The novel adaptivity introduced in this chapter can be viewed as the development of two independent ideas: wavelet multiresolution analysis and the adaptive-scale nonparametric regression estimation.
In nonparametric regression methods, the adaptive-scale selection is a key point. Lepski's approach, which was discussed above, and the ICI rule are the new statistics developed for adaptation. A common feature of these methods is that they all form a number of test estimates Å⋅ h (x) that are different by the scale hâH as well as by the data sets used for estimation, and statistical rules are exploited to select the best scale (window size) of the estimate. Thus, we yield the adaptive varying-scale pointwise adaptation.
The approach presented in this chapter results in a different type of scale-adaptive estimate. Mainly, instead of selecting the estimate with the unique best scale h + (x) , we build a nonlinear estimate using all of the available estimates Å⋅ h (x) , hâH . This estimation is divided into two steps. The first step transforms the data into noisy spectrum coefficients (multiresolution analysis). In the second step, these noisy estimates of the spectrum are filtered by the thresholding procedure and used for estimation (multiresolution synthesis).
The LPA-based filters are exploited for the nonparametric (pointwise) multiresolution spectrum analysis and synthesis. In this way we introduce an extension of the conventional nonparametric local regression concepts and yield a wider class of adaptive-scale multiresolution nonparametric regression techniques with potentially better performance.
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