Model Misspecification Tests, Model Building and Predictability in Complex Systems

In this paper we show that significant advantages can be realized by using functions of simple variables, such as time lags, as directions in reconstruction spaces for complicated time series. First, we show that more sensitive model specification tests can be constructed when the candidate model is used as one of the directions in the reconstruction space. This naturally results in misspecification tests based on the construction of conditional probabilities which can reveal misspecified models even when other powerful methods, such as the BDS test applied to a sequence of residuals, fails. Second, we show that model building and predictability of the time series can be substantially improved by using an informational criterion to determine the functions with which to associate the directions of the reconstruction space. This criterion is of the form of a conditional probability and is related to a measure of the short term predictability of the time series. We consider specifically an example in which the search space of functions is linear combinations of time-lags, and we compare the resulting model with linear least squares fit to the data. We also show that this optimized reconstruction space can improve longer-term predictability. We demonstrate this by comparing longer-term predictions made using the optimized space with predictions made using a simple time-lagged reconstruction space.