The covariance of two random variables X and Y is the expectation of their product, minus the product of their expectations:


Covariance is of course analogous to the variance of a single random variable. The covariance function of a random process is its autocovariance; if this random process forms a vector, then the autocovariance of this vector forms its covariance matrix.

In object-oriented terminology, covariance relates to rewritten methods, and what types are allowed. The Eiffel FAQ provides this example of two classes: a PARENT class and a CHILD class, with the child class rewriting the foo method.

   class PARENT
         foo (arg: A) is ...
   end; -- PARENT

   class CHILD
         PARENT redefine foo
         foo (arg: B) is ...
    end; -- CHILD

The question is: what relationship between the types A and B should be allowed? There are 3 possible answers:

  • They're always the same
  • B is a subtype of A -- this is covariance, since the classes and the method parameters move (vary) in the same "direction", ie down the inheritance hierarchy.
  • A is a subtype of B -- the contravariance rule. The classes and the parameters move in different directions.

Sather is contravariant, while Eiffel is covariant.

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