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

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Principal component analysis (PCA) finds
linear combinations of variables
Suppose we have p features
X  ( X , ..., X )' , E[X]  0, E[XX' ]  Ψ,
and we want a linear combination U with max. variance
U  a' X, a   , a' a  1.
By the spectral decomposition theorem,
Ψ  V Λ V', V  ( V , ..., V ), orthogonal, Λ  diag ( ,..., ),
the first PC, U  V X, has maximal variance, and
var(U )  var( V1' X )    ...  var(U p )  var(Vp' X )   p .
The total variance is preserved,
1
p
p
pxp
1
p
1
p
'
1
1
1
1
   var( X )   var(U ).
2
total
p
i 1
p
i
i 1
i
Dimension reduction: use first k PCs as new “features”
CASC
Sapphire/IKF 10
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