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Flexible Metric NN ClassificationPowerPoint Presentation

Flexible Metric NN Classification

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## PowerPoint Slideshow about ' Flexible Metric NN Classification' - george

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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:

want this

not this

Flexible-metric NN Methods try to capture this idea…

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

- Consider an arbitrary function f on Rp
- If no values of x are known, have:
- Suppose xi=z, then:

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 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

- Recall:

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

- Start with all data points R0
- Compute
- Then:
- Continue until Ri contains K points

M1th order statistic

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