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Download Mining the FIRST Astronomical Survey Imola K. Fodor and Chandrika Kamath

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Transcript
```Principal component analysis (PCA) finds
linear combinations of variables
Suppose we have p features
X ï½ ( X , ..., X )' , E[X] ï½ 0, E[XX' ] ï½ Î¨,
and we want a linear combination U with max. variance
U ï½ a' X, a ï ï , a' a ï½ 1.
By the spectral decomposition theorem,
Î¨ ï½ V Î V', V ï½ ( V , ..., V ), orthogonal, Î ï½ diag (ï¬ ,...,ï¬ ),
the first PC, U ï½ V X, has maximal variance, and
var(U ) ï½ var( V1' X ) ï½ ï¬ ï³ ... ï³ var(U p ) ï½ var(Vp' X ) ï½ ï¬ p .
The total variance is preserved,
1
p
p
pxp
1
p
1
p
'
1
1
1
1
ï³ ï½ ï¥ var( X ) ï½ ï¥ var(U ).
2
total
p
i ï½1
p
i
i ï½1
i
Dimension reduction: use first k PCs as new âfeaturesâ
CASC
Sapphire/IKF 10
```
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