Does high correlation imply proportionality? suppose I produce two vectors d1 and d2 using the below scheme which results in a correlation of 0.8716288 between the two vectors. Can I make a precise statement on the proportionality of d1 and d2.
Say I have:
m <- matrix(rnorm(100*100),nrow=100) 
d1<-as.matrix(dist(m, diag=T, upper = T))
d2<-1.-cor(t(m))
plot(as.vector(d1),as.vector(d2))
cor(as.vector(d1),as.vector(d2))

which results in 
correlation of 0.8716288 between d1 and d2
and plot:

NOTE:


*

*I'm leaving the definition of proportionality open -- since I personally do not know rigorous definitions of the concept beyond what is taught at grade school

*Would appreciate definitions of proportionality as part of the answer

 A: Even if the correlation is high, the two variables maybe far from being "proportional". Here I define "proportionality" as linear dependence ($y=ax+b$), because this is what the correlation coefficient is meant to measure. Look at the following four data sets (Anscombe's quartet) from the Wikipedia page:

All four sets have the same correlation 0.816. For the two sets in the left column, we can say that $x$ and $y$ are approximately linearly dependent (ignoring the outlier in the bottom set). But this is not the case for the two sets in the right column.
Moral: never use the correlation coefficient blindly, check the scatter plot! :)
A: Since you do not provide definition of proportionality, it can be assumed that you mean the regular definition that if $ y = cx $ then we can say $ y \propto x $.
If this is what you mean then if $Y$ is $X$ times constant, then they are linearly dependent and correlation between them is equal to $-1$ or $1$. That is the only relation between the two terms.
As @whuber correctly pointed out, the relation between two terms is not symmetric. Correlation measures linear relationship, so both if $Y = cX$ and $Z = a + cX$, have correlation with $X$ equal to $-1$ or $1$, while only the $Y$ is proportional to $X$.
