The Construction of Smooth Models using Irregular Embeddings Determined by a Gamma Test Analysis

One of the key problems in forming a smooth model from input-output data is the determination of which input variables are relevant in predicting a given output. In this paper, we show how the Gamma test can be used to select that combination of input variables which can best be employed to form a smooth model of an output. For time series prediction this amounts to the selection of an appropriate irregular embedding. We give some simple zero noise examples of time series analysis, and illustrate how using these techniques a binary message encoded into a chaotic carrier can be retrieved without knowledge of the dynamics used to generate the carrier. Provided the underlying dynamics are such as to produce a smooth embedding model with bounded partial derivatives, the sampling distribution is dense in input space, and any associated distribution of measurement error has the first few moments bounded, so that the typical prerequisite conditions of the Gamma test are satisfied, we conclude that the Gamma test is an effective tool in the determination of irregular time series embeddings. These techniques can also be useful in practical applications which involve filtering seismic data to detect anomalous events.