# Statistical Data mining: Wavelets

### Problem: curve fitting

• Splines are classical way of fitting curves
• E.g. piecewise linear
• Often more smoothness, so piecewise cubic say?
• allows derivatives to match

### Where to put the knots?

• Very non-linear model. Moving knots changes fit
• Even counting degrees of freedom isn't easy

### Competitors to splines

• Fourier bases (sines and cosines)
• Polynomials
• wavelets (piecewise Fourier?)
• Both linear

### Fourier can't fit jumps

• Consider jump
• Spline can fit by outting two knots at jump
• Fourier runs into "Gibbs" phenomena
• So to polynomials (reason for knots in the first place)
• Wavelets only get hit with a log badness

### Splines can be approximated by wavelets

Theorem: (DJ94) R(SW,f) < a + b log(n)R(PP,f)
• SW = selective wavelet (best basis possible)
• PP = piecewise polynomial
• note: splines are subset of PP

### Bonferroni can approximate best wavelet

Theorem: (DJ94) R(Bonferroni,f) < (1 + 2 log(n)) * (sigma2 + R(SW,f))
• Identical to Risk inflation

### Can anything beat Bonferroni?

Theorem: (DJ94) For some n: R(anything,f) > (2 - epsilon) log(n)R(SW,f)
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