Questions tagged [covariance]

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

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Decomposition of order 3 tensor symmetric along two dimensions

I have a 3rd order tensor $\mathbf{A}$ consisting of symmetric covariance matrices (with dimensions of space by space) stacked in time. I would like to compute the leading spatial features that ...
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2D Kernel for Gausian Process Regression

I want to use Gaussion Process Regression to fit a function to data. Since I know a little about the target function, I would like to build a suitable kernel from existing kernels. The input vector ...
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Graphical model - log transformed covariance matrix

It is known that the non-zero entries of the inverse covariance matrix constitute an estimate on the edges in a graphical model. I wonder if there is an analog or similar result for the log-...
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Inverse transform of the exponential covariance function

Let $C(h)$ be the exponential covariance function, Bochner's theorem says that $C$ is positive definite if and only if its the Fourier transform with respect to a spectral measure $\chi$. Now, under ...
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Singular vectors from true and sample covariance do not align

Suppose we have a covariance matrix $C\in\mathbb{R}^{p\times p}$ which is diagonal with exponential decaying values. This covariance matrix is used to sample $n$ data points $x\sim\mathcal{N}(0,C)$ ...
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Cross covariance and trace identity

This may be a simple answer but I can't find any proof. I know that the following identities are true $$ E\left \{ \left ( \mathbf{x} - E\left ( \mathbf{x} \right ) \right )^T\mathbf{Q}\left ( \...
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Covariance matrix of Linear Transformation

I need help. I'm looking at matrix-vector multiplication. Both vector (x) and matrix (A) are random and independent. The vector has a mean and a covariance matrix. And the matrix has the mean and ...
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Asymptotic Normality of Weighted LSE (Theorem 3.17, Jun Shao)

I am trying to understand Jun Shao's proof of the asymptotic normality of weighted LSE in his book Mathematical Statistics. The theorem: Consider the model $X = Z\beta + \varepsilon$ with a full rank $...
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Are X, Y correlated and independent?

I have a problem from my textbook: Let $A$ and $B$ be independent events. Show that $X = I_A + I_B$ and $Y=|I_A − I_B|$ are uncorrelated. Are $X$ and $Y$ independent? I've managed to shown that they'...
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Covariance of two additions of random variables

I have a set of independent random variables $A_{1}, A_{2}, ... , A_{n}, n \geq 7$ with $ \forall i \in[1, n], \ \mathbb{E}(A_{i}) = 0 \ \& \text{ Var}(A_{i}) = 1$ I have two other random ...
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Maximal Correlation of two random variables

I am trying to wrap my head around one problem that involves two identically distributed random variables $X$ and $Y$ with distribution $Bernoulli(p)$ assuming that $P(X=Y=1)=\theta$. It then asks a ...
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Self Study: Proof $Cov(X, E[X|Y]) = Var(E[X|Y])$

Like the title, here is what i've done so far: \begin{align*} Cov(X, E[X|Y]) &= E(XE[X|Y]) - E[X]E[E[X|Y]] \\ &= E(XE[X|Y]) - E[E[X|Y]]E[E[X|Y]] \\ &= E(XE[X|Y]) - E[E[X|Y]]^2 \\ \end{...
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What is Cov(X-Y,Z-W)

I know the equality of the covariance $$ \operatorname{Cov}(X+Y, Z+W) = \operatorname{Cov}(X,Z) + \operatorname{Cov}(X,W) + \operatorname{Cov}(Y,Z) + \operatorname{Cov}(Y,W), $$ But I have the doubt ...
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Maximize the trace of the convariance of a bounded random vector

Given a random variable $X$, satifying $P(0\leq X \leq 1)=1$, and $\mathsf{E}[X^2] = \alpha$. We know its maximum variance $\text{Var}(X) = \alpha(1-\alpha)$ achived by a binary random variable $P(X =...
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Another puzzling identity that arose from integrating over eigenvalues of Wishart matrices.

Let $n \ge 2$ and let $T > n $ be integers. We consider a sample covariance matrix, i.e. $c := {\bar C} \cdot Y \cdot Y^T \cdot {\bar C}^T \quad (1)$ where $Y $ is a $n \times T$ random matrix with ...
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How do we compute an integral over a unit simplex?

Let $ n \ge 2 $ and $ T > n $ be integers. The joint-distribution of eigenvalues in the Wishart ensemble subject to the underlying covariance matrix being equal to an identity matrix is given as ...
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Why is the covariance of two indicator functions at most 1/4?

In an answer to this question, it was mentioned that the covariance of two indicators of measurable sets can at most be 1/4, in formulas, $$ | P(A\cap B) - P(A)P(B) | \leq \frac{1}{4}, $$ where $P$ is ...
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Proving that $|{\hat \gamma}(h)| \le {\hat \gamma}(0) $

This is my approach: $|{\hat \gamma}(h)| = \frac{1}{n}\sum_{t=1}^{n-|h|}(X_{t}-{\bar X})(X_{t+|h|}-{\bar X}) \le \frac{1}{n}\sum_{t=1}^{n}(X_{t}-{\bar X})(X_{t+|h|}-{\bar X}) = \frac{1}{n}\sum_{t=1}^{...
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Distribution of doubly stochastic Poisson or Cox process

This is an exercise in Stochastic Processes with Applications (Bhattacharya and Waymire) IV.1.6: Suppose that the parameter $\lambda$ (mean rate) of a homogeneous Poisson process $\{X_t\}$ is random ...
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Demonstration and Interpretation between a Fisher matrix and its dual space which is covariance matrix

I have a simple (maybe not) issue about the interpretation of the link between Fisher information matrix and its inverse which is the covariance matrix. How to formulate that a line of Covariance ...
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Is there a matrix algebra equivalent of finding the cross correlation between two vectors?

To calculate the cross correlation of two vectors we slide the vectors over each other multiply the corresponding elements and sum them. This is an example for two length 8 vectors giving a length 15 ...
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Extended Kalman Filter: Measurement equation and Covariance Matrix

i am trying to implement an EKF for orbit determination of a spacecraft. The state which i am interested to estimate is $x = [r_{SC}\, v_{SC}\, \Delta Cd\, \Delta Cs\, b, d]$, where $\Delta Cd\,,\...
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How to prove $Cov(E(X_1|X_2), X_1 - E(X_1|X_2)) = 0$

Question Determine whether or not the following result is True or False: $$Cov(E(X_1|X_2), X_1 - E(X_1|X_2)) = 0$$ Attempt I tried at first: $=E[((X_1|X_2)(X_1-E(X_1|X_2)) - E(X_1|X_2)E(X_1-E(X_1|X_2))...
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Help with covariance of two different joint distributions

The independent distributions $A_{1}...A_{n}$, and $B_{1}...B_{n}$ have expectations $0$ and variances $1$. $C_{n} = A_{n} + \frac{-1}{12}A_{n-1}$ $D_{n} = B_{n} + \frac{3}{7}B_{n-1}$ What is the ...
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Flattening and projection of a covariance matrix onto a vector in the x-y plane

I am working with a covariance matrix $P \in R^3$ that I need to simultaneously project onto a vector $\mathbf{v}_f \in R^3$ and flatten into the plane $z=0$. A paper in my posession states that the ...
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What does absolute value sign mean when it's around the covariance matrix?

I have the following covariance formula and the probability formula: I am trying to write these formulas in python, however, I couldn't understand that |$\sum|^\frac{1}{2}$ What does that operation ...
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Showing that $cov(X,Y)\leq 0$, where $X=$num. of successes in three trials and $Y=$ num. of trials necessary to obtain the first success

"Consider a sequence of independent trials each with the same probability of success $0\leq p\leq 1.$ Let $X$ be the number of successes in three trials and $Y$ be the number of trials necessary ...
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Covariance of a standard normal variable

Let 𝑋 be a standard normal random variable. Another random variable is determined as follows. We flip a fair coin (independent from 𝑋). In case of Heads, we let 𝑌=𝑋. In case of Tails, we let 𝑌=−𝑋...
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Limit of a covariance sequence

I've been struggling on the following exercice lately : Consider $(X_n)_{n\ge 1}$ a sequence of mutually independent real-valued random variables, standardized, that follow any given law. Let $Y$ be ...
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Covariance of min and max of n i.i.d uniform distributed random variables.

There are n independent, uniform random variables on $[a, b]$: $ξ_1...ξ_n$. How to find $ cov(min(ξ_1...ξ_n), max((ξ_1...ξ_n))$. I know that $E(min) = \frac{b+na}{n+1} $ and $E(max) =\frac{a+bn}{n+1}$
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Effect of increase in variance on covariance

I have that the covariance between two random variables is negative: $cov(x,y)<0$. Suppose we replace $x$ by an rv, $x'$, that has higher variance but the same expectation (and, also, a negative ...
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Covariance function of the square of a Gaussian process

Let $X_t$, $t\in\mathbb{R}$, be a Gaussian process with mean $0$. Prove that $$ Cov(X_s^2,X_t^2)=2Cov(X_s,X_t)^2 $$ I don't know how to handle the squares here.
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Can a sample continuous Gaussian process $(X_t)$ have $Cov(X_t,X_s)=0$ while $Var(X_u)\neq 0$ for all $u \in [s,t]$?

Let $(X_t)$ be a sample continuous Gaussian process on $[0,1]$. We suppose that $(X_t)$ is non-trivial, i.e it is not constant on $[0,1]$. If $Cov(X_s,X_t)=0$ for some $s,t$ in $[0,1]$, does it imply ...
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Nongaussian Wishart?

The Wishart distribution of order N is related to the distribution of the N sample covariance of iid Gaussian random vectors. Do you know the name of the distribution of the N sample covariance for ...
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central limit theorem for sample covariance matrix?

Is there a version of the central limit theorem for the sample covariance matrix? Suppose $\{X_1,\cdots,X_n,\cdots\}$ be a sequence of i.i.d length-$p$ random vectors. Suppose their mean is zero ...
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Covariance of normalized variables

I have 7 variables $A_i$, $i\in\{-3,-2,-1,0,1,2,3\}$. ($A_{-1}$ and $A_1$) are identically distributed. ($A_{-2}$ and $A_2$) are identically distributed. ($A_{-3}$ and $A_{3}$) are identically ...
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What functions preserve symmetry and positive-definiteness of covariance matrices?

Suppose I have covariance matrices $H_1,…,H_n$ (symmetrical, positive-definitive) and corresponding weights $w_1,..,w_n$. We want to find a function $f$, such that $H = f(H_1,…,H_n,w_1,…,w_n)$ is ...
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Calculating Mahalanobis distance

I am slightly confused as to how you calculate Mahalanobis distance given a set of data. I have tried asking my tutor for help but he does not seem interested in helping what so ever and I am ...
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Covariance Matrix outer product of vector itself

Let's define vectors as $x_1,x_2,...,x_m$ with zero mean where $R^d$ The computing covariance matrix is $\sum=\frac{1}{m}\sum_{i=1}^mx_ix_i^T \in R^{dxd}$. We know that the covariance matrix ...
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correlation and covariance of two random variables

I have two continuous random variable X and Y. In U- shape region we have $P_{XY}(X,Y)=\frac{1}{12}$ in other regions we have $P_{XY}(X,Y)=0$. How the correlation and covariance of these two variable ...
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Calculate covariance from given distribution

I have two variables $x, y$, where $x$ follows the distribution $$x = \exp\left(-\frac{a^2}{2\sigma^2}\right).$$ $y$ can be calculated from $x$ via a function: $$y = \arctan(b\tan(x)).$$ How can I ...
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Calculating covariance of two variables given equation

Given $ S_t = S_0 + \mu + \sigma W_t $, I am interested in finding the covariance $ Cov(S_t - S_0, S_T - S_t) $ The formula I am using is $ Cov(x, y) = E(xy) - E(x)E(y)$, having already calculated $ ...
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Row-Wise Group Sparsity in (Covariance) Matrix

Given $N$ groups $G=1,..,N$ with different number of samples, we assume that each sample in a group is generated from a group-specific Multivariate Normal distributions (e.g., here 3 groups): $$ s_{1,...
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Is it possible that $A+A_{\text{off-diag}}\succ0$ , where $A\succneqq0$?

Denote $A_{\text{off-diag}}:=A-\text{diag}(A)$, i.e. setting diagonal elements to be zero. Denote $A\succneqq0$ iff $A$ is positive semidefinite (and not positive definite), while $A\succ0$ iff $A$ is ...
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Kendall's tau coefficent of Bivariate Normal [duplicate]

Let the joint distribution of (𝑿, 𝒀) be bivariate normal with mean vector $\begin{pmatrix} 0 \\ 0\end{pmatrix}$ and variance-covariance matrix $\begin{pmatrix} 1 & 𝝆 \\ 𝝆 & 1 \end{pmatrix}$...
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Finite nonzero covariance but infinite variance

Cauchy-Schwarz tells us that it is not possible for random variables $X,Y$ to both have finite variance while having infinite covariance. Now it is easy to think of examples where $X,Y$ have zero ...
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Transformation step in lag-$l$ autocovariance for linear time series that I can not understand.

We have a linear time series defined as follows: $r_t=\mu+\sum_{i=0}^{\infty}\psi_ia_{t-i}$, where {$a_t$} is a sequence of iid random variables with mean zero and a well-defined dsitribution. The lag-...
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Comparing distributions of binary valued vectors: covariance matrix is enough?

Say we have two discrete distributions, $\vec{y}\sim p_y$ and $\vec{x}\sim p_x$, for both of which, data vectors have binary-valued entries: $\vec{y}\in\{0,1\}^n$ $\vec{x}\in\{0,1\}^n$, where $n$ is ...
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Is true that $\operatorname{Var}(X|Y)=\operatorname{Var}(X-E[X|Y])$? [closed]

Is true that $\operatorname{Var}(X|Y)=\operatorname{Var}\left(X-E[X|Y]\right)$? Also, is true that $\operatorname{Cov}\left(E[X|Z],E[Y|X]\right)=0?$
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Calculating $Cov\left(\overline{Y}_j,\:\overline{Y}\right)$ for a basic one-way model

Consider the basic one-way model: I want to show that $Cov\left(\overline{Y}_j,\:\overline{Y}\right)=\frac{\sigma ^2}{na}$. I derived the following expected values: $$E\left(\overline{Y}_j\right)=\mu ...
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