Last modified: Thu Oct 6 17:07:38 EDT 2005
by Dean Foster

# Statistical Data mining: Wavelets

## Admistrivia

- I put your homework in your mail boxes
- Read, read, read!

### 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)) * (sigma^{2} + 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)

dean@foster.net