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Flexible Metric NN Classification. based on Friedman (1995) David Madigan. Nearest-Neighbor Methods. k -NN assigns an unknown object to the most common class of its k nearest neighbors Choice of k ? (bias-variance tradeoff again) Choice of metric?

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Flexible metric nn classification

Flexible Metric NN Classification

based on Friedman (1995)

David Madigan


Nearest neighbor methods
Nearest-Neighbor Methods

  • k-NN assigns an unknown object to the most common class of its k nearest neighbors

  • Choice of k? (bias-variance tradeoff again)

  • Choice of metric?

  • Need all the training to be present to classify a new point (“lazy methods”)

  • Surprisingly strong asymptotic results (e.g. no decision rule is more than twice as accurate as 1-NN)


Suppose a regression surface looks like this
Suppose a Regression Surface Looks like this:

want this

not this

Flexible-metric NN Methods try to capture this idea…


Flexible metric nn classification
FMNN

  • Predictors may not all be equally relevant for classifying a new object

  • Furthermore, this differential relevance may depend on the location of the new object

  • FMNN attempts to model this phenomenon


Local relevance
Local Relevance

  • Consider an arbitrary function f on Rp

  • If no values of x are known, have:

  • Suppose xi=z, then:


Local relevance cont
Local Relevance cont.

  • The improvement in squared error provided by knowing xi is:

  • I2i(z) reflects the importance of the ith variable on the variation of f(x) at xi=z


Local relevance cont1
Local Relevance cont.

  • Now consider an arbitrary point z=(z1,…,zp)

  • The relative importance of xi to the variation of f at x=z is:

  • R2i(z)=0 when f(x) is independent of xi at z

  • R2i(z)=1 when f(x) depends only on xi at z


Estimation
Estimation

  • Recall:


On to classification
On To Classification

  • For J-class classification have {yj}, j=1,…,J output variables, yje {0,1}, S yj=1.

  • Can compute:

  • Technical point: need to weight the observations to rectify unequal variances


The machete
The Machete

  • Start with all data points R0

  • Compute

  • Then:

  • Continue until Ri contains K points

M1th order statistic