Principal Component Analysis

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Principal Component Analysis
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Consider a collection of points
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Suppose you want to fit a line
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Project onto the Line
Consider variance of distribution on the line
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Different line . . .
different variance
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Maximum Variance
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Minimum Variance
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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

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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

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Covariance

• C, of 2 random variables X and Y

where
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Example
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OOBB
Choose bounding box oriented this way
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OOBB Fitting

• Covariance matrix of
• point coordinates describes
• statistical spread of cloud.
• OBB is aligned with directions of
• greatest and least spread
• (which are guaranteed to be orthogonal).

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OOBB
Good Box
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OOBB
Add points worse Box
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OOBB
More points terrible box
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OOBB