principal component analysis. consider a collection of points
Post on 22-Dec-2015
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Principal Component Analysis
Consider a collection of points
Suppose you want to fit a line
Consider variance ofdistribution on the line
Project onto the Line
different variance
Different line . . .
Maximum Variance
Minimum Variance
Given by eigenvectorsof covariance matrixof coordinatesof original points
PCA notes…
• Input data set• Subtract the mean to get data set with 0-
mean• Compute the covariance matrix• Compute the eigenvalues and
eigenvectors of the covariance matrix• Choose components and form a feature
vector. Order by eigenvalues – highest to lowest
PCA
• To compress, ignore components of lesser significance
• The feature vector F is a matrix is the matrix of ordered eigenvectors
• Derive the data set in the new coordinates:
• new_data = FT old_data
Covariance
• C, of 2 random variables X and Y
),cov(),cov(),cov(
),cov(),cov(),cov(
),cov(),cov(),cov(
zzzyzx
zyyyyx
zxyxxx
C
1
))((),cov( 1
n
yyxxYX
n
iii
where
Example
Choose bounding boxoriented this way
OOBB
OOBB: Fitting
Covariance matrix ofpoint coordinates describesstatistical spread of cloud.
OBB is aligned with directions ofgreatest and least spread (which are guaranteed to be orthogonal).
Good Box
OOBB
Add points:worse Box
OOBB
More points:terrible box
OOBB
OOBB