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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What is the conceptual difference between the marginal variance and the range in the Matérn covariance?

The Matérn covariance kernel is given by: $$ C_\nu(d) = \sigma^2 \frac{2^{1-\nu}}{\Gamma(\nu)}\left(\sqrt{2\nu}\frac{d}{p}\right)^\nu K_\nu\left(\sqrt{2\nu}\frac{d}{p}\right) $$ My question is, what ...
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7 views

SEM: Modification indices suggest covariance between an exogenous variable and an endogenous variable. Is there a way to make this work?

I am fairly new to Structural Equations Modeling and I’m trying to fit a simple path model with observed variables only. My original model is as follows (using code from the R package lavaan): ...
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1answer
48 views

Intuition (geometric or other) for $Var(X+Y) = Var(X)+Var(Y) + 2 \ Cov(X, Y)$

Is there any way to make sense out of this formula intuitively? I rederived it algebraically (took me a while...), which made me happy because I used to be incapable of doing that kind of stuff, but ...
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4 views

Add Structure on Experimental semivariogram and validation of semivariogram [on hold]

im fairly new to this so please be patient with me. I am using surpac 6.3.2 and i want to know when to add a "structure" to the experimental semivar, also, once i have completed a semivar, how to i ...
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18 views

Get covariance from conditional covariance for lognormal (and other) observations?

Consider lognormal random variables $X_1$ and $X_2$ with correlation coefficient $ρ$ and a partial observation sample of them of length N, the sample being partial because it only contains occurrences ...
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11 views

Expected value of a semi-partial correlation

Say I have 4 random variables. $X^{(1)}$ and $X^{(2)}$ are jointly multivariate normal with mean 0 and covariance $\Sigma_X$, and $Y^{(1)}$ and $Y^{(2)}$ are jointly multivariate normal with mean 0 ...
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70 views

factor model and large covariance matrices

Imagine the number of variables $\textbf{p}$ is 5 times larger than the number of observations $\textbf{n}$ and the sample covariance matrix is almost block diagonal (with low off block diagonal ...
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9 views

Which average to use when calculateing covariation in one pass?

I want to calculate covariance in one pass through my data set. John Coook has a recipe for one pass calculation of variance. It goes like this: Initialize $M_1 = x_1$ and $S_1 = 0$. For ...
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1answer
203 views

How to define the covariance for a finite set of vectors in an inner product space space V? What object is it? [closed]

Motivation: This question is motivated by a type of problems in medical imaging and computer vision as follows: suppose we've a set $A$ of points ("shapes") $\{x_1, ...x_d\} $in a Riemannian ...
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8 views

Covariance of GMM-distributed vector

Let $X\sim~GMM(\pi,\theta)$, i.e. $$p(X) = \sum_i \pi_i g(X;\theta_i )$$ where $g(X;\theta_i)$ is a Gaussian density with parameters $\mu_i,\Lambda_i$ and $\sum \pi_i=1$. Is there a formula for the ...
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35 views

Generating samples from high-dimensional multivariate Gaussian with few training samples

Say I have a $n\times d$ dataset $D$ where $n\ll d$ ($n$ number of observations, $d$ number of dimensions). Currently, if I want $m$ samples from $D$ assuming it is multivariate Gaussian, I can do ...
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36 views

Covariance matrix through bootstrapping - close to zero determinant

I have a set of 260 sets of measurements (for each set of measurements there is an amplitude measured as a function of 8 radii). Since I do not get measurement errors and I am interested in the ...
3
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2answers
102 views

Are there ways in which perfect correlation (i.e., $\text{Cov}(X,Y) = 1$) can still be a deceiving statistic?

Are there ways in which $\text{Cov}(X,Y) = 1$ can hold and yet "something surprising" can still also be true? For example: it is possible for the mean depth of a pool to be only $10 \text{ ft}$, yet ...
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53 views

Euclidean distance between the unbiased covariance estimator and the true covariance

Let X be a n*p matrix whose p columns are sampled from a centred normal multivariate distribution with true covariance matrix $\Sigma$ and let $\Sigma_{n}$ be its unbiased sample covariance estimator ...
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1answer
69 views

Covariance of two variables that are products of shared random variables

How to analytically express cov(X,Y), when: X=C*A/(A+B) and Y=C*B/(A+B) Here C, A and B are independent variables with ...
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12 views

How to make a covariance matrix from multiple observations of different objects?

I have $N$ objects. From each object, I sample $M$ values $(x,y)$ like so: ...
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10 views

plots of covariance with jackknifed confidence intervals in OPLS

The SIMCA-P software is able to produce plots of covariances with jackknifed confidence intervals for OPLSDA models. Is there any R package implementing the same feature? Or is there other means of ...
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1answer
41 views

Covariance and correlation in multivariate random variables

I have this experiment where there are two random vectors $P_1 = (x_1,y_1)$ and $P_2 = (x_2,y_2)$. These two vectors represents two measurements for the location of two nearby points ($10$ meters ...
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17 views

Covariance matrix of complex random variables

Let's say we have datapoints $x_i \in \mathbb{C}^N$, like this: $x_1 = \begin{pmatrix}z_1 & z_2 & ... & z_N \end{pmatrix}^T$ and $x_2=\begin{pmatrix}z_1 & z_2 & ... & z_N \end{...
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1answer
38 views

How to prove that $Cov(\hat{\beta},\bar{Y}) = 0 $ using given covarience properties

To quote: It is well known that, if $W_1, ..., W_n, Z_1, ..., Z_m$ are random variables and $a_1, ..., a_n, b_1, ..., b_m$ are constants, then $Cov ( \sum_{i=1}^n a_iW_i, \sum_{j=1}^m b_jZ_j) = \...
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19 views

Error propagation with correlation

I have two different multidimensional objects in two different conditions, therefore four different vectors of observations: $$V1_{cond1} = v1_{1,1},v1_{1,2},\dots , v1_{1,n}$$ $$V1_{cond2} = v1_{2,1},...
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2answers
23 views

Co-Variance of Zero and Non-Zero mean random variables

Is the value of co-variance function for non-zero mean random variables different from the value of co-variance function when random variables have a zero-mean? I think yes based on this: For a ...
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114 views

Covariance of order statistics

I'm a researcher in social science and I have encountered the following math formulation of a problem in my field. Note that I'm relatively new to stack exchange and I have already posted this on math....
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26 views

What is the variance of the sum of Yi's

Seems a simple enough question, and I presume that, if Yi are normally distributed, Var(Sum(Yi)) = Sum(Var(Yi)) This feels like I'm jumping to the wrong conclusion though. Any help would be ...
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1answer
94 views

What is the difference between the anti-image covariance and the anti-image correlation?

What is the difference between the anti-image covariance and the anti-image correlation? How are the matrices of these coefficients computed, and what is the meaning of their elements?
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1answer
26 views

Prove that $Var(\hat {Y_i})=\sigma^2h_{ii}$

I just got a simple question. In general linear model, we have $$\hat Y=HY$$ where $H=X(X^TX)^{-1}X^T$ and the residual $$E=Y-\hat Y.$$ Now I want to prove that $$Var(\hat {Y_i})=\sigma^2h_{ii}$$ ...
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36 views

$c(n)$ is trend, $r(n)$ is fluctuation. Should $\text{cov}[c(n),r(n)]/\text{var}[r(n)]$ be close to zero?

Suppose $y(n)$ is a random time series given as function of the discrete-time variable $n$. Suppose we can decompose it into $y(n) = c(n) + r(n)$, where $r(n)$ is a strict stationary residual ...
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13 views

Using Covariance and Correlation for Similarity matching

I am trying to find if a particular pattern exists in a time series. I have found that I could try using Covariance or correlation for the task. I have used a sliding window technique for doing this. ...
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38 views

Is there any point to Reverse Engineering the Fisher Information Matrix from an Inverse Covariance Matrix?

Would there be any advantage in deriving a Fisher Information Matrix backwards from an inverse covariance matrix? I've discovered that this is much easier to do on the SQL Server platform I use than ...
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7 views

longitudinal correlations matrices (structural covariance)

I am interested in doing structural covariance analyses in a longitudinal manner. In structural covariance analyses in our field, we correlate grey matter volumes of various regions of the brain with ...
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3 views

Calculate covariance of slow and fast variables

Say you have two time series $X_t$ and $Y_t$ where $X_t$ is given by an $AR(1)$-process and $Y_t$ is a deterministic function of $X_t$: $$Y_t = f(X_t).$$ Also assume that the fluctuations of (the ...
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36 views

What's the relationship between covariance, shared variance, and common variance?

I've generally assumed that shared variance and common variance were the same thing. However, here it is written that "Common variance is the realm of total collinearity. On the other hand, the term "...
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7 views

Transformations and covariances

I am looking for some potential results regarding the relationship between the covariance of $X,Y$ versus the covariance of $T_1(X),T_2(Y)$, where $T_i$ is some map. I know the relation when $T_i$ is ...
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75 views

Why do we batch normalize the pre-synaptic values rather than the activations when using batch normalization?

I'm trying to make sense of this batch normalization (1) paper, in Section 3.2, it says We could have also normalized the layer inputs u, but since u is likely the output of another nonlinearity,...
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37 views

Splitting up the variance of Z for Z = X*Y

$Z$ is a function of two dependent random variables, e.g. $X \cdot Y$. Here it is shown that $$var(Z) = var(XY)=(cov(X^2,Y^2)+E[X^2]E[Y^2])-(cov(X,Y)+E[X]E[Y])^2$$ I am interested in a metric that ...
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22 views

Covariances of a random variable and its subset

Suppose event B is a subset of event A. $P(A) = p$ and $P(B) = q$. What is the Covariance of indicator functions from A and B. $$ \mathbb{1}_{A}(\omega) = \begin{cases} 1 & \omega \in A\\ 0 &...
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1answer
23 views

Covariance Decomposition

I have the returns of three stocks, $R_{1t}$, $R_{2t}$, $R_{3t}$, with 100 monthly observations for each return series. Lets suppose that I create a portfolio consisting of stocks 1 and 2, $P_t=w_{1t}...
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1answer
74 views

Proving that $y_t = \beta_1 x_t + \beta_2 y_{t-1} + u_t$ parameters are biased when $u_t$ is autocorrelated

How do you prove the result that for equation: $$y_t = \beta_1 x_t + \beta_2 y_{t-1} + u_t$$ the beta parameters are biased when $u_t$ is autocorrelated? In other words, that$$ \text{Cov}(u_t, y_{t-...
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18 views

What is a good estimator for the reciprocal of covariance?

Let $X,Y$ be random variables with unknown but nonnegative covariance. What is a good estimator for $1/\operatorname{Cov}[X,Y]$? Specifically, how does one deal with negative sample covariance when it ...
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15 views

Factor analysis Geometric interpretation of common variance [duplicate]

I am trying to understand the fundamental differences between PCA and Factor Analysis. PCA is straight forward in that you take the eigenvectors of the data's variance (and hence considering all the ...
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24 views

ANCOVA vs two-way ANOVA

I'm a little new to R and I haven't done stats in a while. I know a one way ANOVA is the same as a linear regression, but is there a difference between a two way ANOVA and a linear regression with two ...
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19 views

Aggregating covariances

Suppose I have 2 data sets too large to combine. Each data set has one row per user, $i$, and two columns $(x, y)$. Users can be in both data sets but have different column values, $(x_{i1}, y_{i1})$...
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1answer
58 views

Numerically stable correlation coefficient calculation

I have been trying to calculate the correlation coefficient $(\rho)$ of two variables, and noticed that in cases where either $var(X)$ or $var(Y)$ are very small, the correlation coefficient ...
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16 views

What test is appropiate to compare two variables having the same pattern?

we have an experiment with two variables that behave very similar (see attached picture). We think a correlation test would be suitable, but we are not sure which one. What would you recomend? ...
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26 views

Cov(y,z)? when z=min(Y)

For independently distributed normal random variables $y_i$ ~ $N(\mu_i, \sigma_i ^2 )$ Let $z =$ min$(Y) $, where $Y=${$y_1,y_2,y_3...y_n$}. How to calculate cov$(y_j,z)$ ? I tried to calculate it ...
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17 views

A function that measures covariance and size of random variables [closed]

I am trying to implement a function of two random variables that assigns higher values if the values of the variables are higher AND their (cross-)covariance or correlation is too. Does such a ...
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36 views

Why are principal components of the residuals from a multivariate regression correlated with the estimated coefficients?

Say I have some data that follows a general linear model: $$ Y = XB + E $$ for which: $Y \in \Re^{n \times m}$, $X \in \Re^{n \times p}$ and $B \in \Re^{p \times m}$ Further, let's assume (1) that ...
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27 views

Trasformation to Positive or Negative Quadrant Dependent Random Variables

Can we transform two statistically dependent random variables $X_1,X_2$ such that the covariance $Cov(X_1,X_2) \geq 0$? Can we transform $X_1,X_2$ such that $Cov(f(X_1), g(X_2))\ge0$ for all real ...
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24 views

Independence of Gaussian surrogate model

For a Gaussian process model, $$ f (x) ∼ GP(m(x), k(x, x' )) $$ $m$ is mean and $k$ is covariance function. The predictive distribution of unknown $x$ is, $$ \hat{y}(x) \sim N(\mu(x), \sigma(x)^2) $...
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1answer
33 views

Law of total covariance for products of random variables

I have two sets of random variables. $X_i \sim N( \theta, \sigma^2 )$, where $X_i$ are i.i.d. $Z_j$ which are simply iid binary random variables with success probability $p$. I want to find $\text{...