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Statistical Methods for Fuzzy Data by Reinhard Viertl

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25.3 Covariance and correlation

Definition 25.5:

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):

Unnumbered Display Equation

The correlation Corr (X*, Y*) is defined by

Unnumbered Display Equation

This covariance can be decomposed into a local and a fuzzy part. With X0* = we obtain

Unnumbered Display Equation

This is proved by using the definition of the covariance.

Lemma 25.2:

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.

Proposition 25.1:

For fuzzy random quantities X*, X1*, X2* and Y*, and the following holds:

1. Cov

2.

3. Cov for with

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