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Questions tagged [covariance]

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

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Is a symmetric positive definite matrix always diagonally dominant?

A Hermitian diagonally dominant matrix $A$ with real non-negative diagonal entries is positive semidefinite. Is it possible to have a Hermitian matrix be positive semidefinite/definite and not be ...
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computing the inversion of a matrix which is the sum of a Kronecker product and an identity matrix

I'd like to evaluate a single entry $s_{ik}$ of the $\mathbf{S}$ matrix, using Markov chain Monte Carlo approach. The posterior of $\mathbf{S}$ has a Gaussian likelihood with a covariance matrix $$\...
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Applying PCA on covariance matrix in order to generate a new random variable.

Let $\mathbf{x}$ be a random $n\times1$ real vector, $\mathbf{x}\in\Bbb{R}^n$, which is distributed normally with mean $\bar{\mathbf{x}}$ and covariance matrix $\Sigma_x\in\Bbb{R}^{n\times n}$, i.e. $\...
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The variance of a covariance

The expectation over $x$ of a covariance between variables $A$ and $B$ (where the distribution of $B$ varies according to $x$) is equal to the covariance of $A$ with the expectation of $B$ over $x$: $...
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Deriving $Cov(X,Y,Z)$, is it even a thing?

So I am trying to derive a nice general formula for $Cov(X,Y,Z)$ and $Corr(X,Y,Z)$, I defined it as such $$ Cov(X,Y,Z) = E[(X-E[X])(Y-E[Y])(Z-E[Z])] $$ $$ Corr(X,Y,Z) = \frac {Cov(X,Y,Z)} {\sqrt{Var[...
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What is ${\rm cov}(e_i, \hat y_i)$ in simple linear regression?

The model is $y_i = \beta_0 + \beta_1x_i + \epsilon_i$ What is ${\rm cov}(e_i, \hat y_i)$? What is ${\rm cov}(\epsilon_i, \hat \beta_1)$? What is ${\rm cov}(e_i, \epsilon_i)$? For 1, I am writing $...
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Fourier transform and covariant/contravariant vectors

A Fourier transform of a function of $x$ is related by the following equation: $$φ'(p) = \int_{}^{}φ(x) e^{ix·p} \mathrm{d}x.$$ Let's say that $x$ is a contravariant vector. Does it follow that $p$ ...
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Is the off-diagonal part of a covariance matrix, $M = \Sigma -\operatorname{ diag}(\Sigma)$ studied?

If $\Sigma$ is a real, symmetric, positive semidefinite matrix (a covariance matrix), then we can construct $M = \Sigma - \operatorname{diag}(\Sigma)$, where we essentially take the covariance matrix ...
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Explicitly proving that the Hamiltonian is Lorentz covariant

I want to show explicitly that the Hamiltonian $$ H = -\Omega V + \int d\tilde{\textbf{k}}\ \omega (a^\dagger(\textbf{k}) a(\textbf{k}) + b(\textbf{k}) b^\dagger(\textbf{k}) ) $$ is Lorentz ...
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Are autocovariance operators trace class?

Suppose that $\{X_k\}_{k\in\mathbb Z}$ is a weakly stationary sequence of random elements with values in a complex separable Hilbert space $\mathbb H$ and let us define the sequence of autocovariance ...
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AR(p) Covariance Matrix

So I have seen specific derivations of covariance matrices for AR(2) and AR(1) processes. However, I have not seen one for a general AR(p) process. Suppose I know the coefficients of a given AR(p) ...
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Why is the range of covariance infinity?

I don't understand why the range of covariance is $+\infty,-\infty$ $$\operatorname{Cov} (X,Y)= \Bbb E\left[(X-\mu_x)(Y-\mu_y)\right] .$$ Can explain why this is true? All I can think of is if one ...
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covariance and expectional in proccess

Show that the process $X=(W_{\sqrt{t}}I_{(1,2)}(t))_{t \ge 0} \in \mathcal{L}_3^2$. ($W$- Wiener) Additionally calculate, for $t,s \in [1,2]$, $EX_t$ and $Cov(X_t,X_s)$ I have no idea how to start ...
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Compute the Var(x1+x2+x3) and other Variances.

The problem said; Let X1, x2, X3 be independent and identically distributed random variables each with mean 0 and variance 1. Below I state the work I did so far, I need help specificaly in point b....
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Linear space transform transformation based on covariance?

I have a linear space of n dimensions with non-overlapping groups characterized by different variation (different covariance matrices). Is there a way to deform non-linearly the space according to an ...
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Tensor notation generally

I'm pretty new to tensors in differential geometry and I have a basic question about the notation used. In general a vector field $X$ can be expressed as $$X=\sum_{i=1}^n X^i \partial_i,$$ where $X^...
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Finite Moments of complicated Stochastic Differential Equation

Suppose I have a SDE of the form: $$dx_i = x_i\left(b_i-\sum_{j=1}^n a_{ij}x_j\right) \,dt + \sigma_i x_i \, d\eta(t)$$ where $\eta$ solves the Ornstein-Uhlenbeck process: $$d\eta(t) = \lambda \eta(...
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sampling from a multivariate guassian: intuition behind using cholesky decomposition

I'm trying to understand how sampling from a multivariate gaussian works and why the cholesky decomposition is a way to do it. Let's say we have a 25 dimensional multivariate with a 25x25 covariance ...
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How does 2D kriging interpolation work?

I have a grid of points Example ...
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767 views

If Gaussian random vector has singular covariance matrix, isn't there probability density function?

I got a complicated problem. Suppose that Gaussian random vector having Covariance matrix; $$K_X=\left[\begin{matrix}1 &-\frac12 &-\frac12 \\ -\frac12 &1 &-\frac12 \\ -\frac12 &-\...
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Is the positive part of a covariance stationary process also stationary?

I am wondering if it is possible to derive a result on the stationarity of the positive or negative part of a covariance stationary process. Namely, consider $\{ X_t \}, t=1,2,3,...,$ a covariance ...
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Covariance matrix and Gaussian i.i.d. random variables

I have a set $X = \left \{ X_i | i \in (1,n) \wedge X_i \text{ is a random variable} \right \} $ Does $\forall i \in (1,n ), X_i \text{ follows a normal distribution} $ implies that $\...
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Covariance of errors $\text{cov}(\hat{e_{ij}},\hat{e_{i\ell}})$ in Two-Way Anova model

Exercise : Consider the Two-Way Anova model $Y_{ij} = \mu + a_i + b_j + e_{ij}$ with $i = 1, \dots, p$ and $j=1,\dots,q$. Show that : $$\text{cov}(\hat{e_{ij}},\hat{e_{i\ell}}) = -\sigma^2\left(\...
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A question about a generalization of covariance

Suppose, $H$ is a Hilbert space over $\mathbb{R}$. Suppose, $X$ and $Y$ are random vectors in $H$. Let’s define Hilbert expectation of a random vector $X$ in a Hilbert space $H$ as a vector $v \in H$, ...
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Expectation of the product of three normal variables

Let $(X_1, X_2, X_3)\sim N(\mu,\Sigma)$ be a three-dimensional random variable where each coordinates are dependent (i.e. $\Sigma$ has non-zero values outside of its diagonal) I want to know how to ...
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The covariance between sum of random variables and maximum of random variables

Let $X_1,\ldots,X_n,\; n\ge1$ — independent random variables $U(0,1),$ $S_n=\sum_{i=1}^n X_i,$ $Z_n=\max(X_1,\ldots,X_n).$ Calculate $\mathrm{cov}(S_n,Z_n).$ Solution: $\mathrm{cov}(S_n,Z_n)=\...
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Mercer decomposition of transformed covariance kernel

I'm currently trying to determine the spectral representation of covariance kernels with the structure: $$K(s,t) = C \cdot [F(\min\{s,t\}) - F(s)F(t)], \quad s,t \in \mathbb{R}$$ of centered and ...
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Building covariance matrix from paths in the inverse covariance

The inverse of the covariance matrix for a multivariate normal random vector can be thought of as holding some measure of conditional dependence between two of the variables in the vector. Consider ...
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Find the variance of Y

In an old probability test: $$ P(X_i=1)=n^{-\frac{1}{2}} $$ $$ P(X_i=0)=1-n^{-\frac{1}{2}} $$ $$ S_{i,j,k} = 1 \text{ if }X_i=X_j=X_k=1 \text{ (and 0 otherwise)}$$ $$ Y = \text{Number of }S_{i,j,k}\...
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Change order of eigenvalues and correspoding eigenvector

I'm struggling with some bugs in my program and try to find the mistakes. The situation is as follows: I have a covariance Matrix on which I perform a principal component decomposition into a ...
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Do we really need finite variances $\mathrm{Var}(X)$ and $\mathrm{Var}(Y)$ in the definition of covariance $\mathrm{Cov}(X,Y)$?

You know that covariance of jointly distributed random variables $X$ and $Y$ is $$\mathrm{Cov}(X,Y) = \mathrm{E}[XY] - \mathrm{E}[X] \mathrm{E}[Y].$$ It is clear that we should require finiteness of $...
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Recovering the covariance matrix of pixel coordinate from normalised camera coordinate

lets assume we have the pixel value of some interest points as well as their covariance matrices that defined as follows: $$ \mathbf x^{\prime} = \begin{bmatrix} x^{\prime}\\ y^...
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Covariance of independent random variables.

I've been fighting with following problem: Problem: Let $N, X_1, X_2, \ldots $ be independent random variables with given $\Lambda = \lambda$. Variables $X_i, i=1,2,\ldots$ have exponential ...
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Is this matrix singular and if yes, why?

Consider the following matrix \begin{align} \begin{pmatrix} p_1(1-p_1) & -p_1p_2 & \cdots & -p_1p_k \\ \vdots & \vdots & \ddots & \vdots \\ -p_kp_1 & -p_kp_2 & \cdots &...
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Is $X_t Y_t$ stationary?

If $X_t$ and $Y_t$ are independent and both are second order stationary processes, is $X_tY_t$ also stationary? I need to show that i) $E(X_tY_t)$ is time independent ii) $Var(X_tY_t)<\infty$ ...
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Joint optimization of precision matrices for common sparsity pattern

This question is motivated from paper by Cai, 2016 on joint estimation of multiple (K) precision matrices from K datasets. Let $X^{(k)} \sim N(\mu^{(k)}, \Sigma^{(k)})$ be a p-dimensional random ...
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Make a covariance matrix have a larger eigenvalue for one eigenvector

Suppose I have a covariance matrix $\Sigma$, with eigenvalues $u_1,u_2,u_3$ and eigenvectors $v_1,v_2,v_3$. How can I use that information to generate a new covariance matrix $\Sigma_2$ with the same ...
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Norm of covariance between scalar and vector

I would like to show that if $Y\in\Re$ is a random real vector such that $\|Y\|≤ \bar Y$ and $X \in \mathcal H$ where $\mathcal H$ is a Hilbert space (whose field is $\Re$), then $$ \|\operatorname{...
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Conditional PDF on Gaussian random vectors

Suppose the Gaussian random vector $\mathbf{X}\sim\mathcal{N}(\mathbf{\mu_X},\Sigma_\mathbf{X})$ where $$\mathbf{\mu_X}=\begin{bmatrix}1\\5\\2\end{bmatrix}$$ and $$\Sigma_\mathbf{X}=\begin{bmatrix}1&...
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Finding subset of uncorrelated variables

Assume I have $n$ random variables with covariance matrix $\Sigma$. Now, I want to find $m$ groups of variables such that they very correlated inside each group, but their correlation between groups ...
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When can I be sure that the state values estimated from the Kalman Filter have approached the actual values? Is it from the state co-variance matrix?

Below are the equations for state estimation using Kalman Filter. Here are the first few equations, and the rest follows in a link below: $$ \newcommand{\blue}{\color{blue}} \newcommand{\grey}{\color{...
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Simplify variance expression, taking into account covariances:$\operatorname{var}(Y_i-\bar Y-\hat {\beta}_1(x_i-\bar x))$

Find $\operatorname{var}(Y_i-\bar Y-\hat {\beta}_1(x_i-\bar x))$ where $\hat {\beta}_1=S_{xy}/S_{xx}$ is the least square estimator and $Y_i$ a random variable. I know that I can't simply split the ...
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643 views

Variance of a sum of identically distributed random variables that are not independent

I am "new" to probability/statistics and I was hoping someone could verify that this is correct. Let $Y_1,\ldots,Y_n$ be random variables that follow a common distribution with mean $\mu$ and variance ...
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368 views

Stationary Gaussian process whose correlation parameter approaches zero.

Consider a mean-zero ($\mu = 0$), unit-variance ($\sigma^2$) Gaussian random process $X(t)$. This process is strictly stationary (the joint-probability distribution does not vary with $t$). The ...
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What does it mean if $cov(f(x1), f(x2))$ is positive in the context of LHS sampling?

If cov(f(x1),f(x2)) is positive, does that mean f is close to symmetric along x1 and x2? I am struggling to put this into understandable terms. Edit: The context is equation 6 in this paper: http://...
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Means and Covariances of powers of a normal distribution

Let $X$ be a normally distributed random variable, with mean $\mu$ and variance $\sigma^2$. Consider a random vector $$V = \left[ X^n, X^{n-1}, \dots, X^2, X, 1 \right]^T $$ What is the expected ...
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89 views

Basic MVUE Application

I am having some trouble with the following problem: Let $X = (X_{1}, . . . , X_{n})$ a random sample from $f_{\theta}$, where $\theta \in \Theta$. Suppose that $W$ is the MVUE for $\theta$. Let $Z$ ...
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468 views

Expected value of the sample covariance

Let $X = (X_1,\dots,X_p)$ is a random (column) vector with values in $\mathbb R^p$. The covariance matrix $\mathrm{Cov}(X,X)$ is defined by $$\mathrm{Cov}(X) := E[(X-E[X])(X-E[X])^T]$$ By definition ...
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157 views

If a Stochastic Process has Variance linear with t, how to prove it is not Wide Sense Stationary?

For my study, as a part of a Matlab exercise, the following question is asked: Using the results of the estimated standard deviations of the random variable $x(k)$ for $k = 10^3; 10^4; 10^5$ ...
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63 views

Variance and covariance

I'm practicing for an exam and a mock question has me completely stumped. If someone could show me the steps I would be very grateful! There are two random variables, $A$ and $B$. $Var(A) = 9$, and $...