A note on the Gamma test analysis of noisy input/output data and noisy time series

In a smooth input/output process y=f(x), if the input data x∈Rd is noise free and only the output data yy is corrupted by noise, then a near optimal smooth model gˆ will be a close approximation to ff. However, as previously observed, for example in [H. Kantz, T. Schreiber, Nonlinear Time Series Analysis, 2nd ed., Cambridge Univ. Press, 2004], if the input data is also corrupted by noise then this is no longer the case. With noise on the inputs, the best predictive smooth model based on noisy data need not be an approximation to the actual underlying process; rather, the best predictive model depends on both the underlying process and the noise. A corollary of this observation is that one cannot readily infer the nature of a process from noisy data. Since almost all data has associated noise this conclusion has some unsettling implications. In this note we show how these effects can be quantified using the Gamma test.In particular we examine the Gamma test analysis of noisy time series data. We show that the noise level on the best predictive smooth model (based on the noisy data) can be much higher than the noise level on individual time series measurements, and we give an upper bound for the first in terms of the second.