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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 ...
super_mario's user avatar
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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 ...
mather's user avatar
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Compacting Notation of Propagation of Uncertainty Formula into Vectors with Covariance?

This question has a nice solution to compacting the propagation of uncertainty formula when all terms are uncorrelated. To re-iterate, they found that the combined uncertainty, which I will denote $...
javery's user avatar
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Lower bound for variance of sum of with functions containing gaussian random vectors.

in my current research i am stuck with the following problem and search for nice lower bounds: Assume that the n-dimensional random $X_j$ is normal distributed, i.e. $X_j \sim N(0,\Sigma_j)$ where $\...
statuser123's user avatar
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4 answers
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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 $X_1, X_2, ..., X_n$ ...
HappyFace's user avatar
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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 ...
batman247's user avatar
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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 ...
knrumsey's user avatar
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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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3 answers
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Covariant (absolute) derivative of a vector along a curve -- compare cartesian vs. polar coordinates

BACKGROUND: Suppose $A^μ$ is a vector field and $x^μ(λ)$ is a curve in spacetime. A first guess at measuring the change in $A^μ$ along the curve might be $$\frac{dA^μ(x(λ))}{dλ} = \frac{∂A^μ}{∂x^ν} \...
Khun Chang's user avatar
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Variogram matrix and covariance matrix

Let $\textbf{W}$ be a mean-zero Gaussian random vector in $\mathbb{R}^d$ with the covariance matrix $A$. Let $\vec{1} \in \mathbb{R}^d$ be a vector of $1$s. We define its variogram matrix by $\Gamma_{...
Phil's user avatar
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Is Cov (X,XY) positive if X,Y >0

if X and Y are random variables which both only take values greater than 0 is their covariance $Cov(X,XY)>0$. I was able to use the law of total covariance to get to: $Cov(X,XY)=E[Cov(X,XY|Y)]+Cov(...
Raghav Malhotra's user avatar
2 votes
1 answer
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Probability of coin flip given forecasts

Suppose you have a coin that flips $H$ or $T$ with some unknown probability. You also have access to two devices, $A$ and $B$, where $A$ correctly predicts the outcome of the coin with $p = 0.7$ and $...
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Covariance for two correlated Ornstein Uhlenbeck processes [closed]

Given two Ornstein Uhlenbeck stochastic differential equations: $$ dX^1_t = \theta_1(X_t^1 - \mu_1)dt + \sigma_1 dW^1_t $$ $$ dX^2_t = \theta_2(X_t^2 - \mu_2)dt + \sigma_2 dW^2_t $$ Where $W^1_t$ and $...
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Modified Markov chain

Let $x_t, y_t$ be to independent Markov chains with non-zero transition probabilities and two states {0, 1}. $t$ is a positive integer. $x_0 = 0, y_0 = 1$ . Process $a_t$ is defined as $a_t = c^{x_t} ...
ArtBac's user avatar
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Inner products induced by inner product and multiplication of vectors, e.g. $f(A,B) = tr(A^TB)-tr(A)tr(B)$ as an inner product?

It just came to my attention that both the expected value $\mathbb{E}(XY)$ and the covariance $\text{Cov}(X,Y)$ can be understood as scalar products. This is a consequence of the linearity of the ...
theta_phi's user avatar
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Covariance estimation via bootstrap

Given two samples : an i.i.d. $n_u$-sample $(u_{j})_{1 \leq j \leq n_u}$ and an i.i.d. $n_v$-sample $(v_{i})_{1 \leq i \leq n_v}$. Note : The populations of the two samples are disjoint (let's say we ...
Steve R. NUNES's user avatar
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How does correlation impact parameter estimation?

I'm trying to get an intuitive understanding of how the kalman filter is able to update variables indirectly during the measurement process. Suppose we have two random variables, $X$ and $Y$ with ...
maxical's user avatar
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2 votes
2 answers
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A "proof" of negative variance

In trying to solve the problem below, I got a negative variance. What is the error in this "false proof" of negative variance? And why does it produce the correct variance times $-1$? Let $...
SRobertJames's user avatar
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Finding the joint distribution of two dependent variables $X_1$ and $X_2=(X_1)^2$

Let $X_1$ be a uniform RV along the interval $[-a;a]$ and define $X_2=X_1^2$. We can obviously see that $X_1$ and $X_2$ are dependent, upon calculating their covariance we find that it is equal to $0$,...
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If X and Y are uncorrelated, i.e., $Cov(X,Y) = 0$, then $E[Y | X] = c$ for some $c \in \mathbb{R}$.

It is given that X and Y are random variables on a common probability space $(\Omega, \mathcal{A}, P)$. Prove or disprove the following statement: If X and Y are uncorrelated, i.e., $Cov(X,Y) = 0$, ...
clementine1001's user avatar
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Why the first matrix is not always psd but the second matrix is always psd?

I learned that one important motivation for the Newey-West covariance estimator is that naive estimator of the covariance matrix is not necessarily positive semidefinite(psd from now on) (see the ...
ExcitedSnail's user avatar
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Definition of Covariance operator.

Let $x\in H$ for some Hilbertspace $H$. Recall the Covariance operator $\mathbb{E}[x\otimes x]$, where $x \otimes x := \langle x, .\rangle x$. How is actually the expectation defined in this case. For ...
emma bernd's user avatar
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1 answer
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How to calculate the variance of this random variable?

$\theta$ is a random variable whose distribution is a normal distribution with mean $m_0$ and variance $\sigma_0^2$. Let $u$ be a $k\times1$ random vector following a normal distribution where the ...
Ypbor's user avatar
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Expectation of $u^\top(u + Ax)$, when $A$ and $u$ are nonlinear functions of $x$

Let $x\in\mathbb R^d$, and $s=\operatorname{softmax}(x)$. Let $y$ be a fixed one-hot vector such that $$u = s-y \\ v =(\operatorname{diag}(s) - ss^\top)x$$ I am interested in the inequality $u^\top (u ...
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Conditions such that $\text{Cov}(X, ZY)= 0$ for $X\sim Y \sim U(-1,1)$ and $P(Z=1) = P(Z=-1) =1/2$

Say I have three random variables $X\sim Y \sim U(-1,1)$, and a third one $P(Z=1) = P(Z=-1) =1/2$. What condition on these three variables must I have so that I can say $\text{Cov}(X, ZY)= 0$? I know $...
matte_studenten's user avatar
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state space descretization of covariance matrix propagation

The descretization of a state space model is well defined. I understand that this descretization results in the same state vector values at the sample instants -- so long as the input is constant ...
jekain314's user avatar
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Change in classwise distribution of hidden layer outputs given categorical crossentropy loss in a single layer linear neural network

Given a linear neural net with a single hidden layer, and a set of input samples $\mathbf X\in\mathbb R^{m\times n}$. Consider an input $\mathbf x\in \mathbf X$, such that output $$\mathbf z=\mathbf{...
Phoenix's user avatar
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0 answers
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Linear Combinations of Sample Mean Difference

What is the standard error of the mean difference between 2 variables (a & b)? I have the following data: $$ \sigma_{A} = 1.813529 \nonumber \\ \sigma_{B} = 1.932183 \nonumber \\ cov(A,B) = 2....
joelleoqiyi's user avatar
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writing empirical covariance operator as the multiplication of sampling operator (elaboration on a paper)

I have been reading this paper (page 85) and have difficulty to understand that how the empirical version of the covariance operator is $\hat{C}_{XX} = \frac{1}{n} S_x^\ast S_x$ can be written as ...
domath's user avatar
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0 votes
1 answer
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Does covariances uniquely determine the joint distribution?

We know that if we have a random vector $x \in \mathbb{R}^k$ following a multivariate normal distribution $\mathcal{N}(0, \Sigma)$, then we have (1) each $x_i$ follows a normal distribution $\mathcal{...
Vezen BU's user avatar
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1 answer
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Probability: Question about expected value with coins

In a coin purse, there are two one-dollar coins and one five-dollar coin. One randomly draws without replacement two coins and sets X = the value of the first coin, Y = the value of the second coin. ...
Need_MathHelp's user avatar
1 vote
1 answer
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$Cov(\max(x, 0), \min(x, 0))$ where $x \sim N(0, 1)$

I've been stuck on this problem. Let $y \sim \max(x, 0) \quad z \sim \min(x, 0)$ $$Cov(y, z) = E[z \cdot y] - E[y]\cdot E[z]$$ It's clear that $E[z \cdot y] = 0$, but what is $E[y]\cdot E[z]$? Is ...
user2330624's user avatar
0 votes
1 answer
55 views

Gradient with respect to $LDL^\prime$ parameterization of covariance matrix

I have been working with the matrix-variate normal distribution (a.k.a., matrix normal distribution) $\mathbf{X} \sim \text{Normal}_{nm}\big(\mathbf{M},\;\mathbf{I}_n,\mathbf{V}\big)$, such that (...
ChewysCaretaker's user avatar
-2 votes
1 answer
37 views

How to compute Covariance?

Flip a fair coin $10$ times. Let $X$ be the number of heads, $Y$ the number of tails. Calculate Cov($X,Y$) I worked out that $E(X) = E(Y) = 5$ Also, Cov($X,Y$) $= E(XY) - E(X)E(Y) = E(XY) - 25$ How do ...
adisnjo's user avatar
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Proving inequality involving mean, covariance and their estimate.

Let $$ A = \left( I - \frac{\Sigma \iota \iota'}{\iota' \Sigma \iota} \right) (\mu - \hat \mu) + \gamma \left( I - \frac{\Sigma \iota \iota'}{\iota' \Sigma \iota} \right)(\hat \Sigma \iota), \...
alejandroll10's user avatar
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1 answer
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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 ...
alejandroll10's user avatar
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0 answers
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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 ...
Fernando's user avatar
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0 answers
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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 ...
Josh Pilipovsky's user avatar
6 votes
1 answer
73 views

deriving covariance of SDE from fokker-planck

In the book 1 the covariance of an SDE is derived. I am not sure about a particular step. Let me describe it in a TLDR version, then in a longer version. We have an SDE $$dx = f(x,t) dt + L(x,t) d\...
black's user avatar
  • 279
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1 answer
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Covariance between functions of same $Z_k$ = 0

Let's assume we are faced with showing that the following covariance between two functions $g(\cdot)$ and $f(\cdot)$ of the same set of $Z_1,...,Z_K$ i.i.d. standard normal random variables is equal ...
BMBE's user avatar
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1 vote
0 answers
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Expected value of the product of two dependent discrete random variables

I do realize that there were several questions regarding the expected value of the product of two discrete variables. However, I have not found any natural derivation of the covariance terms that ...
Tomasz P's user avatar
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0 answers
43 views

Is this particular function a positive semidefinite kernel function?

Curious if there is a proof or counter for whether the following function is a positive semidefinite kernel function. $K(x,y) = \max\left(b_0 - b_1 \frac{|x-y|}{x+y}, 0\right)$ with $x > 0, y > ...
pandasdataframe's user avatar
1 vote
1 answer
37 views

(Why) is the norm of a RKHS positive definite?

$ \newcommand{\real}{\mathbb{R}} $ A $\color{red}{\text{(strictly)}}$ positive definite kernel $k: \real^d\times \real^d \to \real$ satisfies for all $x_i \in \real^d$, $a=(a_1,\dots, a_n)\in \real^d$ ...
Felix B.'s user avatar
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0 votes
1 answer
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Sum of autocorrelation coefficients

This is a follow-up to this thread: Proof that sum over autocorrelations is -1/2 I am posting a new thread as that was posted 6 years ago. In that threat the stackexhange author (Kuhlambo) lists some ...
Bazool's user avatar
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0 answers
29 views

How to show that the Gaussian Process parameters decreases with more training data

The posterior mean prediction of a Gaussian Process is given by $$\mu(x_*) = \sum_{i=1}^n\alpha_i k(\mathbf{x}_i,\mathbf{x}_*) $$ where $$\alpha = (K + \sigma_n^2I)^{-1} \mathbf{y}$$ Can we show that $...
Tee's user avatar
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3 votes
1 answer
82 views

Sampling from Gaussian with very large covariance matrix in block form

I'm interested in sampling from a Gaussian with zero-mean and covariance given by: $$ \Sigma = \begin{bmatrix} \Sigma_{11} & \Sigma_{12} & \cdots &\Sigma_{1,100}\\ \Sigma_{21} & \...
WeakLearner's user avatar
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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 ...
Benykō-Zamurai's user avatar
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0 answers
31 views

Variance of sum of products of normally distributed random variables?

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 ...
anonymous 's user avatar
2 votes
1 answer
119 views

Covariance Operator corresponding to multivariate covariance function

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 \...
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