A $k\times k$ matrix of covariances between all pairs of $k$ random variables. It is also called variance-covariance matrix or simply variance matrix.

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Is there any point to Reverse Engineering the Fisher Information Matrix from an Inverse Covariance Matrix?

Would there be any advantage in deriving a Fisher Information Matrix backwards from an inverse covariance matrix? I've discovered that this is much easier to do on the SQL Server platform I use than ...
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1answer
16 views

Why is multinomial variance different from covariance between the same two random variables?

We know that if $\big(X_1,X_2...X_k) \sim multinomial(n;p_1,p_2...p_k)$ then $X_i \sim bin(n;p_i) $ Then, $var(X_i) = np_i(1-p_i)$. But we have $cov(X_i,X_j) = -np_ip_j$. So doesnt that imply ...
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29 views

Covariance matrix of multivariate multiple regression coefficients

I would like to perform a regression analysis on a dataset comprising one independent variable (X) and two dependent variables (Y1 and Y2) which may be affected by correlated errors. R's stats::lm ...
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29 views

Why estimated covariance matrix by glasso is always zero?

I am using glasso function from glasso package, as follow: ...
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1answer
36 views

Intuition for near-decorrelation through centering

Consider a $p\times 1$ random vector $\mathbf u = (u_1,...,u_p)'$ with zero mean vector and variance-covariance matrix $$E(\mathbf u \mathbf u')\equiv ...
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1answer
18 views

$LDL^T$ decomposition from Cholesky decomposition

Suppose we have a covariance matrix $\Sigma$. I know that the Cholesky decomposition $A^T A$ can be found from the LDL decomposition using $$ \Sigma = LDL^T = (LD^{\frac 1 2})(LD^{ \frac 1 2 })^T = ...
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3answers
126 views

Decrease of $(X'X)^{-1}$ as n increases

Let $X$ be a $n \times p$ matrix, filled with iid draws, with $n \geq p$ (like a conventional data matrix). I would like to show that, in a sloppy notation, $(X'X)^{-1} \rightarrow 0$ as $n ...
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1answer
25 views

Perform a t-test in multiple regression matrix

I have the following matrices: $$ X'X $$ $$X'y$$$$ y'y $$ I know that the B matrix can be computed as follows : $$ B = (X'X)^{-1}X'y $$ If I want to perform a t test for a specific B, say $$B_{1}$$, ...
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10 views

Calculation of vector norm uncertainty using covariance matrix

I have a vector $\mathbf{v} = [ v_x , v_y ] ^T$ whose PDF is Gaussian, thus it has an associated covariance matrix to represent its uncertainty (which is zero mean): $P = \begin{bmatrix} \sigma_x^2 ...
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20 views

Should I estimate a covariance mattix or a precision matrix (inverse covariance matrix)?

More specifically, what are some pros and cons of estimating a precision matrix over a covariance matrix?
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30 views

If an inverse covariance matrix is sparse, what can I say about the covariance matrix?

How does the sparsity condition on an inverse covariance matrix affect the actual covariance matrix?
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15 views

Factor analysis Geometric interpretation of common variance [duplicate]

I am trying to understand the fundamental differences between PCA and Factor Analysis. PCA is straight forward in that you take the eigenvectors of the data's variance (and hence considering all the ...
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12 views

Unconditioinal covariance of factors in go-GARCH

In this pdf in section 2.4 page 11 Alexios Ghalanos explains the theory behind the go-GARCH model (general orthogonal GARCH). I don't understand why the unconditional covariance is $$\operatorname{E} ...
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2answers
53 views

What is an isotropic covariance matrix?

Could somebody explain to me in simple terms what an isotropic covariance matrix is? I can't find anything online.
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1answer
117 views

A name for operator-dependent cross-product

Suppose that we have a $\\n\times p$ matrix $\mathbf{M}$. Different transformations using different column-wise operators can lead to a new $\\p\times p$ symmetric matrix $\mathbf{S}$. For example, ...
3
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1answer
29 views

Covariance between two random matrices

I have two random matrices (matrix-valued random variables) $X$ and $Y$, both of dimension $n \times n$. Is there a notion of covariance between the two random matrices, i.e., $\text{Cov}(X,Y)$? If ...
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0answers
6 views

Multiple Imputations in Mplus: Asymptomatic Covariance

I ran multiple imputation on a model that created 30 imputations that I then used to analyze a multilevel model that included a three-way cross-level interaction. In order to interpret the interaction ...
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1answer
20 views

Why is the covariance matrix of the OLS regression error written as $\mathrm{E}({\bf e}{\bf e^\prime} | {\bf X})$?

In Hansen (2016) I read that the conditional covariance matrix of the regression error $\bf e$ is $\mathrm{E}({\bf e}{\bf e^\prime} | {\bf X})$. How is this related to the covariance formula: ...
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10 views

Covariance between added harmonic timeseries

I am trying to construct a covariance matrix for an optimization between two added harmonic time series. In general, the basic equation looks like this: $Z(t)=Y(t)-X(t)$ When I break this down into ...
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2answers
59 views

Variance-covariance matrix of a single variable

I want to calculate the variance-covariance matrix of a single variable: $$ \begin{align*} Var({\bf y}) & = E({\bf yy'}) - E({\bf y})E({\bf y'}) \\ & = \left( \begin{array}{cccc} ...
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22 views

Robust Bootstrap Covariance Estimator

I sometimes see particular bootstrap repetitions give "wild" regression coefficient estimates for one or more bootstrap resamples. This occurs more often in binary logistic regression. One or two ...
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21 views

Can PCA be used to reduce estimation error for covariance estimation?

One of the uses of factor models is to estimate covariance matrices. The reason why you might want to do this is to reduce the number of variables that you have to estimate and so avoid accumulating ...
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1answer
45 views

Determinant of a block matrix with sparse elements

I have a positive definite symmetric matrix that looks like $$\pmatrix{A & 0 & 0 & E \\ 0 & B & 0 & F \\ 0 & 0 & C & G \\ E^\prime & F^\prime & G^\prime ...
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1answer
22 views

R unfold a list into a matrix [closed]

I have a list in which each element inside it is a matrix. How can I unfold this list to get a matrix. Example: List[[1]]=matrix A List[[2]]=matrix B List[[3]]=matrix C and I want directly to get a ...
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30 views

Simulating data from a given multivariate covariance matrix - workarounds for a non positive definite covariance matrix?

As part of a simulation study, I would like to create multivariate data that follow a specific covariance matrix in R. In this study I would like to be able to show that my algorithm is able to find ...
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3answers
242 views

Invert a sparse covariance matrix

I have a postive definite symmetric covariance matrix which looks like this: Note that all A,B,C,D,E,F,G are also poitive definite symmetric covariance matrices I want to find an easy way were I can ...
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34 views

Regularization methods for factor analysis (in the $n<p$ situation)

Is there any covariance matrix regularization suitable for factor analysis? I have a data matrix where number of observations is smaller than the number of dimensions: $n<p$. I am thinking of ...
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45 views

Efficient way to solve generalized eigenvalue problem when the number of dimensions is greater than the number of samples

I am trying to solve the generalized eigenvalue problem: $$C_e v = \lambda C_o v.$$ $C_e$ and $C_o$ are both covariance matrices generated from data with $10512$ dimensions and about $2000$ samples. ...
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1answer
30 views

Is the Covariance Matrix a dyad? [closed]

Given a random vector $X\in\mathbb{R}^n$ with random scalar entries $X_i$, why isn't the covariance matrix $\Sigma$ of the random vector $X$ a dyad? In other words, why isn't the rank of the ...
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0answers
45 views

Why is covariance matrix not positive-definite when number of observations is less than number of dimensions?

I have a data matrix $X$ of size $n\times p$ with $n < p$, where $n$ is the number of observations and $p$ is the number of dimensions. My question is: why $n < p$ results in not a ...
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1answer
101 views

Sampling from Gaussian Process Posterior

Anyone know of a Python package that both fits a Gaussian Process to data, and also lets you sample paths from the posterior? I'm interested in sampling the colorful lines on right (b) of the ...
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83 views

Graphical LASSO Interpretation

I have a conceptual question about graphical LASSO interpretation. I have been using the huge package in R to estimate an association network for a matrix of node ...
2
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1answer
75 views

Comparing four formulations of class scatter matrices

I am trying to decide between / reconcile four formulations for class scatter matrices. The first from Duda et al. (2012), p.544 has (with symbols modified): $$m_i = \frac{1}{n_i} \sum_{x\in ...
3
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1answer
148 views

Why I got different variance-covariance matrices for different subjects from getVarCov function from R nlme package?

I fit a linear model using generalized least squares with gls {nlme} function in R. Then I use a ...
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24 views

How to represent the sum of two linear model predictions in same unit

I have built two linear regressions independently of one another, and $Y_1$ and $Y_2$ are in the same units. I am interested in using the sum of $\widehat{Y}_1$ + $\widehat{Y}_2$ (the predictions) to ...
4
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1answer
54 views

Looking for the distribution of the difference of two Gaussians in a weird relationship

I have a covariate $B$ (let's say age) and two different responses $T_1$ and $T_2$. The bivariate distributions of $B,T_1$ as well as $B,T_2$ are bivariate normal and known: $$ ...
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1answer
60 views

glasso- Assumptions of Meinhausen-Buhlmann approximation?

First off, This is not an R question - it is a conceptual question, so there is no need to perform any code. Problem: I'm trying to invert a large dimensional covariance matrix of p features. For ...
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3answers
75 views

Express correlation matrix of $X$ in terms of $X^{T}X$ (in the OLS context)

In least squares estimation where $Y = \beta X$, how can we find the correlation matrix of $X$ in terms of $X^{T}X$? It seems that $X^{T}X$ is very close in structure to the correlation matrix, but ...
5
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2answers
433 views

Why does the correlation function in R, cor() return a matrix with fewer rows that you started with?

I have a matrix $G$ in which I would like to compute the correlation matrix of. The matrix in R looks like: ...
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1answer
36 views

How to get the determinant of a covariance matrix from its diagonal elements

I am trying to implement a speaker recognition system in MATLAB. I am using Gaussian Mixture Models (GMM) for speaker modelling and maximizing the posterior probabilities for classification. The ...
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0answers
11 views

SSCP vs correlation matrix [duplicate]

Lets say I have a matrix X, where each row is a subject and each column is a variable. I have a few questions about manipulating this matrix, and what the result is: If I calculate X'X, am I correct ...
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1answer
60 views

Variance of $a^TX$ for MVN X

How do you show that the variance of $a^TX$ for multivariate normal X is $a^T\Sigma a$? I have $V(a'X)=E(a'X-E[a'X])^2$, but it seems like the dimensions get messed up or something after that. So I'm ...
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19 views

Distribution of a transformation of a inverse Wishart distributed covariance matrix

Suppose to have a covariance matrix $\boldsymbol{\Sigma}$ of dimension $n \times n$ with $\boldsymbol{\Sigma} \sim IW(\nu, \mathbf{I}_{n} )$. Then $$ f(\boldsymbol{\Sigma}) \propto ...
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26 views

Do I distort information stored in covariance matrix if I normalize the matrix to range of $[-1 , 1]$

I am using covariance matrix to maximize mutual information between samples that I selected and sample that I did not selected. The optimization algorithm that I use is sfo_greedy_lazy. It computes ...
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60 views

Is the covariance matrix the equivalent of standard deviation for a 2d matrix?

I want to measure the scale of a 2d matrix that contains the coordinates (x,y) of a set of point. I know that the standard deviation of the x coordinates of these points is the scale of the x ...
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23 views

Practicality of sparse inverse covariance matrix assumptions

For a set of $p$ datapoints in $m$ dimensional space, if the features are packed in a $p\times m$ matrix $X$, then $C = XX^T$ is the covariance matrix and $K = C^{-1}$ is the inverse covariance ...
3
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1answer
47 views

How to compute the variances of PCA-transformed variables?

In principal component analysis (PCA), what is the covariance matrix in the new space of reduced dimensionality, i.e. the covariance matrix after multiplying the data by the eigenvector matrix? I ...
3
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1answer
87 views

Correlated features in dataset

I undestand in general that it is important to take correlational structure into account while applying almost any statistical techniques. First question - could you help with the examples why it is ...
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21 views

Estimating residual variance with covariance matrix and mean vector

I'm currently working on estimating a multiple regression solely with the covariance matrix and the mean vector. The example I'm working with gives the following formulas for the bivariate case of ...
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1answer
70 views

Find covariance if given mean and variance

I have a signal x that I want to classify in one of the classes A and B in which the means are Ma=[0.5,0.6] and Mb=[2,2] and with variances ...