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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35 views

Can you use Kendall's Tau to compute covariance matrix?

I am working with multivariate archimedean copulas, and I am wondering how I can extract a covariance matrix out of them? I can get Kendall's Tau matrix of correlation so I was thinking that maybe I ...
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Multiplying by vectors to assess covariance is zero

I want to prove the following but am unsure how. Show that if: For all fixed vectors $c$, $Cov(X,c'Yc) = 0$ Where $X$ and $Y$ are matrices of random variables, then it must be true that: $Cov(X,Y)=...
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45 views

Inverse Wishart Prior for linear model

I know some bayesian methods employ an inverse wishart distribution for the prior distribution of the covariance matrix in a linear regression. I.e. for the model: $$Y=X\beta+\epsilon$$ Where $\...
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Are the conditional expectation values of y and f necessarily equivalent in Gaussian processes?

Suppose $y$ is a Gaussian process given by $y \sim f + \epsilon$, where $\epsilon$ is a Gaussian noise model with zero mean, and $f$ is a deterministic yet unknown mean function (or a Gaussian process ...
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Proof that covariance of RV and group average is less than variance of RV

I have a random variable $X_i$ and a group of $N_j$ other random variables that includes $X_i$. Let's just call this group $J$. There are no distributional assumptions made on these RVs (other than ...
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How to make a scree plot out of SVD data to validate PCA

After doing a singular value decomposition (SVD) of a data set, I'm left with three matrices: 1. An orthogonal Left Singular Vector (U) 2. diagonal matrix with elements in descending order (S) 3. ...
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Question on a proof in Bickel's covariance estimation paper

Recently, I am reading Peter J. Bickel's paper Regularized Estimation of Large Covariance Matrices. In that paper, the author tries to prove lemma A.3, which goes as follows: Let $Z_{i}$ be i.i.d. $\...
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In a repeated measures model where assessments are performed over time, should baseline data be excluded if baseline covariate is used?

In a model where subjects are evaluated over time and a baseline (time=0) covariate is used (eg, ...
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How does a covariance intensity function measure clustering?

I was taught in a class on spatial statistics that the covariance intensity function (defined below) measured clustering and inhibition in a point process, but isn't used because good test statistics ...
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Moderation and Ancova

I conducted a repeated measures anova with accent (British versus American) as 1 level and dress style (business, super casual, and business casual). My dv was how professional each was rated (1 to 7)....
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Covariance between two binomial random variables

Consider a binomial random variable $X$ with parameter $p$ and another binomial random variable $Y$ with parameter $q$. What is the covariance of $X$ and $Y$? How well does the proof generalize to $n$...
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What statistic to use in testing the variance of maximum likelihood estimators

(A physicist self-studying statistics here) I was previously confused about the meaning of the standard error in a maximum likelihood estimate. Certain stack exchange posts (linked below) have gone ...
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the accuracy of covariance between two high-dimensional vectors

Question Is the covariance between high-dimensional vectors less accruate than covariance between two vectors in low-dimensional vecotrs? I am asking this questio to check if there is a need for '...
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Covariance correction for linearly mixed signals

Consider the simple linear mixing model: $$ X = AS + v $$ where: $X$ is N-by-T, $A$ is N-by-M, $S$ is M-by-T, and noise $v\sim\mathcal{N}(0,\Sigma_v)$. Assume that we know the matrices $X, A,...
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What is the formula for the conditional variance when taking the derivative of a Gaussian process?

The formulae for the conditional mean and variance of a Gaussian process is given by equations (2.23) and (2.24): Also, the formula for the covariance of the derivative of a Gaussian process is given ...
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Is the spherical covariance function not positive definite for d > 3?

I read in a textbook (Japanese one) that the spherical covariance function is only valid for dimensions $d = 1,$ $2,$ and $3.$ I have the following questions: Does that mean the spherical covariance ...
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Sign of Covariance and of Spearman's Rho

Has anybody available a proof that the covariance between two variables has always the same sign as Spearman's Rho, assuming that both are not zero, or an explanation / counterexample to show why this ...
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Repeated-measures ANCOVA using aov AND Error function

I have looked into other answers found on SE but still do not feel that my case matches other posters' data. I would like to perform a RM-ANCOVA on the following data: ...
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1answer
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In a Multiple Linear Regression Model, is $Cov(\pmb{\hat{y}}, \pmb{e}) = 0$?

I'm solving the problem 3.33, from the book "Introduction to Linear Regression Analysis (5th edition)", by Montgomery and I got a doubt. 3.33) Prove that $R^2$ is the square of the correlation ...
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MCMCglmm extract covariances

I am trying to extract the covariance between changes in the expression of attributes. I have tried to do this with the help of: https://besjournals.onlinelibrary.wiley.com/doi/epdf/10.1111/1365-2656....
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Why do you need a variogram for Kriging? goldingn/gpe package?

I am using golingn/gpe (github) package, and it does not provide a variogram and instead look at co-variances. Is it possible to do kriging without providing variograms?
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Variance of X-Y and X-Z when Z and Y are correlated

I have a hard time solving the following issue, so hopefully someone is willing to help. I believe I am almost there but just missing a single step. There are three random variables $X$, $Y$, and $Z$ ...
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Covariance as Measure of Linear Fit? Circle as “Anti-Line”

I am trying to figure out how one can deduce from the formula for Covariance: $ Cov(X,Y) \:=E[(X-\mu_X)(Y-\mu_Y)]= E(XY)-\mu_X\mu_Y $ Correlation measures the degree of linear dependence. Does ...
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Which robust covariance estimator to use if I want to 1) robust against outlier 2) stay positive definite

I am now working on a very noisy dataset $X$ , which is a m-by-n matrix, with m samples, each contains n features The n-by-n covariance matrix can be naively calculcated as $X^{T}X$ However, my ...
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Why does independence imply zero correlation?

First of all, I'm not asking this: Why does zero correlation not imply independence? This is addressed (rather nicely) here: https://math.stackexchange.com/questions/444408/why-does-zero-correlation-...
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cross-covariance estimation and variance reduction

Let $X,Y$ be two vector variables and $$ \mathrm{Cov}(X,Y) = \mathbb{E}[(X-\mathbb{E}X)(Y-\mathbb{E}Y)^T] $$ their cross-covariance (but I think we could just pretend that's the covariance between two ...
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Estimating state covariance with the Unscented Transform and diffuse prior

I have a set of measurements (assume additive Gaussian noise on each), a non-linear measurement model, and a diffuse prior. The state covariance estimate: $P = (H^T R^{-1} H)^{-1}$ where $H$ is the ...
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Variance of linear combination of AR(1) process

Let $ \{X_t\}$ ~ AR(1): $$ X_t=2.62-0.84X_{t-1}+\epsilon_t, \ \ \ \epsilon_t\sim WN(0,2.27)$$ Compute the variance of $$ \overline{X}= \frac{1}{3}\sum_{t=1}^{3} X_t $$ The solution is: Var($\...
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Using indistinguishable subjects as predictors/random effects

I would like to model data where the outcomes are produced, jointly, by a pair of indistinguishable subjects. As an example[*], consider the length of two-participant conversations. These data have ...
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1answer
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How to include off-diagonal elements in covariance matrix in uncertainty of variable

I am performing a linear regression (y = a + bx) and calculate the correlation matrix: \begin{bmatrix} 1 & -0.84 \\ -0.84 & 1 \\ \end{bmatrix} and covariance matrix: ...
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Conditional Covariance Problem

Suppose we have independent (not necessarily identical) normally distributed random variables X, Y. If we're given that, upon sampling each variable, X is some multiple a of Y (i.e. x = ay), what is ...
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1answer
188 views

Gaussian process - Why adding data points cannot increase the predictive bias?

I've seen this question here: How to increase variance in Gaussian Process regression? And trying to complete the proof. I'm looking at this book: Rasmussen & Williams 2006: Gaussian Processes ...
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Lars alphas_ results

Could any one explain how the results on alphas_ attribute in Lars model are calculated? In the definition: alphas_ is the maximum covariance (abs value) in each iteration. But when I look into ...
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1answer
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Heteroskedasticity-Consistent Covariance Matrix Estimation

I would like to ask about the difference between the vcovHC and vcov in R. The former is described as the Heteroskedasticity-Consistent Covariance Matrix Estimation. What is the difference between ...
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1answer
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Covariance of an estimate from optimization

Consider a standard linear regression model, $\boldsymbol y = X \boldsymbol \beta + \boldsymbol \epsilon$. $\boldsymbol y$ is a vector of $m$ responses, $X$ is a design matrix with $m$ rows and $p$ ...
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Relationship between CART single node tree, variable correlation and variable variance

I am currently working on finding a pattern of when CART produces single node trees after cost-complexity pruning. I am most interested in the effect of variable variance and variable correlation. ...
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2answers
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Covariance of two normally distributed variables

I saw in a statistic book that "It can be prooved that if two normally distributed variables have covariance = 0, they are independent". How can I start this proof? Can I say that $cov(X,Y) = E(XY) ...
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Finite second moments inhertitable to conditional variables?

Assume a random vector $\mathbf{x}=(x_1,\ldots,x_n)^\top$ that has finite second moments, i.e., $$\int\mathbf{x}\mathbf{x}^\top\rho(\mathbf{x})\,\text{d}\mathbf{x} < \infty.$$ Does it follow that ...
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1answer
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Minimum variance of the mean for $n$ correlated random variables

If $X_1,\cdots,X_n$ all have the same variance equal to 1, then $0\leq \mbox{Var}[\bar{X}]\leq 1$ where $\bar{X}=(X_1 + \cdots + X_n)/n$. The upper bound is attained if $\mbox{Cov}[X_k,X_l]=1$ for all ...
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1answer
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Variance and covariance inequality

Given a real-valued random variable $X$, is $$2\mathbb E[X] \mathrm{Var}(X) \geq \mathrm{Cov}(X, X^2)$$ true? Any pointers for how to tackle this problem would be immensely helpful.
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Does $\text{cov}(a_1' X, a_2' X) = 0$ imply $a_1 \cdot a_2 = 0$?

Let $X$ be a $p$-dimensional random vector with $p$ principal components $y_1, y_2, \dots, y_p$. By definition, a restriction put on the second principal component $y_2 = a_2'X$ is $$ \text{cov}(y_1, ...
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Variance of ratio of a time series of random numbers

Say we have a time series($X_0, X_1, ..., X_n$) with only the first element $X_0$ known, rest being random. How can we express: $$ {\rm var} \frac{X_n}{X_0} = f({\rm var} \frac{X_i}{X_{i-1}})$$ Don'...
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Notation problem with sparse regularized correlation matrix

I am trying to apply a specific method to obtain a sparse correlation matrix $R$ from a regularized correlation matrix $\Sigma^{\delta}$, which was computed from $N$ observations of a multivariate ...
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Interpretation of $det(X'X)$ in MLR

I would like to understand the interpretation of $det(X'X)$ in case of multiple regressors. $Var(x) = \sum_i^n(x_i-\bar{x})^2 = \frac{1}{n}\sum_i^nx_i^2 - \bar{x}^2 = \frac{1}{n}\sum_i^nx_i^2 - \frac{...
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Variance of a hypergeometric distribution

I'm trying to answer the following question from Ross's book: A pond contains 100 fish, of which 30 are carp. If 20 fish are caught, what are the mean and variance of the number of carp among the 20? ...
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Convergence of covariance matrix

I was looking for a simple way to find the number of samples $n$ needed to get a decent approximation to the covariance matrix $\boldsymbol{\Sigma}$. Given a random sample $\{ \mathbf{X}_1,\mathbf{X}...
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1answer
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Usage of sample covariance and sample mean

I understand the difference between sample mean/covariance and population mean/covariance and how to calculate them. However, I'm a bit unsure about what happens afterwards. If I only have the sample ...
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Misunderstanding of time series autocovariance

I'm reading the "Time Series: Theory and Methods (2nd ed.)" by P.J.Brockwell and R.A.Davis. I've stopped at the one moment at pp.218-219 (Chapter 7 "Estimation of the mean and the Autocovariance ...
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Using coeftest results in predict.lm() in R [closed]

I am analyzing a dataset in which the variance of the error term in my regression is not constant for all observations. For this, I re-built the model, estimating heteroskedasticity-robust (Huber-...
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Method of moment through covariance derivation

Given a Bivariate INAR(1) Poisson Process: $Y_t^1 = \rho_1 * Y_{t-1}^1+R_t^1$ $Y_t^2 = \rho_2 * Y_{t-1}^1+R_t^2$ Where $R_t^1$ and $R_t^2$ are the innovation terms and follow the bivariate Poisson ...