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Questions tagged [covariance]

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

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Determinants of the covariance between two random variables

I want to know if the covariance between two random variables is always determined by the two terms of its formula or if it can be only determined by the E(XY). If E(Y)=0, then Cov(X,Y)=E(XY)-E(X)E(Y)...
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Given that $X_{i+1} = \rho X_{i}$, determine the dispersion matrix $Var[\textbf{X}]$

If $X_{1},X_{2},\ldots,X_{n}$ are random variables and $X_{i+1} = \rho X_{i}$ $(i = 1,2,\ldots,n)$, where $\rho$ is constant, and $var[X_{1}] = \sigma^{2}$, find $Var[\textbf{X}]$. MY ATTEMPT ...
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Given two random vectors, determine the dispersion matrix $Var[\textbf{X}]$.

Let $\textbf{X} = (X_{1},X_{2},\ldots,X_{n})^{\prime}$ be a vector of random variables, and let $Y_{1} = X_{1}$ and $Y_{i} = X_{i}-X_{i-1}$ where $i = 2,3,\ldots,n$. If $Y_{i}$ are mutually ...
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Calculate covariance given correlation, problem with percentages

The question is: find the covariance of ABC stock returns with the original portfolio returns. Pretty straightforward. However I get confused working between percentages and units. The ...
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Covariance matrix of uniform distribution in $L^p$ Euclidean ball [on hold]

Let $$X=\operatorname{Unif}\left(\left\{x:\ \sum_{i=1}^n |x_i|^p\le 1\right\}\right)$$ Is there some method to calculate the covariance matrix of $X$? Thank you!
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Using Cholesky decomposition to compute covariance matrix determinant

I am reading through this paper to try and code the model myself. The specifics of the paper don't matter, however in the authors matlab code I noticed they use a Cholesky decomposition instead of ...
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truncation degree of decomposed covariance matrix

I have a covariance matrix of a standardized data set. Doing a singular value decomposition i find near zero singular values and would therefore like to truncate it. I know of Picard plots which ...
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Covariance matrix and projection

I have troubles understanding a geometrical meaning of a covariance matrix. Let's say we have a data set containing two points (-1,1), (-1,2) and write them in to the matrix $$D = \begin{bmatrix} -...
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How to compute $cov(\frac{1}{2}Z'AZ,\frac{1}{2}Z'BZ)$?

I want to show that $cov(\frac{1}{2}\textbf{Z}'A\textbf{Z},\frac{1}{2}\textbf{Z}'B\textbf{Z})=\frac{1}{2}tr(AR(\theta)BR(\theta))$ where $A,B \in Sym(p)$ (real symmetric $p \times p$ matrices) and $\...
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Example of “The eigenvalues of data covariance matrix, $\Phi^T\Phi$ measure the curvature of the likelihood function.”

I am reading PRML, Chapter 3.5.3, screen shot attached. I can understand the derivation and maths but hard to understand the meaning of "The eigenvalues of data co-variance, $\Phi^T\Phi$ matrix ...
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Rank of covariance matrix

I am having a problem with rank deficiency in a covariance matrix. I have a data-set of M variables and N observations, M>N. Calculating the singular value decomposition of the data-sets covariance ...
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How to calculate $\text{cov}(\hat{Y}_{ij}, \hat{Y}_{kj})$ if $Y_{ij} = \mu + a_i + b_j + e_{ij}$?

Let's assume we have the model following Two-Factor model without replications : $$Y_{ij} = \mu + a_i + b_j + e_{ij}, \; i=1,\dots,p \; \text{and} \; j=1,\dots, q $$ I am interested in calculating ...
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1answer
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Dimensional properties derived from PCA eigenvectors

Background Let's assume I'm using principal component analysis to carry out clustering of a 2-d data set, using a non-normalized covariance matrix to carry out the operation. I then solve for the ...
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Understanding Weird Relationship Between Hamming Weights

I have two binary "mapping" matrices $\delta_0$ and $\delta_1$ $ \delta_0 = \begin{bmatrix} 1 0 1 0 0 0 0 0\\ 1 1 0 1 1 1 1 0\\ 0 0 0 0 1 1 0 0\\ 0 1 1 1 0 0 0 0\\ 0 1 1 0 1 0 0 0\\ 1 0 0 1 1 1 0 0\...
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Covariance of errors $\text{cov}(\hat{e_{ij}},\hat{e_{i\ell}})$ in Two-Way Anova model

Exercise : Consider the Two-Way Anova model $Y_{ij} = \mu + a_i + b_j + e_{ij}$ with $i = 1, \dots, p$ and $j=1,\dots,q$. Show that : $$\text{cov}(\hat{e_{ij}},\hat{e_{i\ell}}) = -\sigma^2\left(\...
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Covariance of square root for two bins of a multinomial

Take $(X_1, \dots, X_k) \sim Multinomial(n, (p_1, \dots, p_k))$. Do we have a closed form expression for $\mathbb{E}[\sqrt{X_i X_j}], i\neq j$ ?
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A box contains 3 red balls and 2 white balls

A box contains 3 red balls and 2 white balls. Two balls are picked randomly from the box without replacement. The random variable 𝑿 is the number of red balls, and 𝒀 is the number of white balls. ...
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Normal Random Variables that all correlate with a single time series but not necessarily with each other

I have a sequence of normally distributed random variables. Let's call it $S_1$. I want to generate 4 more series, each of which has its own correlation with $S_1$ and its own variance. Let's call ...
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Scalar product induced by covariance matrix

Suppose that $n$-dimensional random vector $Y$ has covariance matrix $\Sigma$. It is well known that for any $a\in\mathbb{R}^n$ we have \begin{align} var(a^TY)=a^T\Sigma a. \end{align} Is there any ...
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Finding $\text{Cov}(X,Y)$ when $(X,Y)$ has joint density $\frac{1}{2}\sin(x+y)\mathbf1_{0\le x,y\le\pi/2}$

Joint probability density: \begin{equation} P_{x,y}(x,y) \begin{cases} \frac{1}{2}\sin(x + y) & , \text{if}\ 0\leq x \leq \frac{\pi}{2}, 0 \leq y \leq \frac{\pi}{2} \\ 0 & ...
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Variance and covariances from linear mixed model for power simulation using R

I am working with longitudinal data where the outcome is the number of steps per minute. My LMM fit would look like: ...
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1answer
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Constructing a probability measure on the Hypercube with given moments

Let $H = [-1, 1]^d$ be the $d$-dimensional hypercube, and let $\mu \in \text{int} H$. Under these conditions, I can explicitly construct a tractable probability measure $P$, supported on on $H$, ...
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1answer
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How to show that the error variance of the best linear predictor is inferior to the proportional predictor?

Let's consider the 1D case. How do we prove that the error variance of the Best Linear Predictor (BLP) is inferior than the Proportional Predictor (i.e. the Linear Predictor without the intercept)? ...
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1answer
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Stretching the covariance of a daily change to the year

I'm diving into financial mathematics and have calculated a matrix that gives me the daily change of four given securities: The start of it looks like this: My Tutorial is now calculating the ...
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We have an urn with 6 red balls and 4 green balls.

We have an urn with 6 red balls and 4 green balls. We draw balls from the urn one by one without replacement, noting the order of the colors, until the urn is empty. Let X be the number of red balls ...
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Testing cross-covariance in the residuals of a VAR(p) model

Suppose I have a vector autoregressive model of order $p$: $$y_t = c + A_1 y_{t-1} + ... + A_p y_{t-p} + u_t$$, where $y_t$ is a $K\times 1$ vector and $A_i$ are $K\times K$ matrices. We assume the ...
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Prove a time series to be NOT identically independent distributed

I am trying to prove that this time series (given that $X_{t}$ and $M_{t}$ are iid and independent of each other) $$ Y_{t} = X_{t}(1-X_{t-1})M_{t} $$ is not i.i.d, so my understanding is that I need ...
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Finding the probablity density of a point for a multivariate gaussian distribution

I just learnt this and it's formula but don't know how to use it. The question i've been given is: "What is the density probability of a point at x=[1,2] for a multivariate Gaussian distribution ...
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1answer
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Convergence in mean of a series of random variables

I'm stuck on the following proof: Let $(X_1,.. X_n)$ random variables so that $E(Xi) = \mu$, $V(Xi) = \sigma^2$ final for all $1 \leq i \leq n$. It is also given that for all $i \neq j$, $Cov(Xi, Xj) ...
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Expectation of cubic form (covariance between a quadratic form and a dot product)

Consider a $n$-dimensional random vector $X$ with mean $\mathbb{E}X=\mathbf{0}$ and covariance matrix $\Sigma$. Given a $n\times n$ matrix $A$ and a $n$-dimensional vector $b$, is there any explicit ...
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Covariance of assets with different probabilities for each scenario.

There are three assets given and for each asset there are three scenarios with their respective probabilities. Asset 1: $$\begin{array}{c|c|c|} & \text{Return} & \text{Probability} \\ \...
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1answer
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Standard normal covariance

I am given that X is a standard normal distribution. Why is $Cov(X, -X) = -1$? I know that $Cov(X,X) = Var(X)$ and that the $Var(X) = 1$. Is the $Var(-X) = -1$?
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A question about a generalization of covariance

Suppose, $H$ is a Hilbert space over $\mathbb{R}$. Suppose, $X$ and $Y$ are random vectors in $H$. Let’s define Hilbert expectation of a random vector $X$ in a Hilbert space $H$ as a vector $v \in H$, ...
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How to compute the covariance of a linear equation

Given the equation: $V_k = M v_k - A M v_{k-1}$ with $ M = [C^T C]^{-1} C^T,$ where capital letters represent matrices. A paper that I am reading ( see equation (27) here) states that you can ...
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Log det of covariance and entropy

I understand log of determinant of covariance matrix bounds entropy for gaussian distributed data. Is this the case for non gaussian data as well and if so, why? What does Determinant of Covariance ...
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Obtaining inverse of root square of a matrix using singular decomposition value

I have a variance-covariance matrix $\Sigma$ and I'd like to obtain its inverse of root square. How to use the singular decomposition values to solve this problem?
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Adding the variances of 2 dependent variables and covariance

$ E(\hat \theta_1 ) = E(\hat \theta_2) = \theta_1 $ , $ Var (\hat \theta_1) = \sigma_1 , Var(\hat \theta_2) = \sigma_2, Cov(\hat \theta_1, \hat \theta_2) = \sigma_{12}$ $\hat \theta_3 = a \hat \...
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Using covariance function to normalise a function

In the statistical inversion setting it is common to adopt a Gaussian Process (GP) Prior with a Gaussian kernel to preferentially treat smooth parameter fields. With covariance matrix, $$S_{ij}=a \; ...
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Fill a joint table distribution, find covariance and check if two variable are independent.

Choose numbers from {2, 3} by tossing a fair coin; the coin is tossed twice. Choose 2 if the coin turns up Heads and 3 if the coin is Tails. So the possible outcomes are {(2, 2),(2, 3),(3, 2),(3, 3)}. ...
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Covariance matrix under multiplication with independent random variable.

assume two dependent random variables X and Y with their covariance matrix $\Sigma_{ij}$. Now assume one multiplies X with another independent random variable Z. The variance of the resulting random ...
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Approximation of a positive definite matrix

I have a covariance matrix (A), which is positive definite. I would like to approximate matrix A by another positive definite matrix B in such a way, that the eigenvalues of B span only 2 orders of ...
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If $X \sim U(-2,2)$ ,find the correlation between $X,|X|$

If $X \sim U(-2,2)$ ,find the correlation between $X,|X|$ It is clear that $|X| \sim U(0,2)$. Obviously , $E(X)=0$ Then $E(X|X|)=E(X^2 \cap X>0)-E(X^2 \cap X \le 0)=0$ due to symmetry. Thus, the ...
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Correlation between 2 signals of uneven dimensions

As a part of my work, I am trying to correlate the audio signal in a video with the pixels of each frame. The steps I follow are: 1] Audio sampling rate and frame rate of the video are known. So, ...
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Demonstration for covariance with linear combination of 2 variables

I don't know how to prove this relation : I know that : $$\text{Var}\bigg(\sum_{i}\,a_{i}\,X_{i}\bigg)=\sum_{i}a_{i}^{2}\,\text{Var}(X_{i})+\sum_{1\leq i \leq j\leq n} 2\,a_{i}a_{j}\text{Cov}(X_{i},...
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1answer
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Comparing diagonal elements between two inverse matrices

I would like to compare diagonal elements between inverse matrices. Suppose that we have three real block matrices as follows: $$ \underbrace{\begin{bmatrix}\mathbf A & \mathbf B^T \\ \mathbf B &...
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central limit theorem: what is the variance?

This is a very basic question that I'm pretty sure I understand but I wanted to double check. Given a regression model of: $$y_t = \mathbf{x_{t}^{\prime}}\beta + u_t$$ We can use one of the CLT ...
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1answer
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Showing that an estimator for covariance is consistent?

I'm having trouble proving that a certain estimator is consistent. I know that to show an estimator is consistent, I have to show that the variance of the estimator approaches zero as n grows large/...
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1answer
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Hyperparameter gradients for Matérn covariance

Similarly to this question I am seeking the derivative of a covariance function with respect to its parameters. However, that question is specifically about the squared exponential kernel. How can I ...
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Under what condition are $U$ and $V$ uncorrelated?

Let $X$ and $Y$ be independent random variables with finite variances, and let $U = X + Y$ and $V = XY$. Under what condition are $U$ and $V$ uncorrelated? MY ATTEMPT We say that two variables are ...
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2answers
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Let $U = \min\{X,Y\}$ and $V = \max\{X,Y\}$. Find $\textbf{E}(U)$, and hence calculate $\textbf{Cov}(U,V)$.

Let $X$ and $Y$ be independent random variables each having the uniform distribution on $[0,1]$. Let $U = \min\{X,Y\}$ and $V = \max\{X,Y\}$. Find $\textbf{E}(U)$, and hence calculate $\textbf{Cov}(U,...