Karhunen–Loève expansion revisited for vector-valued random fields: Scaling, errors and optimal basis.

Due to scaling effects, when dealing with vector-valued random fields, the classical Karhunen–Loève expansion, which is optimal with respect to the total mean square error, tends to favorize the components of the random field that have the highest signal energy. When these random fields are to be used in mechanical systems, this phenomenon can introduce undesired biases for the results. This paper presents therefore an adaptation of the Karhunen–Loève expansion that allows us to control these biases and to minimize them. This original decomposition is first analyzed from a theoretical point of view, and is then illustrated on a numerical example.

► A review of the Karhunen–Loève expansion (KLE) is first presented.
► Truncating the KLE tends to favor the components of highest signal energy.
► When dealing with vector-valued random fields, undesired biases can be introduced.
► A scaled KLE is thus introduced to control and minimize these biases.
► The possibilities of such an expansion are illustrated on a numerical example.