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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Covariance matrix of autoregressive process
I am learning about autoregressive processes and there is something that I find unclear about the structure of their covariance matrix. Some sources (e.g. Box and Jenkins, 2016) describe the ...
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Estimating covariance matrix of irregularly updated time series
I would like to estimate the covariance matrix of returns from a set of time series that don't get updated regularly. To be precise, in my case all of the series fall into 2 classes. Class A gets ...
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Dealing with covariance
Within my dataset, I employ a variable termed "Instagram reach," quantifying the audience size exposed to a particular post. Simultaneously, "engagement" denotes the count of ...
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Endogeneity Analysis without the access of raw data?
I currently have the correlation/covariance matrix for a set of variables, as well as the output from a regression analysis, but lack access to the underlying raw dataset. Given these constraints, ...
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Correlation between variable and variable conditioned on the sign agreement [closed]
Suppose I have two variables X1 and X2, following a bivariate standard normal distribution with a correlation coefficient of 0.2, if I create a new variable X3 that is equal to X2 if X1 and X2 have ...
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Is $E((\Sigma_i^n u_i)^2) = 0$ or $n\sigma^2$ (in OLS)?
Consider an OLS estimator,
$$y= \beta_0 + \beta_1 x_i + u_i $$
I think $E((\Sigma_i^n \: u_i)^2) $ is equal to zero because simply $\Sigma_i^n \: u_i$ is always zero.
But my professor in class showed ...
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Including covariates in simultaneous vs. hierarchical regression
If you are computing a regression, want to know if specific IVs predict the DV when controlling for covariates (one included because age was different between IV groups, another because it is ...
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Clarification on covariance matrix for multidimensional Gaussian distributions
It is a well known property of Gaussian distributions that if $Y = (Y_1, \ldots, Y_n)$, where each $Y_i$ is a real Gaussian random variable, then the components of $Y$ are independent if and only if ...
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Parameter estimation of a WSS process
As my research revolves around parameter estimation from signals that evolve in time in a random fashion, I am curious to know what features/ retrievals people normally use to determine the parameters ...
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Covariance of Best Linear Unbiased Estimators and arbitrary LUE
I'm working on a problem involving two linear unbiased estimators $T$ and $T'$ of a parameter $\theta$, defined from a sample $\{X_1, \dots, X_n\}$ with mean $\theta$ and finite variance. I aim to ...
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Is the Between-Groups Variance a Covariance?
I am currently working through a book/class in quantitative genetics, and in Falconer and Mackay's Introduction to Quantitative Genetics, the following line stumped me:
"The between-group ...
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Can I covary exogenous and endogenous latent variables in SEM in lavaan
I am editing the question as it was too meandering. I am sorry about that.
I am running an extended Theory of planned behavior (TPB) model of digital piracy and I have a question that boils down to - ...
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Decompose covariance matrix into uncorrelated and correlated part
I have an almost diagonal covariance matrix, I would like to decompose it in an uncorrelated and correlated part:
$$
\Sigma = \Sigma_U + \Sigma_C
$$
where all the matrices above are covariance ...
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Adding and interpreting covariates in logistic regression
I have a dataset and I want to do a logistic regression between the continuous variable "A" and the categorical variable "B". However, I also wanted to include "age" and &...
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Matern covariance increasing with smoothness
Consider the Matern covariance function with the following parametrization:
$$C_{\nu,\phi,\sigma}(h) = \sigma^2 \frac{2^{1-\nu}}{\Gamma(\nu)}\left(\sqrt{2\nu}\frac{h}{\phi} \right)^\nu K_\nu (\sqrt{2\...
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Prove that the equality holds [closed]
How to prove that for any random variables $X$, $Y$ and $Z$ with finite variances, we have $Cov(X,Y)=E(Cov(X,Y|Z))+Cov(E(X|Z),E(Y|Z))$?
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Imaginary numbers in PCA output
Using PCA manually on correlation matrix, I'm getting imaginary numbers in both eigenvalues and eigenvectors. Is this expected behavior?
I understand that when interpreting a matrix as a linear ...
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If $A^2$, and $B^2$ are DEPENDENT random variables, will $A$, and $B$ be necessarily DEPENDENT too?
I know that if $A$, and $B$ are independent, the independence is preserved for $A^c$, and $B^c$, where $c$ is a constant. I am wondering if the same applies to the case where the random variables are ...
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Is this formula the calculation of covariance matrix
I came across this formula in a text that says $S$ is the sample covariance matrix where
$$S = \sum_{j=1}^n(\mathbf{X}_j - \bar{\mathbf{X}})(\mathbf{X}_j-\bar{\mathbf{X}})'$$.
What I am trying to ...
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Literature on GP with kernels having no closed form
I have to use GP regression on a complex time series, and the kernel function is not known in closed form. I have found a numerical approximation with the Gauss-Laugerre quadrature. It takes the ...
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Multivariate sample covariance
I have a set of $X_1,...,X_n$ samples with covariance $\Sigma_1,...,\Sigma_n$.
The multivariate sample mean is then $$ \left(\sum_{i=1}^n \Sigma_i^{-1} \right)^{-1} \left(\sum_{i=1}^n \Sigma_i^{-1} ...
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Cartesian precisions from spherical standard deviations
I have to simulate a simple sensor, which has 3 standard deviations defined in spherical frame : sigma-azimuth, sigma-elevation, sigma-distance.
When I simulate a detection, I compute a noisy position ...
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Question about expanding a vector form of covariance of two r.v
I am trying to proof why the two lines of the following expression are equal. I found it in one of the books on machine learning
$$
\begin{align}
\operatorname{Cov}[\mathbf{x},\mathbf{y}]
&= \...
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Joint density of two functions of a uniformly distributed random variable
I'd like to work out $\operatorname{Cov}(\cos(2U), \cos(3U))$ where $U$ is uniformly distributed on $[0, \pi]$.
I believe this involves computing $\mathbb{E}[\cos(2U)\cos(3U)]$. If so, then I first ...
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If Cov(X,Y)=Var(Y), what is the dependence between X and Y?
In a problem I have found that
$$Cov(X,Y)=Var(Y),$$
where $X$ and $Y$ are random variables.
What can I conclude on the linear dependence between $X$ and $Y$?
Thank you!
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What is the appropriate normalization for finding correlations between Poisson distributions? [closed]
I am interested in using this algorithm, glm-pca, to find a lower dimensional embedding in time series data, specifically neuronal spiking data, which is Poisson distributed. I have looked at some ...
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Conditions of the covariance matrix between discrete and continuous variables
Does the covariance matrix for a discrete variable and a set of continuous variables have extra constraints beyond being positive semi-definite as in the case of a real-valued random vector?
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If $corr(A,B)=1$, and $cov(A,C) \neq 0$, do we have $cov(B,C)\neq 0$ as well?
If $corr(A,B)=1$, and $cov(A,C) \neq 0$, do we have $cov(B,C)\neq 0$ as well?
I understand $cov(A,B) \neq 0$ and $cov(A,C) \neq 0$ don't imply $cov(B,C) \neq 0$. And I am looking for a counter-example ...
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What are the mechanisms for the propagation of effects across stages in a structural model?
I have an SEM question to which I have not found an answer anywhere. I have used SEM a bit already but my difficulties with this question suggest I lack some basic understanding.
Question:
In a ...
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Determinant of High Dimensional Correlation Matrix
According to [1] :
The determinant of the correlation matrix will equal 1.0 only if all correlations equal 0, otherwise the determinant will be less than 1 [...] When the measures are uncorrelated, ...
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Covariance calculation in a DiD estimation
I am estimating a difference-in-difference model estimating the effect of a parental leave reform on female wages. It is not possible to take the logarithm of the varaibles as a lot of wages are 0.
I ...
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Is the expectation of a random vector multiplied by its transpose equal to the product of the expectation of the vector and that of the transpose
I'm taking a course in advance statistics and we have to prove whether the following expression is true: $E[zz^T]=E[z]E[z^T]$. I am assuming it is not, since the formula of the covariante matrix is $...
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Recover Variances From GLS Model (Phylogenetic Least Squares on Evolutionary Tree)
I don't know how to calculate the variance of a variable when all of its observations have an arbitrary correlation structure.
I am simulating the evolution of animals as they branch apart into ...
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Proof of general linear process autocovariance
I am struggling to get to the general formula of the general linear process autocovariance.
If $Y_t = \mu + \sum_{k=0}^\infty \omega_k e_{t-k}$ where $e \sim WN(0,\sigma_e^2)$ (a.k.a. the general ...
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Is this the formula for the conditional covariance of normally distributed random variables? [duplicate]
Assuming $X$, $Y$, and $Z$ are normally distributed random variables, is it true that:
$Cov[X, Y | Z] = Cov[X, Y] - Cov[X, Z]Cov[Y, Z] / Var[Z]$
Could you provide a simple derivation?
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If A and B are correlated, what is the correlation between |A1 + A2| and |B1 + B2|?
To preface, let random variables $X = A_1 + A_2$ and $Y = B_1 + B_2$. $A_1$ and $A_2$ are copies of $A \sim N(0, 1)$, and $B_1$ and $B_2$ are copies of $B \sim N(0, 1)$. In this situation, if $A$ and $...
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On the estimated formula of covariance of two random variables
We define the covariance of two random variables $X$ and $Y$ as $Cov(X,Y) = E[(X - \mu_X)(Y - \mu_Y)]$. The covariance measures the "linear dependence" between the two r.v s.
But in a lot of ...
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How to sample covariance matrices? [duplicate]
Is there a function (preferably in python ecosystem) to sample covariance matrices?
I’m aware of an LKJ prior, so I might use this. (I believe this is a prior in correlation not covariance but the ...
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Minimum Pearson's correlation between $X$ and sign($X$)$\cdot X^2$
Suppose $X$ is a (whatever bounded or not) continuous random variable ($X$ is not constant) with an arbitrary distribution. Is it possible to construct a distribution s.t. $\text{Corr}(X, \text{sign}(...
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Add covariates to the random-intercept mixed effect model
I'm working on a study where I want to explore whether two covariates, let's call them C and D, can help explain the differences we observe in our dependent variable, which we'll call B (depression ...
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Can a sequence with increasing variance satisfy weak stationarity
In Section 3 of Page 3, the notes that one of the conditions for weak stationarity is that $\gamma_{X}(h)=Cov(X_t, X_{t+h})$, essentially that the covariance of $X_s$ and $X_t$ depend only on $t-s$ ...
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Betas from precision matrix?
This is really 2 questions.
I would like to do a linear multi-regression on a large quantity of data, so large that I cannot really store it (it’s about 1e10 observations across 2500 features). Hence ...
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Does $\operatorname{Cov}(X,Y)>0$ imply $\operatorname{Cov}(X^2,Y^2)>0$ if $X$ and $Y$ are positive random variables?
Does $\operatorname{Cov}(X,Y)>0$ imply $\operatorname{Cov}(X^2,Y^2)>0$ if $X$ and $Y$ are positive random variables?
My intuition would say yes since if $X$ increasing means $Y$ is likely to ...
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Is $P(\frac{X}{Y}<a)$ greater when $Cov(X,Y)>0$ than when $Cov(X,Y)=0$ if $X$ and $Y$ are positive random variables?
As the question says, is $P(\frac{X}{Y}<a)$ greater when X and Y have positive covariance than when they are independent (both positive random variables)?
My intuition is that if Y is likely to ...
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Weighted least squares estimator variance using noisy weights
I have a linear system with uncorrelated, heteroscedastic noise, $Y \propto \mathcal{N}(Xβ,Σ)$ where $Σ$ is a diagonal matrix with elements $σ_{ii}^2$. The MLE is given by weighted least squares (WLS) ...
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Is there anything interesting to be taken from the fact that $E[(X-E[X])(Y-E[X])] = E[(X-E[X])(Y-E[Y])]$?
While playing around with the formula for covariance, I discovered something I wasn't expecting. Replacing the $E[Y]$ in the definition of covariance with an $E[X]$ appears to simplify back down to ...
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Weighted average of covariance matrixes
My issue is as follows:
In my model there are 4 different states, which each have a calculated probability of happening. I also have calculated covariance matrixes for my variables in each of these ...
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What affects correlation in this situation?
Suppose a dataset I with 20 samples pairs of x and y collected from 20 different farms in State A.
Another dataset II with 20 samples pairs of x and y collected from 20 different farms in State A, B, ...
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Is there a simpler proof than mine for this obvious proposition about correlations?
$\newcommand{\e}{\operatorname E}$"Obviously" if $g$ is a weakly increasing function and $X$ and $g(X)$ are both random variables with finite variance, then the covariance (and hence the ...
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Expected Squared Error Derivation of Least Square Model (in The Elements of Statistical Learning Book)
I'm studying chapter 2.5 of The Elements of Statistical Learning by T.Hastie.
Here, they assume the ground truth relation between Y and X as
$$
Y=X^T\beta +\varepsilon,
$$
where $\varepsilon\sim \...