I Correlation functions and correlation length

[Frank Castle]

I thought I understood the concept of a correlation function, but I having some doubts.
What exactly does a correlation function quantify and furthermore, what is a correlation length.

As far as I understand, a correlation between two variables $X$ and $Y$ quantifies how much the two variables are related to one another, i.e. a statistical measure of how much they covary (fluctuate together).

With this in mind, does a correlation function, $\langle\xi(x)\chi(y)\rangle$ $^{(\ast)}$ of two spatially dependent variables $\xi(x)$ and $\chi(x)$ measure the extent to which the values of the variables, at spatially separated points $x$ and $y$, are related? In this sense, if the value of $\langle\xi(x)\chi(y)\rangle$ is close to 1 does this indicate a co-dependency between the two variables at spatially separated points, i.e. that if $\xi$ has a particular value at $x$ then $\chi$ will have a very similar value at the point $y$?!
As an example in physics, for spins on a lattice, does the correlation function between spins at different lattice points quantify the extent to which spins at different lattice points are co-aligned?! Another example would be anisotropies in the CMB - does the correlation function between temperature fluctuations in the CMB quantify the extent to which temperature fluctuations at different points in the CMB are similar in value?!

As for correlation length, does this quantify the length at which two variables cease to be similar in value? For example, if one has a surface that has a spatially varying height, does the correlation length quantify the length at which the height of the surface starts to significantly deviate from a given value?! Another example, in the case of spins on a lattice, does the correlation length quantify the length at which spins are no longer similarly aligned (i.e. the length at which they cease to be correlated)?!$^{(\ast)}$ [Does the notation $\langle\;\rangle$ indicate the one is taking the expectation value, for example, does $\langle\xi(x)\chi(y)\rangle =\mathbb{E}\left[\xi(x)\chi(y)\right]$?]

 

[haruspex]

if the value of ⟨ξ(x)χ(y)⟩ is close to 1 does this indicate a co-dependency between the two variables at spatially separated points, i.e. that if ξ has a particular value at x then χ will have a very similar value at the point y?!

Not necssarily similar in value, just closely correlated. That is, knowing the value of one the value of the other is highly predictable, but might be quite different. X and 1-5X are 100% correlated.

As for correlation length, does this quantify the length at which two variables cease to be similar in value?

Again, it refers to strength of correlation, not similarity in value. But yes, it is the separation beyond which the events are poorly correlated. I have not been able to find a quantitative definition.

 

[Frank Castle]

Not necssarily similar in value, just closely correlated. That is, knowing the value of one the value of the other is highly predictable, but might be quite different. X and 1-5X are 100% correlated.

So does it simply mean that if the variables closely correlated then they are related to one another, such that knowing the value of one enables one to predict the value of the other to a reasonable degree of accuracy?!

If the values of two variables at different spatial points are correlated, in simple terms does this mean that their values at these two points are related to one another to some extent?!

 

[haruspex]

So does it simply mean that if the variables closely correlated then they are related to one another, such that knowing the value of one enables one to predict the value of the other to a reasonable degree of accuracy?!

If the values of two variables at different spatial points are correlated, in simple terms does this mean that their values at these two points are related to one another to some extent?!

Yes, and yes.
Edit: no, sorry - correction.
A perfect correlation would imply a linear relationship, i.e. a straight line graph. So X² can be exactly predicted from X, but it would not yield a 100% correlation.

 

[Frank Castle]

Yes, and yes.
Edit: no, sorry - correction.
A perfect correlation would imply a linear relationship, i.e. a straight line graph. So X² can be exactly predicted from X, but it would not yield a 100% correlation.

Can't two things be correlated without there being a linear relationship between them, for example, one could have a Gaussian curve relating their values at different points (consider the heights of a rough surface above its mean height, heights at neighbouring points will be highly correlated, but the correlation will fall of as one moves further away either side from a specific point and so this will form a Gaussian-like distribution).

From my naive understanding, I thought a correlation function of two variables was simply a statistical measure of how related their values are at different points?! For example, with spins on a lattice, doesn't a correlation function between spin values at different lattice points measure how their values are related, i.e. whether their alignments are related?

 

[mfb]

X^2 and X have a non-zero correlation coefficient (apart from a few special ranges maybe), but the correlation coefficient is not sufficient to check if two things are independent. There are other models like Spearman's rank correlation coefficient or distance correlation to look for more general forms of dependency.

 

[FactChecker]

The correlation is more about general tenancies rather than specific relationships. A correlation near 1 says that one variable being higher than its average implies that the other variable tends to be above it's average. A correlation near 0 says that on average one variable being higher than its average implies very little about the other variable.
A pseudo-random variable is an extreme example of something where one value completely determines the other (the next) value, but the correlation is near 0. It is possible for the series from a pseudo-random number generator to look like there are no correlations unless very sophisticated tests are applied.

 

[Frank Castle]

X^2 and X have a non-zero correlation coefficient (apart from a few special ranges maybe), but the correlation coefficient is not sufficient to check if two things are independent. There are other models like Spearman's rank correlation coefficient or distance correlation to look for more general forms of dependency.

Is any of what I wrote a correct intuition for what correlation functions quantify?!

I've always understood a correlation between two variables to be a statistical measure of how much the variables covary (or in other words, a measure of how their fluctuations around their respective mean values are related)... but now I'm not so sure

 

[mfb]

That is what the correlation coefficient measures, and if you have a roughly linear relation on average the coefficient will tell you how strong this is - but it is a single number, it cannot tell you anything about more sophisticated dependencies.

 

[Frank Castle]

That is what the correlation coefficient measures, and if you have a roughly linear relation on average the coefficient will tell you how strong this is - but it is a single number, it cannot tell you anything about more sophisticated dependencies.

Ah ok. So have I at least understood the general notion of a correlation between two variables?!

As far as correlation functions in physics are concerned, in what sense to they measure a correlation between physical variables? As examples, what does a correlation function between spins on a lattice quantify? And what does a correlation function between temperature anisotropies in the CMB quantify? And what does a correlation function between galaxies at different spatial points quantify?!

 

[mfb]

As far as correlation functions in physics are concerned, in what sense to they measure a correlation between physical variables?

In the same way they do that in mathematics.

A positive correlation between adjacent spins means spins next to each other are more likely to point in the same direction.
A positive correlation between CMB temperature at 1 degree separation means if one point is warmer, then points 1 degree away from that point are also more likely to be warmer.

 

[Frank Castle]

In the same way they do that in mathematics.

A positive correlation between adjacent spins means spins next to each other are more likely to point in the same direction.
A positive correlation between CMB temperature at 1 degree separation means if one point is warmer, then points 1 degree away from that point are also more likely to be warmer.

That makes sense. So is it simply a measure of how the values of the variables at different points are co-dependent, in the sense that if the values of variables at different points are correlated, then if you know the value at one point then one can predict the value of variable at the other point?!

Have I understood the concept of correlation correctly at all?

 

[mfb]

Yes.

 

[Frank Castle]

Yes.

Just to clarify, is that "yes" to both

That makes sense. So is it simply a measure of how the values of the variables at different points are co-dependent, in the sense that if the values of variables at different points are correlated, then if you know the value at one point then one can predict the value of variable at the other point?!

And this

Have I understood the concept of correlation correctly at all?

 

[mfb]

I can't look into your head, but your description here is right.

 

[Frank Castle]

I can't look into your head, but your description here is right.

Ok cool, thanks for your help!