EE16B Designing Information Devices and Systems II
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1 EE16B Designing Information Devices and Systems II Lecture 9A Geometry of SVD, PCA
2 Intro Last time: Described the SVD in Compact matrix form: U1SV1 T Full form: UΣV T Showed a procedure to SVD via A T A Today: Show procedure via AA T Continue proofs (symmetric matrices) PCA
3 Computing the SVD with A T A Proof concept: let Show that, where Show that
4 Alternate Procedure using AA T Step 1: Find eigenvalues of AA T and order s.t. Step 2: Find orthonormal eigenvectors of AA T : Step 3: Set,
5 Example Signs of u 1,v 1 (u 2,v 2 ) can be flipped!
6 Uniqueness of the SVD Find SVD of A
7 Uniqueness of the SVD Find SVD of A
8 Uniqueness of the SVD Find SVD of A
9 Uniqueness of the SVD Find SVD of A
10 Accuracy with Finite Precision Consider matrix A R 512x256 with the following singular values: Precision error A T A single LAPACK single A T A double LAPACK double
11 Accuracy with Finite Precision Consider matrix A R 512x256 with the following singular values: Precision error A T A single LAPACK single A T A double LAPACK double
12 Full Matrix Form of SVD
13 Unitary Matrices Multiplying with unitary matrices does not change the length Example: Rotation, or reflection matrices
14 Geometric Interpretation 1) re-orients without changing length. 2) Stretches along the axis with singlular values 3) re-orients again without changing length
15 Geometric Interpretation Q: What vector would amplify the most?
16 Symmetric Matrices We assumed before that, A T A has only real eigenvalues, r of them are positive and the rest are zero A T A has orthonormal eigenvectors (to be proven next time) For symmetric matrices:
17 Properties of Symmetric Matrices 1) A real-valued symmetric matrix has real eigenvalues and eigenvectors Somehow we need to use the symmetric and real-ness property of Q to show that b==0
18 Properties of Symmetric Matrices real So x is real as well A real-valued symmetric matrix has real eigenvalues and eigenvectors
19 Properties of Symmetric Matrices 2) Eigenvectors of a symmetrix matrix can be chosen to be orthonormal Choose two distinct eigenvalues and vectors Eigenvectors of a symmetrix matrix can be chosen to be orthonormal
20 Positiveness of Eigenvalues 3) If Q can be written as Q = R T R for real R, then Q is positive semidefinite eigenvalues greater of equal to zero If Q can be written as Q = R T R for real R, then Q is positive semidefinite eigenvalues greater of equal to zero
21 Principal Component Analysis Application of the SVD to datasets to learn features PCA is a tool in statistics and machine learning, which can be computed using SVD movies viewers
22 Example -- PCA Consider data s.t. SVD SVD
23 Example -- PCA Consider miterm data mid1 mid2 mid2 students mid1
24 Example -- PCA Consider miterm data mid1 mid2 mid2 students mid1
25 Example -- PCA Consider miterm data consistency mid1 mid2 up-down mobility mid2 students mid1
26 PCA Procedure Remove averages from column of A From A T A, find mid1 mid2 mid2 are principal components! students mid1
27 A T A as sample covariance matrix
28 Example midterm
29 PCA in Genetics Reveals Geography Genes mirror geography within Europe Study: Nature 456, (6 November 2008) Characterized genetic variatios in 3,000 Europeans from 36 Countries Built a matrix of 200K SNPs (single nucleotide polymorphisms) Computed largest 2 principle components Projected subjects on 2 dimentional data Overlayed the result on the map of Europe
30 PC1 could be associated with food PC2 associated with west migration
31
32
33 Interesting conclusions The results have implications for a lot of biomedical research. Many scientists are scanning entire genomes on a hunt for SNPs that affect a person s risk of diseases like cancer or their reaction to drugs. Novembre says that researchers who are running these whole-genome studies need to bear in mind where their sample has come from. Even if a study looks at a small and seemingly related parts of Europe, it would have to adjust for any geographical influences in the genetic variations it uncovers.
34 23 and me
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