Questions tagged [covariance]
Questions about covariance, a measure of (linear) association between two random variables.
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Does Covariance inequality still hold when variable is a vector?
When I were studying probability theory, I came across the following conclusion ("Covariance inequality"):
If $x$ and $y$ are two random variable (scalar), the covariance of $x$
and $y$ is: ...
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Covariance of spatial derivative of a spatially stationary delta correlated process
If K is a spatially stationary delta correlated process, what is cov(∂K/∂x(t);∂K/∂x(t-τ))? Here t is time. I understand that expected value of ∂K/∂x is zero.
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Covariance matrix of an anisotropic normalized Gaussian random vector
Let $\xi \sim \mathcal{N}(0, \Sigma)$ be a Gaussian random vector in $\mathbb{R}^n$. I would like to calculate the covariance matrix of the normalized vector $\frac{\xi}{\lVert \xi \rVert}$, i.e.:
$$ ...
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Positive covariance with all non-decreasing functions implies function is non-decreasing
Consider a random variable $X$ with continuous CDF $F$ over $[0,1]$ admitting strictly positive density: $f(x)>0$ for all $x$ in $[0,1]$.
Take a continuous function $h(x)$ over $[0,1]$. Assume that ...
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Show that $(M_1,\cdots,M_p)$ and $(N_1,\cdots,N_q)$ are independent
Let $(\Omega,\Sigma,\mathbb{P})$ be a probability space and $X_1,\cdots,X_n:\Omega\to\mathbb{R}$ be independent normally distributed random variables. Show that if $M_1,\cdots,M_p,N_1,\cdots,N_q\in\...
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Calculating Covariance of two exponentially distributed random variable with their respective probability density functions
I need to calculate the Covariance of two random variables $X$ and $Y$ with exponential distributions.
The probability density functions are $f(x)=\lambda \cdot e^{-\lambda x}$ for $x \in [0, \infty)$
...
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Confusion with Complex Gaussian process with Auto-covariance
I have a complex sequence $z(t)$ in time which I know to be a Gaussian process. I read that the complex Gaussian process is not only characterized by the covariance, but also the pseudo-covariance ...
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Covariation function of $\int_{0}^{t}e^{-W_s}ds$ [closed]
The problem is to find covariation function of $\int_{0}^{t}e^{-W_s}ds$, where $W_t$ is a Wiener process (Brownian motion). Could anybody give a hint on how to approach this problem?
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Joint Gaussians of three random variables
Suppose that
$(X,Y)$ are joint Gaussian with $Cov(X,Y)=\theta_1$,
$(Y,Z)$ are joint Gaussian with $Cov(Y,Z)=\theta_2$.
Are $(X,Z)$ also joint Gaussian? What is $Cov(X,Z)$?
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Cross-covariance matrix calculation based on known covariance matrices
I have a very common problem, but couldn't find any proper evidence why it isn't possible (with proof) or how to do it (with a method). The problem is that 2 random vectors are modeled with white ...
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Why does a covariance matrix have to be positive semi definite on an intuitive level
If I have a covariance matrix of some random vector $X$ with expectation $\mu \in \mathbb{R}^n$ it is not that difficult to show that its covariance matrix is positive semi definite; given
$$
Cov(X) = ...
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Matrix Vector multiplication of Gaussian Matrix and Gaussian Vector
I am curious about how $y$ is distributed if:
$$y=Ax,$$
with Gaussian Matrix $A \in \mathbb{R}^{m \times n}$, every entry of the matrix $e_{ij} \sim N(0, 1)$ and Gaussian Vector $x \in R^n, \mathbb E(...
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Maximum value of quadratic form over unit hypercube
Let $$f(A) = x^T A^{-1} x$$ where the elements of vector $x\in \mathbb{R^{n\times 1}}$ are $0\le x_i \le 1$ and $A\in\mathbb{R}^{n\times n}$ is a covariance matrix (i.e. $A$ is symmetric positive ...
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How to merge 2 multivariate Gaussians with different probabilities of occurring?
So, have a system that estimates a covariance matrix. The scenario occurs where at time step $t$, I will either have a Gaussian with covariance matrix $\Sigma_A$ with probability $p$ or $\Sigma_B$ ...
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Use Covariance Matrix to Convert Ellipsoid to Sphere?
If I have an unbiased ellipsoid with a known covariance, is there a way I can use that covariance matrix to transform all known points on the given ellipsoid to trace a sphere instead?
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Proving a property about the Covariance of two random variables.
I am struggling with this problem from an introduction to Machine Learning Course at a my University that I never fully wrapped my head around. Here's the problem as it was written:
For any two random ...
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Calculating expectation of dependent variables
Suppose $E[X] = 3$, $Var(X) = 1$, $E[Y] = 2$, $Var(Y) = \frac14$, $Corr(X,Y) = \frac12$
What is $E((X+2Y)^2)$?
We can rewrite $E((X+2Y)^2)$ as $E(X^2+4XY+4Y^2)$
We can solve for $E(X^2)$ using $Var(X)...
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Correlation of X and Y, when E[X] = 0
X and Y are random variables and it is known that E(X) is zero. Then how can I prove that corr(X,Y)is zero?
Here is what I tried:
The covariance between two random variables X and Y is defined as: Cov(...
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Lower bounding $\mathbb{P}(X>a, Y<b)$ given $|\text{Corr}(X, Y)|$ is small
Suppose we want to get a lower bound for the probability $$\mathbb{P}(X>a, Y<b),$$ and let us assume $\mathbb{P}(X>a) + \mathbb{P}(Y<b) <1$, so a direct union bound would not work.
If $...
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Expectation of muliplication E(xf(x)) vs E(x)E(f(x)
Assume $X$ is a random variable with realization always positive. $f(x)$ is increasing in $x$ and also positive.
Is there any proposition regarding $E(X\cdot f(X))$ and $E(X)E(f(X))$?
My naive ...
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Calculating covariance for 10 dice rolls
I have two variables, X and Y. I also have a dice. I roll the dice 10 times. X is defined as the number of times I get a result greater than 3, meaning 4,5,6 and Y is defined as the number of times I ...
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Uncertainty of the area of a Gaussian curve atop a linear background
I have some data from a counting-based spectroscopy experiment. Each data point is an (Energy, Rate) pair.
I'd like to fit the dataset to a Gaussian curve (signal) plus a slope (background):
$$
I(E) =...
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Is there anything to be said on the inverse of the covariance matrix of a reconstructed matrix using PCA or SVD decomposition?
Let $X$ be a real $n \times p$ data matrix where we can assume $n > p$. The SVD decomposition is $X = UDV'$ where $U$ is $n \times p$, $U'U = I_p$, $D$ is $p \times p$ diagonal with singular values ...
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Bounds for the covariance of a functional
Let $X$ and $Y$ be two positive continuous random variable. I have noticed that
$$cov(\sqrt{X}, \sqrt{Y}) \leq cov(X,Y) \leq cov(Y^2, X^2).$$
I am wondering if there is a general rule that say that if ...
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Rewrite Matrix normal distribution likelihood
I have already posted a similar question on cross-validated but with no answer so far https://stats.stackexchange.com/questions/618972/how-to-write-the-likelihood-for-a-multivariate-gaussian-linear-...
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Why does $cov(a, Bc) = cov(a,c)B^T$
Suppose I have a matrix, whereby $$cov(a, Bc) = cov(a,c)B^T$$
Why do we have $B^T$ from the covariance equation, and is this condition strict, when do I know that $$cov(a,Bc) = cov(a,c)B$$
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Concentration of inverse sample covariance matrix
Let $X_1, X_2, \dots, X_n \sim X$ be i.i.d. random vectors in $\mathbb{R}^d$ with $\mathbb{E}[XX^{\top}] = \Lambda$. Let $\hat{\Lambda} = \sum_{j = 1}^{n} X_j X_j^{\top}$ be the empirical covariance ...
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covariance of coefficient estimates in a linear regression model
paper
I am confused about how the above paper gets the correlation of the coefficient estimates $Cor^2(\hat{\gamma_j}, \hat{\gamma_k})$. The covariance matrix of the coefficient would be $(Z^TZ)^{-1}\...
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Covariance of functions of two random variables
I am trying to show the following: Suppose that x and y follow identical and independent distribution. F(x, y) is an increasing function of x and y. Is there any way to show that the covariance ...
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Variance of a random one-hot vector.
Question: Given a one-hot random vector (a vector with one entry taking one and the rest taking zeros) $B\in\{0,1\}^L$ that follows the Categorical distribution with probability $\pi\in\mathbb{R}^L$, ...
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How to interpret covariance(A,B) divided by percentile rank of std dev(A)?
Is there a way to make meaning of it? I was thinking that maybe because the covariance is divided by the percentile rank of std dev of A, the magnitude of A is to a degree bounded. But how does it ...
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Fitting a non-linear curve in symmetric positive definite matrix manifold
I have a variable $x \in \mathbb{R}$. For some values of $x$, $\{x_1, ..., x_n\}$, I have measured a covariance matrix of the variable $y \in \mathbb{R}^n$, conditional on these values of the variable ...
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Eigenvalues of the covariance matrix of the Wiener process
Let $t_1< \dots < t_p$ be distinct points within $(0,1)$ and consider the $p \times p$ matrix $\mathbf{C}$ with entries $$c_{ij} = \min(t_i,t_j)$$ Please note that this is a well-known matrix as ...
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Probability - Cov of two random variable $(X,Y)$
Let $(X,Y)$ be a 2d random variable.
Suppose:
$\mathbb{P}(X\in[-1,15])=1$
$\mathbb{E}[Y|X=t]=t^2$
What above could be value for the common $\mathbb{V}$ar of X,Y, which means: $\mathbb{C}_{ov}(X,Y)$
A: ...
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Covariance matrix of a random vector?
$\newcommand{\cov}{\mathrm{cov}}$ I'd like to ask a question similar to this. The question I've linked to is about $X = (X_1,X_2,\dots,X_n)^T$, a random vector in $\Bbb R^n$ and $A$, an $m \times n$ ...
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Bound on inverse covariance from covariance in regularized covariance estimation problem
In this paper by Bickel and Levina, I am confused about result (A15) which claims that since
$$
(A14) \qquad \| \text{Var}(\mathbf{X}) - \widehat{\text{Var}}(\mathbf{X})\|_{\max} = O_P(n^{-1/2} \log^{...
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Computing covariances of features/landmarks in environment in scan-matching algorithm
I am reading a paper titled "Real-Time Correlative Scan Matching" which explains a scan-matching algorithm used to obtain pose-pose constraints in pose-graph Simultaneous Localization And ...
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strong mixing and correlation decay
Given a sequence $\{X_n\}_{n=1}^\infty$, suppose there exists a function $\alpha$ such that for each $n$ and $N$,
\begin{equation}
\sup \left\{ |P(A\cap B) - P(A)P(B)| : A \in \mathcal{F}_{X_1, \dots ...
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Computing a covariance when one random variable has a concave transformation
Suppose we have two random variables $X$ and $Y$ where $\mathrm{cov}(X,Y) = a > 0$. Let $U(X)$ be an increasing and concave function. I am interested in whether we ever can say something about the ...
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Balls covariance
A bag contains 52 balls: 13 red, 13 blue, 13 green and 13 yellow. I draw 5 balls without replacement. $X$ is the number of red balls I have drawn and $Y$ is the number of blue balls I have drawn. What ...
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Order of the vectors in principal component analysis
For a given matrix $A$, the Principle Component Analysis (PCA) is done by finding the eigenvalues/eigenvectors of the covariance matrix associated with $A$. However, the entries of the covariance ...
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Drawing an distribution of measurments in an XY plan
I have a localization problem where I got 100 measurements of x and y points.
From these points, a mean and matrix of covariance were calculated, from which, I extracted the individual means and ...
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Continuity of Mean Function Slope in Gaussian Process Regression with RBF and Exponential Kernels
I'm working with Gaussian process regression (GPR) and have a question about the behavior of the mean function when using the exponentiated quadratic kernel (or RBF) kernel.
RBF Formula:
$K(x, y) = \...
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What does is mean to center variables?
I was reading this answer that is talking about properties of $AA^T$.
If you center columns (variables) of $\bf A$, then $\bf A'A$ is the
scatter (or co-scatter, if to be rigorous) matrix and $\...
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Largest eigenvalue of the covariance matrix of a smooth unimodal probability distribution on the sphere?
For any probability distribution $P$ on the sphere $\Bbb S^n\subset \Bbb R^{n+1}$ let $\lambda_{max}(P)$ denote the largest eigenvalue of its covariance matrix
$$
Cov(P)=\left(\int_{\Bbb S^n} ...
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Largest possible eigenvalue of the covariance matrix of a smooth probability distribution on the sphere is attained for the uniform distribution?
Let $\mathcal P$ the set of all probability distributions on the sphere $\Bbb S^n\subset \Bbb R^{n+1}$ that have smooth densities with respect to the Lebesgue surface measure. For all $P\in\mathcal P$ ...
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Minimizing the Mahalanobis distance
Definitions
Consider the following optimization problem
\begin{equation*}\arg \min_{x\in\mathbb{R}^n} \lVert y-x\rVert_{P}^2\end{equation*}
where $y,P$ are given parameters and
\begin{equation*}
\...
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Minimize covariance
So, given -
$\vec{x}, \vec{u}$ and $\vec{v}$ which are N-dimensional, $\left<x_i\right> = 0$ for every $i$, and $\vec{u}$ is constant, there exist $y_1, y_2$ so that $y_1 = \vec{x}\vec{u}$ and $...
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Is it possible to calculate covariance of two normal distributions?
Given the mean and standard deviation of the normally distributed changes in two measurements from before and after treatment from the same population, is it possible to estimate the covariance of the ...
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Calculate the correlation coefficient
Every day man puts 10 1 £ coins and 8 2 £ coins in his pocket. Each day, in the morning he losses 8 coins in the afternoon he losses 6 coins. Find its morning and afternoon loss correlation ...
