Is the correlation coefficient better than we think? In order to measure precisely the influence of one variable $Y$ over $X$ I wanted to use the mutual information because so far I believed that correlation coefficient (Pearson) was only limited to linear relation.
But I realized that even in non-linear cases it gives good results.
Example in Matlab:
$X$ is a uniform random variable and $I$ is the range of $X$ 
$Y=X^{2},I=[0,100]: \rho=0.96$
$Y=\sqrt{X} ,I=[0,100]: \rho=0.94$
$Y=exp(X),I=[0,10]: \rho=0.70$
$Y=log(X),I=[0,100]: \rho=0.84$
Why the correlation is quite high in all those non-linear cases? I am beginning to think that correlation is only weak when the monotony of $X$ and $Y$ is not the same. Otherwise it is not so bad.
Do some people have a point of view on this?
 A: A couple of monotonic examples in R (but very similar code works in Matlab):
> x=(1:100)
> y=exp(x)
> z=log(x)
> cor(z,y,method="kendall")
[1] 1
> cor(z,y,method="spearman")
[1] 1
> cor(z,y)
[1] 0.1549211

That's pretty low!

It's quite possible to make it much lower:
> cor(xxx,yyy,method="spearman")
[1] 1
> cor(xxx,yyy,method="kendall")
[1] 1
> cor(xxx,yyy)
[1] 0.010647

and in fact that very low Pearson correlation is caused by only a single outlier in x and a single outlier in y:
> cor(xxx[2:99],yyy[2:99])
[1] 1

So even when 98% of the points lie on a perfectly straight line, and the relationship is perfectly monotonic, the Pearson correlation can still be really close to zero.


In the case where $x$ is restricted to take the values 1, 2, ..., 100, the correlation can still be very low. Here's an example:
 cor(1:100,exp(1:100))
[1] 0.252032

I'm not sure how you got 0.7 - perhaps I misunderstood. I have another example with equispaced $x$ that is about 0.175.
A: The more a function deviates from linearity, the smaller the correlation coefficient is.

Try this one:
> cor(1:100,factorial(1:100))
[1] 0.1740601

The factorial function is $f(x)=x!=\Gamma(x+1)$, monotonically increasing $\forall\space x\in\mathbb{N}$
