Smooth functions and local extreme values

Given a sample of n observations y1, . . . , yn at time points t1, . . . , tn we consider the problem of specifying a function ˜f such that ˜f • is smooth, • fits the data in the sense that the residuals yi − ˜f(ti) satisfy the multiresolution criterion 1 √ k − j + 1 k i=j yi − ˜f(ti) < 2 log(n)σ 1 ≤ j ≤ k ≤ n, (1) • is as simple as possible so that ˜f exhibits the minimum number of local extreme values. We analyse in particular a fast method which is based on minimising n i=1 (yi − f(ti))2 + n−1 i=1 λi (fi+1 − fi)2 + (ti+1 − ti)2 where the λi are chosen automatically. The new method can also be applied to density estimation. Key words: Nonparametric regression, modality, smoothness, total variation. 1 Research supported in part by Sonderforschungsbereich 475, University of Dortmund. Preprint submitted to Elsevier Science 30 January 2006 

• Type: Journal
• Year: 2007
• Where: CSDA
• Authors: A. Kovac
• Keywords: CSDA 2007, I=j Yi, Multiresolution Criterion, Time Points