# Perfectly correlated (normal) random variables

I am not sure in the terminology, so I will simply try to explain the situation that I would like to model as I see it. Suppose there is a set of random variables. The variables are correlated in such a way that they deviate from their expected values into the same direction all together. By this I mean that they can be either all together larger then their expectation or all together lower. Is it possible to model such a dependency with a multivariate normal random variable $\mathbf{X} \sim \mathcal{N}(\mathbf{\mu}, \mathbf{\Sigma})$, assuming the knowledge about the marginal distributions of the components $\mathbf{X}_i \sim \mathcal{N}(\mu_i, \sigma^2_i)$? How to construct $\mathbf{\Sigma}$ is this situation? Thank you.

Best wishes, Ivan

• If you require that there be no possibility of deviations from this correlated behavior, then these variables cannot be multivariate normal unless they all are linear combinations of a constant and one (univariate) normal. But perhaps your restriction is not as severe as this? – whuber Feb 17 '12 at 15:10
• Unless I misunderstand the question, can't you just fill $\mathbf{\Sigma}$ in with the covariances from your dataset? – rm999 Feb 17 '12 at 15:14
• Thank you for the replies. The point is that I am trying to build a model of a quite complicated system and, in order to be able to at least start from something, I can make some assumptions that later, I hope, I will be able to get rid of or relax. Since multivariate r.v.'s are quite comfortable to deal with (and the nature is supposed to be fair), I have decided to start from them, but the correlation mentioned earlier is a must, it should be in the model from the very beginning. – Ivan Feb 17 '12 at 15:36
• @whuber, I also though about this possibility of having just one normal r.v. $\xi$, the rest are just linear transformations of it like $\mathbf{X}_i = a_i \xi + b_i$. May be I should start from this simple scenario. – Ivan Feb 17 '12 at 15:39
• @rm999, probably, you are also right. If one has a bunch of data, based on which s/he estimates $\mu_i$ and $\sigma^2_i$, s/he can also try to estimate the covariance matrix. – Ivan Feb 17 '12 at 15:42

Consider this very simple snippet:

    m1 <- 0
m2 <- 0
cov <- 0.8
x1 <- rnorm(100, mean=m1)
x2 <- cov*x1 + rnorm(100,mean=m2-cov*m1,sd=sqrt(1-cov*cov))
plot(x1,x2)
x2a <- x2*sign(x1-m1)*sign(x2-m2)
plot(x1,x2a)


It folds the distribution of x2 around its mean, aligning its deviations from the mean to those of x1 from its mean. Of course the resulting distribution cannot be characterized as a multivariate normal, although each margin is normal:

    plot( density(x1), ylim=c(0,0.5) ) Contours of the density of (x1, x2a): the probability that would ordinarily be associated with values in quadrants II or IV has been symmetrically displaced into quadrants I and III, leaving the marginal distributions undisturbed.

This is a classic (counter)example of a distribution that has normal margins, yet is not a multivariate normal; frankly, I don't know how to build any other ones.

The transformation increases the correlation somewhat:

    > cor(x1,x2)
 0.7999774
> cor(x1,x2a)
 0.8575814


You would've seen a much stronger effect with lower cov, of course: you can start with cov=0 and still get the correlation of the resulting variables above 0.6.

• When starting with a bivariate normal distribution of correlation $\rho\ge 0$, this procedure produces a bivariate distribution of correlation $\frac{2}{\pi}(\sqrt{1-\rho^2}+\rho \arcsin(\rho))$, which ranges from $2/\pi\approx 0.637$ up to $1$. – whuber Feb 17 '12 at 18:57
• @whuber, do you have a reference? Not that I am not ready to take your word for it, but to have a complete response. It would also be great if you could post the contour plot code if you have it at your fingertips. – StasK Feb 17 '12 at 20:04
• The results in the comment are just a calculation. (And you shouldn't take my word for it: I make mistakes. But at least the formula agrees with your results! It gives 0.854 for $\rho=0.8$.) For the plot, here's Mathematica code: – whuber Feb 17 '12 at 20:18
• $\text{With}[\{\mu=0,\nu=0, \sigma=1, \tau=1, \rho=\frac{4}{5}\},$ $g[\text{x_},\text{y_}] \text{:=}\text{With}[\{f = \text{BinormalDistribution}[\{\mu , \nu \},\{\sigma , \tau \},\rho ]\},$ $\quad\text{Piecewise}[\{\{\text{PDF}[f, \{x,y\}] + \text{PDF}[f, \{2\mu -x,y\}], (x-\mu )(y-\nu )\geq 0\}\}, 0]];$ $\text{ContourPlot}[g[x,y], \{x,\mu -2.5\sigma ,\mu +2.5\sigma \}, \{y,\nu -2.5\tau ,\nu +2.5\tau \},$ $\quad\text{PlotRange}\to \{\text{Full}, \text{Full}, \text{Full}\}, \text{Contours}\to 12,$ $\text{PlotPoints}\to 100, \text{AspectRatio}\to \tau /\sigma ]]$ – whuber Feb 17 '12 at 20:18
• Related to @whuber's calculation: Recent MO question and answer. – cardinal Feb 17 '12 at 22:05