Error criteria for cross validation in the context of chaotic time series prediction

The prediction of a chaotic time series over a long horizon is commonly done by iterating one-step-ahead prediction. Prediction can be implemented using machine learning methods, such as radial basis function networks. Typically, cross validation is used to select prediction models based on mean squared error. The bias-variance dilemma dictates that there is an inevitable tradeoff between bias and variance. However, invariants of chaotic systems are unchanged by linear transformations; thus, the bias component may be irrelevant to model selection in the context of chaotic time series prediction. Hence, the use of error variance for model selection, instead of mean squared error, is examined. Clipping is introduced, as a simple way to stabilize iterated predictions. It is shown that using the error variance for model selection, in combination with clipping, may result in better models.