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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Correlation of multivariate distributions without "slope"

Wikipedia has this image showing different correlations: It says: The correlation reflects the noisiness and direction of a linear relationship (top row), but not the slope of that relationship (...
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Demonstration and Interpretation between a Fisher matrix and its dual space which is covariance matrix

I have a simple (maybe not) issue about the interpretation of the link between Fisher information matrix and its inverse which is the covariance matrix. How to formulate that a line of Covariance ...
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The Variance Covariance Matrix of an Estimator Stacking Two OLS Estimators

I am looking for how to derive the variance covariance matrix (henceforth, VCOV) of an estimator stacking two OLS estimators. Suppose that we have two OLS estimators: $$\hat{\alpha}\sim N(\alpha,\;\...
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When does $cov(X,Y)\neq 0$ imply $cov(XZ,ZY)\neq 0$?

For random variables $X$,$Y$,$Z$, when does $cov(X,Y)\neq 0$ imply $cov(XZ,ZY)\neq 0$? It would be great if you could provide some sufficient conditions or give some special cases such that this hold.
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Estimate the mean vector and the covariance matrix using the simple returns

I would appreciate help with how to to estimate the mean vector and the covariance matrix using the simple returns in R. I have historical (weekly) values of five stocks from a capital market for a ...
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Relationship between $Var(X)$, $Var(Y)$ and $Cov(X,Y)$ for random variables with zero mean

I have two correlated random variables $X$ and $Y$, both with zero mean. Are there any relationship/constraints between $Var(X)$, $Var(Y)$ and $Cov(X,Y)$, apart from the obvious $Var(X) > 0$ and $...
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Use of PCA to model variance for dependent variables

I am working on a math problem with some friend and there is some disagreement on the meaning of what we are doing. We have 3 independent variables measured tens of thousands times and we have a model ...
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Find the principal component and the proportion of the total population variance explained by each when the variance covariance matrix is given

I can understand the part where we have to find the principal component from the variane covariance matrix- find eigen values, make eigen vector and normalise. The principal component would be ...
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What is the meaning of the inner product between two regression variables?

I have been analyzing the effect of design matrix columns on the contour line of the least squares regression. These contours obviously are ellipses when only two columns $\phi_1$ and $\phi_2$ are ...
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Covariance of Least Squares Estimators

I am trying to solve the problem shown below: To solve (a), I defined the sum of squared errors $f(\hat{\beta}) = \sum_{k = 1}^{n} (y_k - \hat{\beta})^2$. This allows us to identify the least-squares ...
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Covariance of a convolution between a gaussian random walk and white noise [closed]

I want to compute the covariance of $$U_t:=\sum_{l=-L}^{l=L} (X_l-X_{l-1})X_{l-t}$$ with $X_t$ defined as : \begin{align*} X_0&=0 \\ X_t&=X_{t-1}+\epsilon_t \end{align*} $t=1,2,...$...
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Covariance of two random variables which share a constant component

Say I had two random variables, X and Y. I also have variables A and B, where: A = X + Z; B = Y + Z; and Z is a constant > 0 Am I correct to say cov(X,Y) = cov(A,B)? I'm certain I am overthinking ...
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Why are Kernels said to be a measure of Covariance? [duplicate]

I have always heard that Kernels are said to be a measure of Covariance - intuitively, I can somewhat understand why this argument is being made; however, I would like to confirm if my understanding ...
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Mean term in simple linear regression

I am trying to derive the expression for the $E(y_i \epsilon_i)$ in simple linear regression. I substitute using $Cov(X,Y) = E(XY) - E(X)E(Y)$, so $E(y_i \epsilon_i) = Cov(y_i , \epsilon_i)- (E(y_i)...
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Interpretation of GLM models with covariance

Interpretation of both GLM-models. How is cognitive function associated to PTSD symptom severity outcome over time between the two groups? Groups exists of treatment and control group. Because of a ...
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How does covariance stationarity even exist?

I've been wondering recently about covariance stationarity. Say we have a stationary series with statsmodels' ADF and KPSS results: ...
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Covariance based SEM and GOF indexes

I have some questions all related to SEM models. I am just estimating a SEM model in R using the function sem. The model looks like this: ...
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Covariance of a sample statistic for two independent bivariate random variables

I have a somewhat convoluted question here. Suppose I have paired random variables $X$ and $Y$. That is, when I draw samples, I get one instance of $X$ and an associated instance of $Y$. Then I can ...
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How is $\text{Cov}(\bar{Y}, Y_i - \bar{Y}) = \dfrac{1}{n^2} \text{Cov} \left( \sum_{j = 1}^n Y_j, nY_i - \sum_{j = 1}^n Y_j \right)$?

I have this example of sufficiency: Let $Y_1, \dots, Y_n$ be i.i.d. $N(\mu, \sigma^2)$. Note that $\sum_{i = 1}^n (y_i - \mu)^2 = \sum_{i = 1}^n (y_i - \bar{y})^2 + n(\bar{y} - \mu)^2$. Hence $$\...
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covariance of lognormal random variables

I am trying to find the variance of b*log(x+y) - log(x), where x and y are independent and identically distributed lognormal random variables, the range for log(x) and log(y) is negative infinity to ...
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Cross-covariance in context of Andrews plot

As shown in this Cross-Validated post Close curves on an Andrews plot I don't understand how, in the accepted answer, the cross-covariance can be defined as, $$\int_{-\pi}^{\pi}f_xf_ydt$$ Considering ...
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ANOVA or ANCOVA? I have a control variable

In my study, I am analyzing 4 groups, as it is a 2 (primed or not primed) × 2 (high efficacy or low efficacy) experimental design. I am planning on using ANOVA between-subject analyses to test for ...
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Is there any way of simplifying the covariance of a function of a random variable and a random variable?

For example, if we had random variables $X$ and $Y$ and we know that $corr(X,Y)=\rho$, how would you solve for $Cov(e^X,Y)$?
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Finding the correlation between comments within the same thread

I have a collection of data of numerical ratings of comments from threads on a website. I want to determine how strongly these ratings are correlated with other comments within the same thread. More ...
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If $(X_t)$ is a Gaussian process, what is $Cov(X_s^2,X_t^2)$?

Let $(X_t)_{t \in \mathbb R^+}$ be a Gaussian process $\mathcal N(\mu(t), \sigma^2(t))$ with covariance function $K$. Let $s \leq t$. Can we express $Cov(X_s^2,X_t^2)$ in terms of $K, \mu(t), \sigma^2(...
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Covariance of ratio of dependent variables?

I am trying to use the Delta method (Please have a look at this link) to compute the covariances between the ratios of random dependent variables. I have 7 dependent variables $A_i$, $i\in\{1,2,3,4,5,...
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Justifying the choice of covariance function

For a Gaussian process I run I have chosen to use the Matern52 covariance function, as from research this is practical to use for physical data as it does not overally smooth the function. However is ...
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When to add covariances of mediators [SEM]

I have just created my first mediation model using sem() with the lavaan package in R. I am using a bootstrapping with 5000 resamples and BCA to calculate the confidence intervals at a 0.9 level. Now, ...
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Confusion on calculating Mahalanobis distance

I am slightly confused as to how you calculate Mahalanobis distance given a set of data. I have tried asking my tutor for help but he does not seem interested in helping what so ever and I am ...
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mean and covarince matrix of AR(1) [closed]

assume I have a price data called pt, I fitted AR(1) model p_t= alpha + beta pt_1 + e_t , ...
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Formula for the variance of the product of two random variables [duplicate]

Let's say I have two random variables $X$ and $Y$. Can I write that: $$VAR \left[XY\right] = \left(E\left[X\right]\right)^2 VAR \left[Y\right] + \left(E\left[Y\right]\right)^2 VAR \left[X\right] + 2 \...
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R function to compute variance of average of correlated random variables

I want to calculate the variance of the average of n correlated variables. I found a formula for that in Borenstein et al. (2009) Introduction to Meta-Analysis. $$\operatorname{Var}\left(\frac{1}{m}\...
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Building Covariance Matrix From Samples and not Features

In most of the applications, I see that the covariance matrix is calculated on feature space, by first applying normalization steps on $X$ then the covariance is calculated by $C = X^{T}X$ where $X \...
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Relationship of covariance and linear regression in context of rank deficiency

Beforehand Usually, we can get intercept and slope of a linear regression from numbers we get from a covariance analysis of same variables. This goes like this: ...
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some thought about independence and orthogonal, please comment on this if it's wrong

It seems that linearly independent is totally different from independent of random variable concept. Non-zero vectors Orthogonality must imply linearly independence. In Statistics, the relation of ...
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What is the relationship between the residuals of an objective function and the uncertainties of the minimizer values?

Consider I have some optimization problem and an objective function $f(x, y, z)$. $f$ is defined using the sum of squared residuals, i.e. for some function $g$, we have $f(x, y, z) = \sum[g(x, y, z) - ...
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Calculating $Cov\left(\overline{Y}_j,\:\overline{Y}\right)$ for a basic one-way model

Consider the basic one-way model: I want to show that $Cov\left(\overline{Y}_j,\:\overline{Y}\right)=\frac{\sigma ^2}{na}$. I derived the following expected values: $$E\left(\overline{Y}_j\right)=\mu ...
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Covariance structure of multilevel random coefficient model

I'm currently trying to follow this tutorial on multilevel random coefficient modelling. It says Finally, a homogenous residual covariance structure is specified, but it is allowed to be different ...
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Correlation of two random variables as functions

Consider 2 random variables $X_1$ and $X_2$: $X_1 = Y_1 \sqrt{\alpha + \beta {X_0}^2}$ $X_2 = Y_2 \sqrt{\alpha + \beta {X_1}^2}$ $ \alpha>0 $ , $\beta \ge 0$ where $Y_1$ and $Y_2$ are independent ...
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Factor Analysis does not give a better covariance estimate than the Empirical Covariance matrix?

I do not see that Factor Analysis gives a better covariance estimate than the empirical covariance estimate, from the toy data simulation with explanation and code below. Am I doing something wrong? ...
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SEM: How to report latent factor relationships (covariance vs. correlation)?

I would like to report results from SEM I did in R using the lavaan package. I have gotten increasingly confused about how papers use different terms and units to describe latent factor relationships ...
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What is the correlation between a random variable and its probability integral transform?

Are there known bounds on the $\operatorname{cor}(X,F(X))$? $X$ is a random variable with CDF $F(X)$. Let $X$ have a fixed variance, for example $\operatorname{var}(X)=1$. What $X$ can maximize or ...
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GEE Explained Variance is Negative

I am creating a series of GEE models and a couple of statistics I normally report are the ICC and the explained variance for both Level 1 and Level 2 of the model. However, I am calculating that for ...
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Show that $\frac{1}{n}\sum_{i=1}^n (x_i -\hat{\mu}^\top)\cdot(x_i -\hat{\mu}^\top)^\top=E[(X-\hat{\mu})(X-\hat{\mu})^\top]$ [closed]

In a lecture my lecturer used the fact that $$\frac{1}{n}\sum_{i=1}^n (x_i -\hat{\mu}^\top)\cdot(x_i -\hat{\mu}^\top)^\top=E[(X-\hat{\mu})(X-\hat{\mu})^\top]$$where $X$ is our data matrix and $\hat{\...
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How to turn many variables into one for the PACF equation

I have been trying to calculate the PACF manually, but I encountered some issues with the following equation: $PACF = \frac{Covariance ([Y_{t}|Y_{t-1}, Y_{t-2},...,Y_{t-k+1}],[Y_{t-k}|Y_{t-1}, Y_{t-2}...
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What is the analog of precision matrix for cross-covariance matrices?

For a covariance matrix, I am aware of the precision matrix, the covariance matrix inverse. What's the analog for that for a cross covariance matrix, i.e. $E[XY^{\top}]-E[X]E[Y^{\top}]$ for two random ...
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Definition of autocorrelation?

EDIT: Apparently I was not thorough enough in my search for previous questions on this topic. This is pretty much the same question, and has been answered: Auto-correlation vs Auto-covariance The ...
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Are my sequential random numbers correlated?

Say I want to generate $n$ random numbers $Y_i, i = 1, ... n$, where $X_i \in [1,..n]$ is my i.i.d. uniform random source and $Y_i$ are drawn sequentially: $Y_0 = X_0$ $Y_i = (Y_{i-1} + X_i)\; \text{...
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How to show $X \text{~Uniform}[-1,1] $ and $Y=-X$ when $X\leq 0$, $Y=X$ when $X \geq 0$, then $Y \text{~Uniform}[0,1] $?

Told to show that: if $X \text{~Uniform}[-1,1] $ and $Y=-X$ when $X\leq 0$, $Y=X$ when $X \geq 0$, then $Y \text{~Uniform}[0,1] $. [where X,Y are continuous random variables] I can see why it holds ...
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Multivariate Regression with Two Different Types of Response

Problem Setting: I have an interesting question related with longitudinal study and multivariate regression. I found that in lots of biomedical studies, multiple discrete and continuous endpoints are ...
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