The role of model validation for assessing the size of the unmodelled dynamics

There are two sources of errors in any identified model: 1) the bias error, due to too simple a model structure where all aspects of the true system cannot be described by any model within the used structure; and 2) the variance error, due to errors and disturbances in the measured data from which the model is constructed. The total model error is the sum of these two contributions, and the objective is to find a structure that makes this error small. While the variance error can be assessed by quite standard statistical methods, the bias error is far more difficult to evaluate. The present paper contains two results that relate to the size of the bias error to that of the variance error: 1) for a typical model that minimizes the total error, the bias error is dominated by the variance error; and 2) for a model that has passed a typical validation test, the bias error is again dominated by the variance error