# Questions tagged [covariance]

Questions about covariance, a measure of (linear) association between two random variables.

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### What are the restrictions on covariance matrices of nonnegative random variables?

If $M \in \mathbb R^{n \times n}$ is the covariance matrix of nonnegative random variables $X_1, \dotsc, X_n$ with $\mathbb E[X_1] = \dotsb = \mathbb E[X_n] = 1$, i.e. $M_{ij} = \mathbb E[X_i X_j]-1$, ...
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### covariance function of sub-VP SDE

For Score-Based Generative Modeling through Stochastic Differential Equations , could anyone help to derive equation (28) which is the covariance function of sub-VP SDE ? Note: I managed to ...
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### Calculating the bias of the inverse of a sample covariance matrix

It's standard in a stats class to calculate the bias of the sample covariance matrix (or lack thereof), but I'm having trouble finding any exact results on how the inverse of the sample covariance ...
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### Is the variance of the mean of a set of possibly dependent random variables less than the average of their respective variances?

Is the variance of the mean of a set of possibly dependent random variables less than or equal to the average of their respective variances? Mathematically, given random variables $X_1, X_2, ..., X_n$ ...
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### Check if 2 point clouds are the same up to coordinate flips and rotation

I want to check if 2 point clouds in N dimensions are the same up to rotations about the origin and coordinate swaps. I define a point cloud as a finite collection of points, already centered at the ...
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### $P=S^{-1}$ and $Q = S[1:k, 1:k]^{-1}$. Can we write $Q$ in terms of $P$?

Let $S$ be an $n\times n$ positive definite matrix. For $k < n$, define $$P= S^{-1}\quad\text{and}\quad Q=S[1\text{:}k,1\text{:}k]^{-1}$$ where $S[1\text{:}k,1\text{:}k]$ is the principal leading ...
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### Covariant and contravariant velocity

I'm facing the following problem in tensor calculus: I want to calculate the velocity of a mass particle in spherical coordinates. So I'm using the following coordinate functions for spherical ...
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### Proving inequality regarding expected error involving covariance and its estimate.

Let $$A = \left( I - \frac{\Sigma \iota \iota'}{\iota' \Sigma \iota} \right)(\hat \Sigma \iota).$$ Where $\Sigma$ is the true $nxn$ covariance matrix of a random vector x that is normally ...
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### Relation between graph Laplacians and covariance matrices

In the "Future challenges" section of the article Dittrich, Thomas, and Gerald Matz. "Signal processing on signed graphs: Fundamentals and potentials." IEEE Signal Processing ...
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### Optimization on Manifolds

I am quite new to the concept of optimization on manifolds, however in my research I have stumbled upon a problem which I believe is amenable to this type of analysis. Specifically, I am concerned ...
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### Covariance between estimated random effects $\hat{\boldsymbol{b}}$ and real idiosyncratic error vector $\boldsymbol{\epsilon}$ in a Linear Mixed Model

Let us assume a linear mixed model of the form $$\boldsymbol{y} = \boldsymbol{X}\boldsymbol{\beta} + \boldsymbol{Z}\boldsymbol{b} + \boldsymbol{\epsilon}$$ where $\boldsymbol{X}$ is the fixed-effect ...
I want to compute $Var[A+B+C]$ where $A$, $B$, and $C$ are not independent of each other. In particular, I don't know how to compute $Cov[A,B]$, $Cov[A,C]$, and $Cov[B,C]$. The model specifications ...
The usual definition of a covariance operator on $L_2(D)$ is: $$C : L_2(D) \to L_2(D), \qquad (C \psi)(x) = \int_D c(x,y) \psi(y) dy \qquad \forall x\in D, ~~\psi \in L_2(D),$$ where \$c(x,y): D \...