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Questions tagged [covariance]

Covariance is a quantity used to measure the strength and direction of the linear relationship between two variables. The covariance is unscaled, & thus often difficult to interpret; when scaled by the variables' SDs, it becomes Pearson's correlation coefficient.

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Proof expression for the autocovariance function of AR(1)

The representation for the model AR(1) is the following: $Y_t=c+ϕY_{t-1}+ε_t$ where $c=(1-ϕ)μ$ ($c$ is a constant). I want to understand the calculations that there are behind the general formula ...
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Average of sample covariances [duplicate]

I have K datasets, each with N variables and M samples and assume they are coming from the same multivariate normal distribution. I am interested in estimating the covariance matrix. Now it can be ...
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Why does $\operatorname E(\varepsilon\mid x) = 0 \implies \operatorname{cov}(\varepsilon,x) = 0$?

I understand the intuition behind the question but I'm trying to prove it to myself with math.
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Convergence of the Matérn covariance function to the squared exponential

The Matérn covariance function converges to the squared exponential covariance function. Many sources, amongst them the GPML book and Wikipedia, state this result. None of them provide details. I ...
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$V(X|Y)=\Sigma_{XX}-\Sigma_{XY}\Sigma_{YY}^{-1}\Sigma_{YX}$

We know that the conditional variance of a multivariate normal vector $(X,Y)$ is equal to the Schur complement: $$V(X|Y)=\Sigma_{XX}-\Sigma_{XY}\Sigma_{YY}^{-1}\Sigma_{YX}$$ However, $\Sigma_{XX}-\...
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Conditional covariance of a multivariate normal vector

We know that the conditional variance of a multivariate normal vector $(X,Y)$ is the Schur complement: $$V(X|Y=(y_1,...,y_n))=\Sigma_{XX}-\Sigma_{XY}\Sigma_{YY}^{-1}\Sigma_{YX}$$ I have the intuition ...
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RCT analysis using ANCOVA for rates

I have a question based on the following approach for the analysis of RCT's. The following works well for the outcome (and baseline) being continuous with normal errors. Expanding upon this, I was ...
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What can be interpreted from (xi-xbar)(yi-ybar) [duplicate]

what is the interpretation of each term in the numerator? why do we multiply the two terms why not add them? what is the physical interpretetation of the these terms?
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Can one usefully specify a multilevel-model with a partially-nested, partially non-nested structure?

Background Gelman and Hill's Data Analysis Using Regression and Multilevel/Hierarchical Models includes an example in section 13.5 of how to model non-nested data. The second example in this section ...
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Quantifying dependence of Cauchy random variables

Given two Cauchy random variables $\theta_1 \sim \mathrm{Cauchy}(x_0^{(1)}, \gamma^{(1)})$ and $\theta_2 \sim \mathrm{Cauchy}(x_0^{(2)}, \gamma^{(2)})$. That are not independent. The dependence ...
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Identifying variables which differentiate 2 groups the most

I have 2 groups of patients (having and not having a disease A), respectively 41 and 19 patients. Set of about 25 different parameters was measured at those patients (some continuous variables and a ...
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Calculate the variance of $\sum\limits_{i=1}^{n-1} \sum\limits_{j=i+1}^n S(X_i - X_j)$ for $X_1,\ldots,X_n$ i.i.d. random variables

In p.88 of Wand & Jones (1995), they asked to show the following result. Let $X_1,\ldots,X_n$ be a set of i.i.d. random variables and define $$U=2n^{-2}\sum_{i=1}^{n-1} \sum_{j=i+1}^n S(X_i - ...
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Are all symmetric matrices with diagonal elements 1 and other values between -1 and 1 correlation matrices?

A question for the statisticians and other math lovers: Are all symmetric matrices with diagonal elements 1 and other values between $-1$ and 1 correlation matrices?
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Correct to evaluate the relation between likert type ratings, response time, and age with repeated measures ANOVA and covariates?

I would like to kindly ask you for your point of view on whether my approach is correct. First I will describe my project, then I will ask a question: I conducted approx. 30 surveys in which 28 to ...
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Analytical solution to the covariance between a continuous and a categorical variable

Let $X$ be a continuous variable with mean $\mu$ and $Y$ be a categorical variable with event probability vector $\mathbf{p}$. I am trying to calculate $Cov(X, Y)$. I have the solution if $\mathbf{p} ...
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Covariance of An Empirical Distribution Function Evaluated at Different Points

The problem is extracted from All of Statistics (Exercise 7.5), Larry Wasserman. I don't have a solution manual to the book so I post here the problem together with my attempted answer: Let $x$ and $...
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How to interpret a given 2D co-variance matrix?

I am trying to solve a problem regarding revision for my Big Data module. I have two main questions. 1) Given a predefined co-variance matrix: A cluster of points is distributed in a two-...
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Why mean in Gaussian Process is not that important? [duplicate]

I would like to seek an explanation to an answer provided in reply to this question- Why is the mean function in Gaussian Process uninteresting?. User: j__ stated: the mean function may not live ...
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How to calculate the covariance matrix for a categorized variable?

Let $X$ and $Y$ be jointly distributed as a multivariate normal with the following parameters: $$ \mu_{XY} = \begin{bmatrix} 0 \\ 0.2 \end{bmatrix} \qquad \Sigma_{XY} = \begin{bmatrix} 1 & 0.05 \...
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Sample weights in covariance matrix estimator

I am using Logistic Regression with count representation. So, for any feature-tuple, I have few 0's(negative class) and few 1's. I duplicate each row, one for target 0 and other for target 1, And I ...
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Autocovariance and autocorrelation function of AR(1) process

I'm preparing the exam about AR models, precisely I have this exercise which I have some issues with points "d" and "e". My try was: Knowing that $W_t=X_t-X_{t-1}$, $h=1$ so: d) $\gamma\left(1\right)...
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Covariance non-zero mean AR(1)

Why when I compute the autocovariance function of a non-zero mean AR(1), X(t)-u=Φ(X(t-1)-u)+ε the presence of the mean does not change my result and so the formula should be the same of a zero-mean ...
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Intuitive understanding of variance of sum vs variance of difference

$\newcommand{\Var}{\operatorname{Var}}\newcommand{Cov}{\operatorname{Cov}}$Mathematically, $\Var(X + Y) = \Var(X) + \Var(Y) + 2\Cov(X,Y)$ and $\Var(X - Y) = \Var(X) + \Var(Y) - 2\Cov(X,Y)$ This ...
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Cauchy Schwarz inequality proof using discriminant

I know the proof but I'm unclear on one thing. Cauchy-Schwarz inequality: Given X,Y are random variables, the following holds: $$ (E[XY])^2 \le E[X^2]E[Y^2] $$ Proof Let $$ u(t) = E[(tX - Y)^2] $$ ...
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What does it mean that if distr. of $X_j$ does not depend on $j$, $E(X_j)=\alpha$, then $Cov(X_i,X_j)$ dep. only on $|i-j|=:d$?

What does it mean that if distr. of $X_j$ does not depend on $j$, $E(X_j)=\alpha$, then $Cov(X_i,X_j)$ dep. only on $|i-j|=:d$? $Cov(X_i,X_j)=E[(X_i-E[X_i])(X_j-E[X_j])]$ $=E[(X_i-\alpha)(X_j-\alpha)...
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Covariance of Two Quadratic Forms

We're looking for the $\operatorname{Cov}\left[x^T A x, ~x^T B x\right]$ where $x$ is random variable and mean-centered, but not independent and $A$ and $B$ are symmetric matrices. The fundamental ...
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How to interpret e.g. pcov returned by numpy.optimize.curve_fit

When doing parameter fits with mathematics frameworks as e.g. numpy, often a covariance matrix is returned. I wonder how to interpret these and if the following is right: The entries of the ...
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What is the meaning of $\sqrt{x^T \Sigma x}$?

I'm reading a paper and this is the main quantity if focuses on. In the above definition x = $[x_1, x_2, \ldots]$ $\Sigma$ is the covariance matrix of variables. What is the meaning of this quantity?...
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Solving a double integral over transformations of joint bivariate standard normal values

My problem is about calculating the covariance between transformations of two test statistics based on the correlation between these test statistics. Let $X$ and $Y$ be two test statistics whose joint ...
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Why does Covariance measure only Linear dependence?

1) What is meant by linear dependence? 2) How can I convince myself that covariance measures linear dependence? 3) How I can convince myself that non-linear dependence is not measured by covariance?...
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Genetic correlation in R (partitioning correlation in mixed models)

I've no issues understanding the ins-and-outs of partitioning variance into random and fixed effects components. But I cannot, for the life of me, visualize partitioning correlation (or covariance) in ...
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Why data points are thought as random variable

I'am currently following a basic statistics course and a machine learning course. I try to understand what is covariance. In general, books define covariance as follows: Covariance is a measure of how ...
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Why Does Second Order Weak Stationarity include Statement on Covariances in addition to Statement on Mean and Variance?

A stochastic process is second order weakly stationary if all random variables have same mean (first moment), and same variance (second moment?), and covariances that are time-invariant (second moment ...
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VAR(p) Model Covariances and Moment Equation

I'm currently going through the book Analysis of Financial Time Series by Ruey S. Tsay and reached the following statement (The book can be found here, with VAR(1) included in the preview): Where: $...
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How to calculate standard error of the difference between two random variables which are dependent?

I know that standard error between two independent variable can be calculated as below. SE(1,2) = SQR(SE1^2 + SE2^2) where SE1 is standard error for variable 1, SE2 is standard error for variable 2. ...
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To find the covariance given the joint probability density function.

Question: I was solving some question papers and got stuck in this problem. My problem: I know how to find "marginal probabilities" from a joint probability density function and also know how to ...
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Maximum likelihood: Why is the number of non-zero eigenvalues equal to $x^T \hat{\Sigma}^{-1} x$

I've been reading this code (based on this R package) and I found that the number of non-zero eigenvalues of the estimated covariance is roughly equal to $x_i^T \hat{\Sigma}^{-1} x_i$. I want to know ...
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Confusion in terminologies for simple linear regression model [closed]

Please go through my draft summary below and let me know if my conventions are correct, comprehensible, and non ambiguous. Simple Linear Regression Model Let given observed sample set be $\{(x_1,...
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Variance of $Y|x$ from regression line

Using simple linear regression model, and sample correlation coefficient $r$, for a sample set $X,Y$, the true regression line could be given as below. $$ \hat{Y}|x = \overline{y} + r\dfrac{s_Y}{...
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Proving covariance equals zero given a specific conditional expectation

I'm trying to prove the following: Given $𝐸[𝑋|𝑌 = 𝛽] = 𝐸[𝑋]$ for any value of $\beta$, prove that $\operatorname{Cov}(𝑋,𝑌) = 0$; So I was thinking to start with the definition of $\...
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Is my Correlation reasoning correct?

I am trying to understand how to arrive at $r = \dfrac{Cov(X,Y)}{\sigma_X\sigma_Y}$ with a logical narrative. This in fact is kind of continuation from my this unanswered question. I see that by ...
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How to derive correlation using regression without empirical proof?

I just finished learning MLE, Regression, Covariance and now in to Correlation.I want to transform logically from Regression to Correlation using Covariance. Regression: A simple regression model ...
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Correlation 2D vector fields

Having multiple (hundreds) of 2D flow maps, ie vector fields how would one find statistical correlation between these? Plotting yields, for visualization purposes only: I am thinking about comparing ...
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1answer
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Interpreting a graphed covariance function

I'm looking through a slide deck (slide 9) about Gaussian Processes, and I came to a slide that describes one example of a covariance function: Matérn $\frac{3}{2}$ Covariance. $$C(x_1,x_2) = (1+\...
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How to find the marginal densities of the given functions

The fraction $X$ of male runners and the fraction $Y$ of female runners who compete in marathon races are described by the joint density function$$f(x,y) = \begin{cases} 8xy & 0 \le x \le y \le1 ...
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1answer
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Linear regression with a given covariance structure in R

I want to fit a linear model in R with a given covariance structure: $$y=X\beta+\epsilon$$ where the covariance matrix of $\epsilon$ is block diagonal by a grouping factor. Suppose there are $B$ ...
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1answer
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Confusion with covariance

For distributions of random variables X and Y, their covariance can be defined as the difference between the multiplication of X and Y, normalized by their joint probability and the multiplication of ...
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GP posterior covariance between f(x) and f(x')

I am working through the examples in the Rasmussen's Gaussian Processes, specifically the GP regression figure attached right -- I can't seem to get the the posterior covariance between f(x) and f(x') ...
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residualized covariance matrix from pca/eigenvalue decomposition

I understand that given N dimensional data you can use PCA to construct an N dimensional orthonormal basis that explains 100% of the variance of the original data. However, you can also construct ...
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1answer
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Mean of two time series

I'm trying to estimate the covariance of two time series using the formula where X and Y are two time series. I don't ...