Correlation Function
A Correlation Function is a statistical correlation between two random variables.
- AKA: Statistical Correlation.
- Context:
- It can be expressed as the ratio between the covariance between two random variables, [math]\displaystyle{ cov(X,Y) }[/math], and product of their individual standard deviations, [math]\displaystyle{ \sigma_X }[/math] and [math]\displaystyle{ \sigma_Y }[/math] , i.e.
- [math]\displaystyle{ corr(X,Y)=\frac{cov(X,Y)}{\sigma_X\sigma_Y}, }[/math]
- See: Correlation Matrix, Cross-Correlation, Auto-Correlation, Convolution, Standard Deviation, Covariance.
References
2015
- (Wikipedia, 2015) ⇒ https://www.wikiwand.com/en/Correlation_function
- A correlation function is a statistical correlation between random variables at two different points in space or time, usually as a function of the spatial or temporal distance between the points. If one considers the correlation function between random variables representing the same quantity measured at two different points then this is often referred to as an autocorrelation function being made up of autocorrelations. Correlation functions of different random variables are sometimes called cross correlation functions to emphasise that different variables are being considered and because they are made up of cross correlations.
Correlation functions are a useful indicator of dependencies as a function of distance in time or space, and they can be used to assess the distance required between sample points for the values to be effectively uncorrelated. In addition, they can form the basis of rules for interpolating values at points for which there are no observations.
Correlation functions used in astronomy, financial analysis, and statistical mechanics differ only in the particular stochastic processes they are applied to. In quantum field theory there are correlation functions over quantum distributions.
Definition: For random variables X(s) and X(t) at different points s and t of some space, the correlation function is
- A correlation function is a statistical correlation between random variables at two different points in space or time, usually as a function of the spatial or temporal distance between the points. If one considers the correlation function between random variables representing the same quantity measured at two different points then this is often referred to as an autocorrelation function being made up of autocorrelations. Correlation functions of different random variables are sometimes called cross correlation functions to emphasise that different variables are being considered and because they are made up of cross correlations.
- [math]\displaystyle{ C(s,t) = \operatorname{corr} (X(s), X(t) ), }[/math]
- where [math]\displaystyle{ \operatorname{corr} }[/math] is described in the article on correlation. In this definition, it has been assumed that the stochastic variable is scalar-valued. If it is not, then more complicated correlation functions can be defined. For example, if X(s) is a vector, then a matrix of correlation functions is defined as
- [math]\displaystyle{ C_{ij}(s,t) = \operatorname{corr}( X_i(s), X_j(t) ) }[/math]
1999
- (Weisstein, 1999) ⇒ Weisstein, Eric W. (1999) "Statistical Correlation." from MathWorl, A Wolfram Web Resource. ⇒ http://mathworld.wolfram.com/StatisticalCorrelation.html
- For two random variates X and Y, the correlation is defined by
- [math]\displaystyle{ cor(X,Y)=\frac{cov(X,Y))}{\sigma_X\sigma_Y} }[/math]
- where [math]\displaystyle{ \sigma_X }[/math] denotes standard deviation and [math]\displaystyle{ cov(X,Y) }[/math] is the covariance of these two variables. For the general case of variables [math]\displaystyle{ X_i }[/math] and [math]\displaystyle{ X_j }[/math], where [math]\displaystyle{ i,j=1, 2, ..., n, }[/math]
- [math]\displaystyle{ cor(X_i,X_j)=\frac{cov(X_i,X_j)}{\sqrt{V_{ii}V_{jj}}} }[/math]
- where [math]\displaystyle{ V_{ii} }[/math] are elements of the covariance matrix. In general, a correlation gives the strength of the relationship between variables. For [math]\displaystyle{ i=j }[/math],
- [math]\displaystyle{ cor(X_i,X_i)=\frac{cov(X_i,X_i)}{\sigma_i^2}=1 }[/math]
- The variance of any quantity is always nonnegative by definition (...)