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**Feature Extraction -definition**
Given a set of features F = {𝒳1,.....,𝒳N}
the **Feature Extraction** ("Construction") problem is to map F to some feature set F" that maximizes the learner's ability to classify patterns.
Find a projection matrix w from N-dimensional to M-dimensional vectors that keeps error low.

Assume that N features are linear combination of M < N vectors
Zi = Wi1𝒳i1 + ......+ Wid𝒳iN
Z = Wt𝒳
What we expect from such basis
- Uncorrelated cannot be reduced further
- Have large variance or otherwise bear no information.
**Algebraic definition of PCs **
PCA
**PCA for image Compression **

**Is PCA a good criterion for classification ?**

- Data variation determines the projection direction
- What's missing ?
- Class information
**What is a good projection ?**
- Similarly, what is a good criterion ?
- Separating different classes

**What class information may be useful ?**

Between-class distance
- Distance between the centroids of different classes
Within-class distance
- Accumulated distance of an instance to the centroid of its class.
Linear discriminant analysis (LDA) finds most discriminant projection by
- maximizing between-class distance
- and minimizing within-class distance

**Linear Discriminant Analysis**

Find a low-dimensional space such that when 𝓍 is projected, classes are well-separated.

**Means and Scatter after projection**

**Good Projection**

- Means are as far away as possible
- Scatter is small as possible
- Fisher Linear Discriminant
J(w) = (m1 - m2)2 /s12 + s2 square

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