# 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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### Best approximation of the Mahalanobis distance by standardized Euclidean distance

I am looking for the best way to approximate the Mahalanobis distance by the standardized Euclidean distance, which would reduce the number of the required multiplications. The easiest way is the ...
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### Covariance of variables [closed]

Let X, Y, and Z be random variables. How do we compute the Cov(aX+b,cY+dZ)? Thank you.
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### Covariance of two or more random variables [closed]

Let X, Y, and Z be random variables. Suppose that the variance of X is a, Cov(X,Y) =b, Cov(X,Z) =c , and Cov(Y,Z) =d. I want to compute Cov(6X+3Y). Is this equal to 63b? Also, to compute the following:...
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### Alternative to Proc IML? Using SAS Viya without IML available [closed]

I'm running the SAS Viya environment and do not have access to IML. However, I need to use the following code, or something similar, to calculate standard errors accurately by combining the covariance ...
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### Unbiased Estimation of Mixed 3rd-Order Moment

Let $X$ be a random variable on the space $\mathcal{X}$, and let $f, g$ be two well-behaved functions $: \mathcal{X} \to \mathbf{R}$ such that \begin{align} \bar{f} = \mathbf{E} [ f(X) ] < \infty \\...
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### covariance of squared projections

Given a vector $x$ of independent mean-zero random variables, and two nonrandom orthogonal unit vectors $u,v$, does $u'v=0$ imply $cov(x'uu'x,x'vv'x)=0$? If so, what is the proof? If not, what happens ...
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### Why is SSXY used in calculating covariance?

I am reading an introductory statistics book and am lost at the authors explanation for covariance. I follow up until the point the author arrives at: $$cov(x,y)=E(xy)-E(x)E(y)$$ However, they then go ...
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### Multiplying vectors by the covariance matrix?

I thought I knew covariance but I'm starting to think that there's more to it. For example, what happens when you multiply observations by their corresponding covariance matrix? ...
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### What is the conditional covariance matrix of $(X_2,X_3)^T$ given $X_1$?

$X=(X_1,X_2,X_3)^T\sim N_3(\mu,\Sigma).$ Suppose $X_1,...,X_{20}$ are i.i.d. observations from $X$. The sample mean vector and the covariance matrix are then defined by  \bar{x} = (1,0,2)^T,\quad S=...
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### Difference between the following two covariance expressions

What is the difference between $\sum_{i=1}^{5} (x_i-\bar{x})*(y_i-\bar{y})$ and $\sum_{i=1}^{5}\sum_{j=1}^{5} (x_i-\bar{x})*(y_j-\bar{y})$? Which of these is proportional to covariance? What is the ...
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### Need help interpreting relationship correlation and covariance with scatter plot

I'm having a bit of trouble interpreting this data. The correlation is very low and I'm assuming that covariance is low given the equation Cor(x,y) = Cov(x,y)/Sd(x)Sd(y). Although, the scatter plots ...
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### Multinomial Probability Model with Correlation between random variables

I'm trying to create a p.m.f. for a Multinomial distribution where the variables are correlated with one another. Let $k$ index the random variables $x$ and their probabilities $p$. There are to be $n$...
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### Relationship of x regress y and y regress x on the slope

Consider a linear regression model y on x and x on y. We have $Y = a'X + a$ where $a' = \frac{cov(X,Y)}{Var(X)}$. Equivalently, we have $X = b'Y+b$ where $b' = \frac{cov(X,Y)}{Var(Y)}$. I am ...
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### Covariance matrix of residuals [duplicate]

I am trying to do an optimization process that minimizes the residuals of an OLS regression. Typically the unbiased OLS residual variance is given by : E(RSS/N−p-1)=σ². Where RSS is the Residual Sum ...
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### Proof: If a matrix is semi-definite and symmetric positive then it is a covariance matrix

Anyone have the following proof? If a matrix is semi-definite positive and symmetric then it is a covariance matrix.
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### Understanding covariance

I came across following problem: A discrete random variable $P$ takes values $-3,-2,0,2,3$ with probability $0.2$. Let $Q=P^2$ be another random variable. What is covariance of $P$ and $Q$? I solved ...
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### Covariance for Random Variables vs. Sample Data

In my textbook, it says that the formula for finding covariance between two random variables is: $Cov(X,Y)=E((X-EX)(Y-EY))$ With $EY$ and $EX$ being the mathematical expectation for the random ...
The original version of the L1 regularization method uses sample covariance matrix ${\mathbf{S}}$ as follows: \begin{equation} \hat{\mathbf{\Omega}}= argmin_{\mathbf{\Theta}\succ 0} \bigg(tr(\...
correlation is computed from covariance so how come covariance can pick up non-linear relationships between variables $X$ and $Y$ but (Pearson's) correlation can't?