# Correlation coefficient between continuous functions

• natski
In summary, the conversation discusses the definition of correlation coefficients in terms of discrete sampling and the possibility of expressing them in terms of continuous probability distributions. The speaker also asks for help in finding a formalism for the correlation coefficient between two continuous probability distributions. A reference is provided and the speaker restates their problem to involve non-independent variables.
natski
Hi all,

The correlation coefficients (Pearson's) is usually defined in terms of discrete sampling of a function. However, I have seen that the mean and standard deviation, for example, are also typically written in terms of discrete variables BUT may also be expressed in terms of a continuous probability distribution. e.g. the mean may be written as \mu_x = \int x p(x) dx.

So my question is, does there exist a similar formalism for the correlation coefficient between two continuous probability distributions? Any help would be greatly appreciated on this issue for which many Google searches came up empty handed. :-)

Natski

I found a reference: Cuadras 2002, On the Covariance between Functions, which has a cited solution form 1940 known as Hoeffding's lemma. This lemma is based on the cumulative distribution functions.

Does anyone know how to go from two independent P(x) and P(y) to C(x,y) where P's are the prob density functions and C is the multivariate cumulative distribution function?

Natski

Actually correction to my last post, the two variables are not independent. Here is the problem restated for my exact case.

P(x) is a uniform distribution and = (1/Pi) for all x between 0 and Pi

Now let u = Sin[x] and v = Cos[x]

What is the joint probability distribution G(x,y)? I know how to do univariate and bivariate transformations by using the Jacobian but to go from univariate to a bivariate seems to require a 2 x 1 Jacobian for which no determinant can be computed. This is where I get stuck.

natski said:
Actually correction to my last post, the two variables are not independent. Here is the problem restated for my exact case.

P(x) is a uniform distribution and = (1/Pi) for all x between 0 and Pi

Now let u = Sin[x] and v = Cos[x]

What is the joint probability distribution G(x,y)? I know how to do univariate and bivariate transformations by using the Jacobian but to go from univariate to a bivariate seems to require a 2 x 1 Jacobian for which no determinant can be computed. This is where I get stuck.

The joint distribution isn't needed because you already have U and V as functions of X, so E(UV) etc are 1d integrals.

## What is a correlation coefficient between continuous functions?

A correlation coefficient between continuous functions is a statistical measure that indicates the strength and direction of the linear relationship between two continuous variables. It is represented by the symbol "r" and can range from -1 to 1, with 0 indicating no correlation and values closer to -1 or 1 indicating a strong negative or positive correlation, respectively.

## How is a correlation coefficient calculated?

A correlation coefficient is calculated by dividing the covariance of the two variables by the product of their standard deviations. This can be done using a formula or by using statistical software.

## What does a correlation coefficient value of 0 mean?

A correlation coefficient of 0 means that there is no linear relationship between the two variables. However, it is important to note that there could still be a non-linear relationship or other types of relationships between the variables that are not captured by the correlation coefficient.

## Is a correlation coefficient of -1 or 1 always indicative of a perfect relationship?

No, a correlation coefficient of -1 or 1 indicates a perfect negative or positive linear relationship, respectively. However, there could still be other types of relationships between the variables that are not captured by the correlation coefficient. Additionally, a perfect correlation does not necessarily imply causation.

## Can a correlation coefficient be used to determine causation?

No, correlation does not imply causation. A high correlation coefficient between two variables only indicates that they are related in a linear manner, but it does not prove that one variable causes the other. It is important to consider other factors and conduct further research before making any causal claims based on a correlation coefficient.

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