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

• SW = selective wavelet (best basis possible)
• PP = piecewise polynomial
• note: splines are subset of PP

Bonferroni can approximate best wavelet

• Identical to Risk inflation

Can anything beat Bonferroni?