/ perturbation values for dataset with small number of observations

In a dataset $D$ of dimension $N \times d$ where the number of observation $N$ is small such that it leads to the covariance matrix being ill - conditioned, a workaround involves introducing ...
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Between class covariance matrix formula

The conditions of a classification problem are as follows: There are $K$ classes we are predicting with the $p$ dimensional predictor vector $X$. Let $Y$ be the $K-1$-dimensional one-hot encoded ...
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Positive definiteness of integral of matrix

I was reading a paper, and did not understand a statement that the author made without further explanation. The author derives the limiting distribution of a non-linear least-squares estimator and ...
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Gaussian Processes: multi-class Laplace approximation

In Chapter 3.5 of Gaussian Process for Machine Learning book by Rasmussen and Williams (R&W 2006), authors present a Laplace approximation for a multi-class Gaussian Process (GP) classifier. ...
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