Questions tagged [covariance]

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

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19 views

What does the symbol X mean in the definition of the covariance matrix specifically?

I just started to learn the covariance matrix in some machine learning online course. The following is the covariance matrix definition from the https://en.wikipedia.org/wiki/Covariance_matrix I can ...
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What happens if order of variables in covariance operation is changed?

Let $X,Y$ be random variables and let $C$ denote the covariance function. $$C(X,Y) := E[(X-m_x)(Y-m_y)] = E[XY]-m_xm_y.$$ $$C(Y,X) := E[(Y-m_y)(X-m_x)] = E[YX]-m_ym_x.$$ Setting these equal to ...
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18 views

How can we interpret a covariance of a data wth zero mean?

Covariance Matrix of Zero Mean Data. The data set has zero mean does it somehow relects in covariance matrix ?
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Is it right to say that vectors $A_1 X$ and $A_2 X$ are independent iff $A_1A_2^\top=0$?

I am new to multivariate normals, so this may seem trivial. Let $X$ be a vector of independent, identically distributed normal random variables. I think that $A_1 X$ and $A_2 X$ are independent iff $...
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Covariance matrix of post-fit residual in Kalman filter

I am interested in deriving the covariance matrix of the post-fit residual in the context of Kalman filter. The usual notations: measurement matrix ($H$), measurement noise covariance ($R$), Kalman ...
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1answer
856 views

Variance of $3$-dimensional vectors

I am currently optimizing some code and thus, I want to replace an inefficient OpenCV function, which calculates a covariance matrix. The thing is, that I only need the trace of this covariance matrix,...
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35 views

A real matrix with missing values multiply with its transpose, would it be a semi-positive matrix? [closed]

For example, there are many stocks, some just come into the market. Thus, if we construct a price matrix, with rows as stocks and columns as time, there would be some missing values in this matrix. ...
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Covariance to Adjacency matrix [closed]

Any suggestions to achive Adjacency matrix of a graph from a given Covariance matrix?
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If $\operatorname{Corr}(X,Y)=1$, then $ \operatorname{Corr}(X,Z)=\operatorname{Corr}(Y,Z)$

$\DeclareMathOperator{\Corr}{Corr}$Given that I have three r.v. $X,Y,Z$ and $\Corr(X,Y)=1$, can I then conclude that $\Corr(X,Z)=\Corr(Y,Z)$? I've tested on some data and found that it was true in my ...
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Itô formula's proof

it's about the proof of the Itô formula Here I give the proof (I marked the lines with my issues in colours) How do we use a/the usual stopping argument? How do we use Lemma 4.51? 2a) We use the ...
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Find $E[X_{(1)}X_{(2)}]$ where $X_1, X_2 \sim N(0,\sigma^2)$

My goal is to find the covariance of $X_{(1)}$ and $X_{(2)}$ and I was able to figure out $$E[X_{(1)}]=-\frac{\sigma}{\sqrt{\pi}} \quad \text{and} \quad E[X_{(2)}]=\frac{\sigma}{\sqrt{\pi}}$$ ...
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What is the sample variance-covariance matrix?

This is a more succinct question from a previous post, but I have arrived at two different answers, and need help determining which - if either - is correct. I start with a 4*3 matrix: ...
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degrees of freedom of sample covariance

For the sample variance we do use the corrected variance using N-1 df as we set the mean constant. However, why do we only use N-1 df for the corrected sample covariance: $S_{XY} = \frac{1}{N-1} \...
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Representing Covariance with Value Approaching Infinity

I am building an agent-based model for cognitive dissonance and want to represent the covariance between a "non-dissonant" state and the dissonance value approaching zero. Problem is I am interpreting ...
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1answer
28 views

Why is the covariance of white noise not finite and not a Kronecker delta?

Suppose we have a Gaussian white noise $f(t)$. Then the covariance $\langle f(t)f(t')\rangle = c\times\delta(t - t')$ for some $c > 0$. However, since $f(t)$ is a Gaussian random variable (with ...
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39 views

Covariant derivate of tensor itself

In the context of Higgs physics when solving the Euler-Lagrange equation I end up with derivatives like the following $\frac{\partial A^\nu}{\partial A^\mu}$ $\frac{\partial A_\mu}{\partial A^\mu}$ ...
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47 views

Finding the covariance of $X$ from $XX^T$ and from its mean.

Given some samples $X(D\times N)$, I am trying to calculate their covariance, however, I only have access to the mean of $X$ ($D\times 1$) and the $D\times D$ matrix $XX^T$. Also $N$ is known. My ...
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Using a Kalman filter with Gauss-Newton solvers

I have two "sensors", that both take the same image $I$, and generate a set of observation points $p_i$ and $q_i$, respectively. These observations are fed into a Gauss-Newton solver from common ...
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989 views

need help to get $Cov(XY,Z)$

I have three variables X, Y, Z. and I know $Corr(X,Y), Corr(X,Z),Corr(Y,Z),Var(X),Var(Y),Var(Z)$. and Cov is Covariance, Corr is correlation, Var is variance, $Corr(X,Y)=\frac{Cov(X,Y)}{\sqrt{Var(X)*...
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Covariance of continuous functions, uniform and normal distribution

For X~Uniform(1, 9.9) and Y|X = x~Normal(1.4, x^2) What is Cov(X, Y) equal to? What I tried was: E[XY] - E[X]E[Y] Where E[X] = 5.45 and E[Y] = 1.4 But for E[XY] I'm a bit clueless. I've considered:...
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Covariance of two R.V's and a specific conditional probability

Question: Suppose that X is a random variable taking on only the values 0 and 1 and that Y is a random variable taking on only the values −1,0,1. It is known that Var(X) = 1/4, Var(Y ) = 1/2 and E[...
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1answer
35 views

Is it possible to recover covariance matrix?

I have the following equation: $$\Sigma= C^T(A^TDA+\Gamma)C$$ where $C$ is a $n\times1$ matrix, $A$ is a $k\times1$ matrix, $D$ is a $k\times k$ matrix, and $\Gamma$ is a $n\times n$ diagonal matrix. ...
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1answer
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Condtional Covarianz Problem

i have on Friday a presentation and i have one small problem. Let $ X_1,...,X_n$ be independent Random Variables. Let $f: \Omega^n \rightarrow \mathbb{R}$ and define $Z:=f(X_1,...,X_n)$. Define also: \...
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24 views

Minimum value of constant correlation

Given a set of random variables $X_1,\dots,X_n$ such that $${\rm cov}(X_i,X_j)=\rho$$ for $i\neq j$. What is the minimum value of $\rho$ possible? Can it be negative? Thanks!
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How to calculate the variance

So i have this question and I wanted to be sure that my calculations were correct. The standard deviations of the market returns is $0.2(20%)$ and the covariance between the return on the market ...
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1answer
42 views

How to calculate the variance of a portfolio?

So I have this question : Florida Company (FC) and Minnesota Company (MC) are both service companies. Their stock returns for the past three years were as follows: FC: -5 percent, 15 percent, 20 ...
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8 views

Variance of two random variables that are defined by independent variables

Let $$Λ_1 = \frac{1}{4} Λ^{(2)} + \frac{3}{4} Λ^{(4)},$$ $Λ^{(2)}, Λ^{(4)}$ are independent random variables. $\mathbb EΛ^{(s)}=\frac{s+5}{s^2}$ and $\mathbb DΛ^{(s)}=\frac{(s+5)^2}{s^2}.$ From the ...
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34 views

Expectations of two random variables

Are the following two statements correct? $\mathbb{E}[X\mathbb{E}(Y)] = \mathbb{E}(X)\mathbb{E}(Y),$ and $\mathbb{E}[X(aY-a\mathbb{E}(Y))] = a\mathbb{E}[XY]-a\mathbb{E}[X]\mathbb{E}[Y] = a Cov(X,Y)$,...
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When is Co-variance multiplication of Standard Deviations?

So I came across this video on which states Co-Varince of $X$ and $Y$ as: $$cov(X,Y) = \sigma_X \sigma_Y$$ I have not come across this formulae anywhere before. The closest is when we define: $$ \rho =...
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Covariance matrix sandwiched between matrix (e.g. matrix $X$) and its transpose ($X'$).

I probably just need more experience in the field of machine learning (and linear algebra in general), but I keep encountering this sort of pattern in some of the equations I have been reviewing ...
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91 views

Covariance between squared bivariate geometrically distributed random variables

If $X$ and $Y$ follows bivariate geometric distribution (where $EX=a$, $EY=b$, $Cov(X,Y)=c$ ) then how to obtain (determine) $Cov(X^2,Y)$?
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$\operatorname{Cov} \, (A,B)\geq 0$ and $\operatorname{Cov}(B,C)\geq 0\Rightarrow \operatorname{Cov}(A,C)\geq 0$?

Let's say we have three random variables $A$, $B$ and $C$. I know that $\DeclareMathOperator{cov}{Cov} \cov(A,B)\geq 0$ and $\cov(B,C)\geq 0$ . Then is it true that $\cov(A,C)\geq 0$?
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Show, that $\frac {E[Xg(X)]}{E[g(X)]} \ge E[X]$ when $g$ strictly monotonic increasing

Let $g:\mathbb R \to (0,\infty), X$ real valued random variable and $g(X) \in \mathcal L^2$ and $g$ strictly monotonic increasing. Show, that $\frac {E[Xg(X)]}{E[g(X)]} \ge E[X]$ I tried something ...
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How to sample from a Gaussian process with only diagonal components of the covariance matrix

This is related to the question asked here, where the answer is that $\boldsymbol{X} = \mu + A\boldsymbol{Z}$. Note that $\boldsymbol{X}$ is the n-dimensional sample we seek, $\mu$ is the mean vector, ...
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Signal-to-noise ratio

Consider the linear structural equation model with known $\beta$ as $$ SEM \quad X_k = \sum_{j=1}^{p}\beta_{jk}X_j + \epsilon_k$$ where $X_k$ is a random variable. I construct a data matrix $D_{m \...
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Find covariance matrix of $\frac{f(x + y) }{x + y}$ function

The conditions are the same, but my task is to find covariance matrix. I only noticed that density function is symmetric so expected values, variance are also the same. But I don't know how to find ...
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Exchange of Covariancematrix and Precisionmatrix in maximum likelihood of Gaussian

If we want to derive the maximum likelihood estimation of the parameters of a multivariate Gaussian, some of the derivations I found claim that we can exchange the covariancematrix with the precision ...
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Prove the negative log-likelihood function is a Lipschitz function

Given $n$ data points $\{(z_i, \phi_i)\}_{i=1,\ldots,n}$, $\phi_i \in \mathbb{R}^d$, $z_i \in \mathbb{R}^m$, consider the negative log-likelihood function $F(X,\Theta) = \left[ \dfrac{1}{n} \sum_{i=1}...
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What is meant by statisticians when they talk about between population differences vs within population differences?

Suppose we have two populations of people in different parts of the world and we want to talk about the variation in heights between the two populations. As I understand it, statisticians are talking ...
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Approximate permutation block diagonalization for correlation matrices.

I'm working with correlation matrices and I'd like to find such permutation of variables that the correlation matrix is 'most block diagonal' (cf. picture below). How does one go about doing this? Is ...
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Finding autocorrelation of seasonal AR model

Problem Find out the autocorrelation function(ACF) $\rho_k$ ($k=1,2,\cdots)$ in the following seasonal AR model. $$ (1 - \phi B)(1 - \Phi B^4)Z_t = \epsilon_t $$ or, $$ Z_t - \phi ...
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Finding the general expression of $\mathbb{E}(Z_t \epsilon_{t-k})$ in $AR(p)$ models

Get the general expression of $\mathbb{E}(Z_t \epsilon_{t-k})$ for $k=0,1,2,\cdots$ in the $AR(p)$ models, $$ Z_t = \phi_1 Z_{t-1} + \phi_2 Z_{t-2} + \cdots + \phi_p Z_{t-p} + \epsilon_t, \forall ...
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51 views

distribute events evenly among persons

I have a real live problem. Excuse me, if i'm not mathematically correct, I'm a developer, not a mathematician. Following situation: I have: a given list of events. eg. each Monday and Wednesday ...
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When is a linear recurrent process stationary?

Let’s call a sequence of random variables $\{X_n\}_{n = 1}^\infty$ stationary, if $\forall n, m, k \in \mathbb{N}$ $EX_n = EX_m$ and $Cov(X_n, X_m) = Cov(X_{n + k}, X_{m + k})$. Let’s call a sequence ...
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Measure of correlation between two vectors that also takes into account their magnitudes?

I am trying to find a measure of correlation between two sets of (2-D) displacement vectors that takes into account not only their directions (in which case, the cosine of the angle between them is ...
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Covariance between lognormal and normal random variable

Good evening, I have the following question: Lets assume X is normally distribued and Y is log normal. Now, i want to obtain the covariance between both random variables. I am aware that: Cov(X,Y) = ...
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Covariance of two r.v. $X\sim B(Z,\alpha)$, $Y\sim B(X\alpha,\delta)$

Suppose $Z_i$ are i.i.d. random variable and $Z_i\alpha$ are positive integers. For the following two random variables $X$ and $Y$, I would like to compute the $\mathrm{Cov}(X,Y)$ where $X \sim B(Z,\...
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Geometric meaning of second Covariant Derivative

This other question exists, but it doesn't answer my question: Geometric interpretation of the second covariant derivative I know the Riemann Tensor can be written as the commutator of the second ...
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Why does the identity $\mathbb{E}(X) = \mathbb{E}\left(\int \mathbb{1}_{u \leq X}du\right)$ hold?

I'm reading on Hoeffding's covariance identity, the proof of which is neatly covered here, or, in a similar manner, in this MSE post, but I can't seem to fully understand the trick/property used there....