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Adaptability is a crucial feature of filtering and differentiating methods. It was demonstrated in the previous chapters that the performance of LPA kernel estimates depends strongly on the selection of the scale parameter. In Fig. 2.6 we saw the Nadaraya-Watson estimates for different h. A small h gives the limit estimate Å⋅ h (x) as a stepwise curve passing exactly through the observations. For large h the same estimator gives a constant value equal to the sample mean of the observations. For intermediate h between these small and large values, we may obtain a large variety of the estimate curves, which are different by their curvature and closeness to the observations.
The idea of the Nadaraya-Watson estimator is natural and fruitful. However, the estimation curve is quite sensitive to h, and the selection of h is a key factor that is able to transform this reasonable idea into an effective working tool. A selection of a proper scale for estimation is a hot topic in both signal processing and nonparametric regression methods. The number of publications in this area is very large and growing quickly.
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