25.3 Covariance and correlation
For two fuzzy random variables X* and Y* the covariance Cov (X*, Y*) is defined using the scalar product of the support functions from (23.9):
The correlation Corr (X*, Y*) is defined by
This covariance can be decomposed into a local and a fuzzy part. With X0* = we obtain
This is proved by using the definition of the covariance.
Covariance and correlation of two fuzzy random quantities X* and Y* fulfill the following:
1. Cov (X*, X*) = VarX*
2. Var(X* ⊕ Y*) = VarX* + VarY* + 2Cov (X*, Y*)
3. Corr (X*, Y*) = 0 if X* and Y* are stochastically independent
4. |Corr (X*, Y*)| ≤ 1
5. Cov .
For the proof see Körner (1997a).
The following proposition gives further properties of covariance and correlation.
For fuzzy random quantities X*, X1*, X2* and Y*, and the following holds:
3. Cov for with