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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In simple linear regression, what is the covariance between the error term and the residual?

In simple linear regression, what is the covariance between the error term and the residual? Model: $y_i = \beta_0 +\beta_1 x_i + \varepsilon_i$ What will be the $\rm {cov}(\varepsilon_i,\ e_i)$, ...
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1answer
39 views

Which error is displayed in an error ellipse?

I have some bivariate data and I have calculated the error ellipse in the following way: I have first calculated the covariance matrix and then to obtain the radii of the ellipse I have taken the ...
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43 views

Variance and covariance notation: $\sigma^2 V_1$, $\sigma^2 V_2$

I am reading an internal paper that says: Let $\sigma^2 V_1$ equal the variance of $\sum_{m\in M}Z_m - Z_0$ and $\sigma^2V_2$ equal the covariance of $||M||^{-1}\sum_{m\in M}Z_m - Z_0$ and $Z_m ...
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12 views

Existence of co-variables

In a study, there are 3 independent variables (A, B, C). I want to manipulate only (A, B). The idea of not manipulating variable C is for it not to interfere with the other two manipulated variables ...
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15 views

In simple linear regression, why the covariance between y bar and beta1 hat is zero? [duplicate]

In simple linear regression, where the model $$\mathbb{E}(y) = \beta_0 + \beta_1 x$$ is estimated using least squares (and the errors are assumed iid of mean $0$), why is the covariance between the ...
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1answer
27 views

Covariance and independence for Student-t distribution

It is well-known that $Cov(X,Y)=0$ means independence if $(X,Y)$ is bi-normal distributed, would this be true for Student-t distributions? If not, could anyone give an counter-example. Great thanks!
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32 views

Parameter uncertainty after non-linear least squares estimation

I've fit a system of non-linear ODE to some experimental data using Levemberg-Marquardt. After the algorithm converged, I estimated the Hessian matrix of the system using: $H = (J^TJ)$ The ...
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1answer
83 views

Covariance term in simple linear regression

I am trying to derive the expression for the variance of $\hat{\beta_0}$ in simple linear regression. I substitute $\bar{y} - \hat{\beta_1} \bar{x}$ for $\hat \beta_0$, but in the intermediate steps ...
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1answer
53 views

Validating Correlation in Excel

I have been using CORREL() function in excel for considerable amount of time; I just wanted semi-manually calculate correlation for fun, and the the value I got did not match the value I'am getting ...
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16 views

Estimation of a vector with a big covariance matrix

I have a Gaussian vector with a known covariance, given by a Covariogram (covariance function). Inverting this matrix (lets say of size 5000x5000 and above) is not reasonable. Is there any known ...
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13 views

Why inverse of the sample covariance matrix is used in Mahalanobis distance calculation? [duplicate]

The Mahalanobis distance of an observation vector $X$ from a set of observations is defined as $d=\sqrt{(\vec{x}-\vec{\mu})^T S^{-1} (\vec{x}-\vec{\mu})}$ where $S$ is the sample covariance matrix and ...
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16 views

Difference between summing and multiplying covariance matrices?

Say we have an RBF covariance matrix A and some periodic covariance matrix B for a given dataset. Covariance matrix A says that you believe that points that are close together are somewhat similar, ...
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11 views

Relation between cross covariance and cross variogram

I'm trying to reproduce a derivation of the cokriging equations that involves the cross covariance and the cross variogram. This derivation is included in the Multivariate Geostatistics by ...
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1answer
55 views

Does A independent of B, and B correlated with C imply that C is independent of A?

Assume 3 random variables, $A, B, C$. If $A \perp B$, but $Cov(B,C) \neq 0$, can we say anything about $Cov(A,C)$? I think it can either be 0 or not 0, but it seems like there should be more general ...
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16 views

Why did fixing the covariance value cause AMOS to mis-estimate variance?

I have 100 data points on two variables, a and b. The correlation between the two is .3 and the SD is 1. When I run the ...
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2answers
115 views

Can a multivariate distribution with a singular covariance matrix have a density function?

Suppose a multivariate distribution over $\mathbb R^n$ has a singular covariance matrix. Can we conclude that it does not have a density function? For example, it is the case for the multivariate ...
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0answers
51 views

In a linear regression, why is the mean of all $y_i$ equal to 0?

I don't understand why the mean for all $y_i$s must equal $0$. What property/properties does the above assertion rely on? I had thought that the mean of $y_i=E(y_i)$, so: $$ E(y_1)=\alpha $$ $$ ...
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36 views

How many samples are needed to estimate a p-dimensional covariance matrix?

In general, how many points are needed to estimate a p-dimensional covariance matrix? Does it depend on how the data are spread out across the different dimensions? Does it depend on the true ...
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1answer
33 views

statsmodels: error in kde on a list of repeated values

I am using Python's statsmodels module to plot a violin/bean plot of some data. I get the error "LinAlgError: singular matrix" in the KDE calculation whenever a single violin plot is drawn from a list ...
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1answer
46 views

Understanding covariance of errors in regression

I am having a hard time understanding the elements of an error covariance matrix for a class. Can someone clarify? First, the diagonal. The variance is $E(e_i^2) - E(e_i)^2$. $E(e_i) = 0$, so it's ...
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2answers
62 views

If x = y*y, and you know var(y), var(z), and cov(y,z), do I know cov(x,z)?

If I know that x = y*y, and I know a whole of statistics pertaining to y, such as the variance and its covariance with other variables, can I analytically solve for the variance and covariance of x? ...
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15 views

Covariance of matrix with element entries as vectors

Suppose there is a matrix A with m rows and n columns. Each element is a 3 dimensional vector. This 3 dimensional vector have values of different orders. Can we calculate covariance of such matrix ?
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1answer
19 views

Variance of an autocorrelated random variable two periods in the future with Bayesian updating

I observe draws of some random variable $Y$ over time where $Y_{t} = aY_{t-1} + \epsilon_{t}$. $\epsilon \sim N(0, 1/\rho_\epsilon)$ and $a$ is an unknown parameter with prior distribution $a \sim ...
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21 views

how to prove this kernel function is positive semidefinite

How to prove $k(x_i,x_j)=e^{-(LR(x_i-x_j))^TLR(x_i-x_j)}$ is a valid kernel function or positive semi definite? $x=(\mu,\lambda)^T$ and R is a 2x2 rotation matrix, L is a 2x2 diagonal scaling matrix ...
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36 views

Correlation of fixed effects with multiple response variables in MCMCglmm

I'm working with a mixed model for which I have several response measurements for every individual. One goal is to determine the sampling variance/covariance of the fixed effect estimates for a ...
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1answer
23 views

$\operatorname{var}[\frac{b}{a} B(a-b)-b B(b)]$ with $b\leq a$ and $b\geq 0$; $B=$brownian motion

I want to calculate: $\operatorname{var}[\frac{b}{a} B(a-b)-b B(b)]$ with $b\leq a$ and $b\geq 0$; $B=$brownian motion. I started like this: $(\frac{b}{a})^2 \operatorname{var}[ B(a-b)]+-b ^2 ...
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28 views

Relationship between $\text{Cov}(x_i^2, e_i^2)$, the asymptotic variance of b under homoscedasticity and heteroscedasticity?

I am trying to figure out the relationship between $\text{Cov}(x_i^2, e_i^2), V$ and $V_0$, where: $V=$ asymptotic variance of $\sqrt{n(\hat{\beta}-β)}$ under heteroskedasticity, and $V_0=$ ...
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1answer
36 views

Changing a multivariate Gaussian along a dimension

I have a multivariate Gaussian parameterised by a mean vector $\mu$ and a precision matrix $\Sigma$. Now, I want to set the Gaussian along a given dimension $i$ to a point mass i.e. I set the ...
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64 views

Marginal distributions of off-diagonal terms in a Wishart-distributed random variable

I am interested in finding expressions for the marginal distributions of the off-diagonal terms in a Wishart-distributed random variable. More specifically, suppose $X$ is an $n \times p$ matrix, ...
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77 views

Multivariate logistic distribution

The normal distribution can be generalized into the multivariate normal distribution. Can the logistic distribution also be generalized into a similar multivariate distribution? Is there a ...
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20 views

Finding the covariance matrix to find the best linear predictor (AR(1) model)?

I need to find the covariance matrix of two given estimates of an AR(1) model $$X_t = \phi X_{t-1} + Z_t$$ to find its best linear predictor of $X_2$, given $X_1$ and $X_3$. Let W = ($X_1, X_3$)' ...
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18 views

Covariances between forecasts from different time-series models

I'm trying to compute the variance of an average of forecasts. But I'm not sure on how to get the covariances required to compute the variances. Here is the situation: Three ARMA(p,q) models for a ...
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1answer
52 views

Inverse of Covariance Matrix

I'm working on some code to find the inverse of a covariance matrix of arbitrary dimension. Obviously, the matrix will always be symmetric, and I think it's that fact that I need to take advantage of ...
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19 views

Covariance between two sample means of correlated data

I have two sets of random data $X=\{x_1,...,x_N\}$ and $Y\{y_1,...,y_N\}$ both of length $N$. The sets are autocorrelated such that the correlation between $x_i$ and $x_j$ depends only on $|i-j|$. ...
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10 views

Increasing function defined as the covariance bivariate normal

Suppose $c>0,\sigma>0$ and $\tau>0$ are fixed real constants. Then I'd like to prove that the function $g_c:(-1,1)\mapsto\mathbb{R}$ defined by \begin{equation} ...
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1answer
46 views

Splitting covariance Cov[x,y+z]

Can I write $\text{Cov}[x,y+z]=\text{Cov}[x,y]+\text{Cov}[x,z]$, where $\text{Cov}(.)$ is referring to the population covariance? $x,y$ and $z$ are random variables. (very fundamental question... :) ...
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13 views

generating covariance matrices from multiple priors

In many optimisation problems, one typically uses many forms of regularisations over the parameters that is being estimated. For example, a typical cost function (to maximise) may look like this: $$ ...
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1answer
33 views

precision vs covariance

I have seen that when modelling the likelihood term with a multivariate Gaussian tend to parameterise the Gaussian with the inverse covariance matrix (precision) rather than the covariance matrix, ...
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27 views

Covariance with a common term

So I've been digging into the math of correlations and covariances and one of my articles mentioned finding the covariance by using the common component, specifically with regards to, say, $cov(A + B, ...
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25 views

Interpretation of posterior variance in gaussian regression

I'm trying to understand how the posterior variance of a Gaussian regression depends on the datapoints. Suppose $y=X\beta + e$, with Gaussian priors over $\beta$ and $e$ and known variance of $e$. ...
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87 views

correlation between adjacent pixels in an image

I would like to know what is the formula for computing the correlation between adjacent values in a 1-D signal. I am reading a paper where there is a formula for converting this to the FWHM of the ...
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1answer
96 views

For PCA prove that covariance matrix is diagonal?

For $z=W^\mathbf{\intercal}(x-m)$ where $k$ columns of $W$ are the top $k$ principal components of a given dataset $x$ and $m$ is the mean, how can I prove that $z$ is diagonal? $x$ is the original ...
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129 views

What are differences between covariate and covariance ?

The terms covariate and covariance appear to mean one and same thing. Is covariance equal to covariate?
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28 views

What is the definition of covariance under the fixed-effects assumption for a meta-analysis of correlation?

There is a doubt about the meaning of covariance and its definition. Does it change under different conditions such as primary data or statistic such as r in case of meta-analysis?
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1answer
52 views

Help with understanding this covariance setup

I have been reading a paper that formulates the problem of image registration as a generative model and I have been having a lot of trouble understanding some concepts and I was wondering if someone ...
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1answer
65 views

Meaning of covariance matrix row sums

Say I have an $n \times n$ covariance matrix for a sample set of $n$ random variables. Is there any meaning if the sum of the rows of this matrix? Is it a meaningful measurement of the contribution ...
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1answer
53 views

Is this the right way to compute covariance-matrix in parallel?

I need to compute a 600x600 covariance matrix, each factor has a sample size of 750. I have a decent cluster computers, so I am thinking about doing the calculation in parallel. The dumb way I am ...
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48 views

What is the distinction between covariance and condtional sampling variance for two random variables in case of meta-analysis of sample correlations?

It seems (I have a vague notion) that there is some similarity between the two measures. Alternatively stated, there is certain relationship between the two? could you please clarify the idea
2
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1answer
69 views

In what situations is covariance preferred to correlation?

If the magnitude of covariance doesn't really convey any reliable information, and correlation provides us the same information but also gives the degree of dependence, then why use covariance at all ...
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28 views

Does (matrix) t-distribution have a conjugate prior?

If my data Y follows matrix T-distribution, i.e. $ Y \sim T_{n,p}(\nu,0,\Sigma,\Phi)$, can I find an appropriate prior for $\Phi$ such that its posterior is in closed form? Inverse Wishart ...