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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Use case for Anomaly Detection using the multivariate Gaussian distribution

We have 5000 vehicles of different classes (trucks, small cars, large cars) with 100 sensors in each car measuring fuel consumption, distance traveled, average speed etc for some time period $t$ that ...
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5 views

DCC models in R: how is the first starting value chosen?

The DCC model is defined through the proxy $Q_t$ as $$Q_t=(1-\alpha-\beta) \overline{Q} +\alpha\epsilon_{t-1}\epsilon_{t-1}' + \beta Q_{t-1}$$which is then normalized to find the correlation matrix ...
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which one is more Efficient estimator on a given data set? [on hold]

I want to know "effiency of estimator" using R or "Relative efficiency of estimators", Here i have 7 co-variance matrix estimators, i want to know which one performed more efficiently then others... ...
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10 views

Chosing an covariance matrix or 'repeated covariance type' spss

I'm struggeling with my choice for a covariance matrix or 'repeated covariance type' for my lineair mixed effects analysis in SPSS. I have a continuous measurement for experienced emotion(0-100), ...
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1answer
18 views

Inverse of block covariance matrix

I have a positive definite symmetric covariance matrix which looks like this: A, B, C, D and E, F, G are MATRICES, also positive definite symmetric covariance What is the inverse of such a matrix? ...
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1answer
19 views

Calculating the Covariance matrix for the mean of variables

Assume that we have 3 3D points to make this questions simpler. $$ pt_1 = \begin{bmatrix} x_1 & y_1 & z_1 \end{bmatrix} \\ pt_2 = \begin{bmatrix} x_2 & y_2 & z_2 \end{bmatrix} \\ pt_3 ...
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3 views

Covariance between non symmetrical matrices with a unidirectional composition

I am trying to explain variation in why some researchers go to certain places and not others by analyzing how many articles scientists from one publish about another location. My matrix is ...
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20 views

Covariance Matrix and Correlation Matrix - Singularity

If a covariance matrix is non-singular, does this implies that correlation matrix is also non-singular. My guess is it depends on mean vector in $K_{X} = R_{X} - m_X.{m_X}^H$ Not sure though.
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17 views

What is the most efficient algorithm for online Non- negative Matrix Factorization (NMF)?

What is the most efficient algorithm for online Non- negative Matrix Factorization (NMF) in recent study? Thanks.
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32 views

Common covariance matrix in linear discriminant analysis

Say I want to perform LDA classification involving three classes with within-class covariance matrices $$ \hat{\Sigma}_1 \,, \hat{\Sigma}_2 \, , \hat{\Sigma}_3$$ and that these matrices are calculated ...
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1answer
17 views

Evaluate high-dimensional Gaussian with variance matrix $\sigma^{2}I_{n_{t}\times n_{t}}+\boldsymbol{\Sigma}_{t}\boldsymbol{\Sigma}_{t}^{'}$

I need to compute the log-likelihood function in a high-dimensional Gaussian time-series. I have the following model: ...
4
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1answer
36 views

Decomposing a locally stationary covariance matrix

Say I have a non-stationary Gaussian Process with a square exponential covariance whose shape varies throughout space. The covariance entries are: $$ K_{ij} = N(|x_i-x_j|,\sigma_i^2+\sigma_j^2) $$ ...
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109 views

Covariance matrix of parameters in logistic regression

Let $$f(y)=\prod_i \binom{n_i}{y_i}\pi(x_i)^{y_i}(1-\pi(x_i))^{n_i-y_i}$$ where $$\pi(x_i)=\frac{e^{\sum_j x_{ij}B_j}}{1+e^{\sum_j x_{ij}B_j}}$$ then the likelihood is $$L\propto ...
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69 views

When can I use $XX^\top/n$ as covariance matrix for PCA?

Given a data matrix $\mathbf X$, can I always obtain its covariance matrix (to use in PCA) by centering (subtracting the column means) and then computing $\mathbf X \mathbf X^\top/n$? Is this always ...
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32 views

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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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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41 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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30 views

Why estimated covariance matrix by glasso is always zero?

I am using glasso function from glasso package, as follow: ...
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42 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
19 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
131 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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26 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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12 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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15 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
87 views

What is an isotropic (spherical) 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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118 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, ...
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1answer
30 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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8 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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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
62 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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24 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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22 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
47 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
24 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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33 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
246 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 ...
2
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35 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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49 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
31 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 ...
2
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0answers
48 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
145 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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87 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 ...
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1answer
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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
153 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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55 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: $$ ...