# Questions tagged [covariance]

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

483 questions
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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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### 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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### 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 ...
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 ... 0answers 117 views ### 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^... 0answers 119 views ### 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 ... 0answers 70 views ### 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 &... 0answers 60 views ### 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 ... 0answers 20 views ### 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 ... 0answers 55 views ### 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 ... 0answers 57 views ### 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{... 0answers 309 views ### 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 $$\Sigma_\mathbf{X}=\begin{bmatrix}1&... 0answers 25 views ### Finding subset of uncorrelated variables Assume I have n random variables with covariance matrix \Sigma. Now, I want to find m groups of variables such that they very correlated inside each group, but their correlation between groups ... 0answers 60 views ### When can I be sure that the state values estimated from the Kalman Filter have approached the actual values? Is it from the state co-variance matrix? Below are the equations for state estimation using Kalman Filter. Here are the first few equations, and the rest follows in a link below:$$ \newcommand{\blue}{\color{blue}} \newcommand{\grey}{\color{... 0answers 74 views ### 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 ... 0answers 677 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 ... 0answers 385 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 ... 0answers 44 views ### 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://... 0answers 87 views ### 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 ... 0answers 101 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$Z$... 0answers 509 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 ... 0answers 161 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$k = 10^3; 10^4; 10^5$... 0answers 66 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$...
Let $\left\{X_t,t\in T\right\}$ be a stationary process such that $\text{Var}(X_t)<\infty$ for each $t\in T$. The autocovariance function $\gamma_X(\cdot)=\gamma(\cdot)$ of $\left\{X_t\right\}$ is ...