STAT 541 Multivariate Analysis Kernels in Multivariate Analysis
- Slides: 22
STAT 541 Multivariate Analysis Kernels in Multivariate Analysis Oleg Melnikov Rice University Department of Statistics Fall 2013 RW
Today’s Agenda • • Inner products Kernels Hilbert space Reproducing kernel map RKHS: Reproducing Kernel Hilbert Space Mercer Theorem (without proof) Kernel Trick Kernel PCA Stat 541, Multivariate Analysis Oleg Melnikov, Rice University, Fall 2013 RW 12/17/2021 2
Inner products • Stat 541, Multivariate Analysis Oleg Melnikov, Rice University, Fall 2013 RW 12/17/2021 3
Inner products • Stat 541, Multivariate Analysis Oleg Melnikov, Rice University, Fall 2013 RW 12/17/2021 4
Concept check: Find imposters • Stat 541, Multivariate Analysis Oleg Melnikov, Rice University, Fall 2013 RW 12/17/2021 5
Kernels (or kernel functions) • Stat 541, Multivariate Analysis Oleg Melnikov, Rice University, Fall 2013 RW 12/17/2021 6
Examples of Kernels • Stat 541, Multivariate Analysis Oleg Melnikov, Rice University, Fall 2013 RW 12/17/2021 7
Positive (semi-)definite kernels • Stat 541, Multivariate Analysis Oleg Melnikov, Rice University, Fall 2013 RW 12/17/2021 8
A space exploration • Stat 541, Multivariate Analysis Oleg Melnikov, Rice University, Fall 2013 RW 12/17/2021 9
• Stat 541, Multivariate Analysis Oleg Melnikov, Rice University, Fall 2013 RW 12/17/2021 10
• Stat 541, Multivariate Analysis Oleg Melnikov, Rice University, Fall 2013 RW 12/17/2021 11
Example: Gaussian RK map • Source: Learning with Kernels… Smola A. J. 2001 Stat 541, Multivariate Analysis Oleg Melnikov, Rice University, Fall 2013 RW 12/17/2021 12
• Stat 541, Multivariate Analysis Oleg Melnikov, Rice University, Fall 2013 RW 12/17/2021 13
Dimension reduction Feature Space , … PCA, CCA, LDA Kernel methods : k. PCA, k. CCA, k. LD A , … • PCA, LDA, CCA: feature extraction and dimension reduction – Note: dimension reduction is in feature space, not the input space • So, the model uses fewer features, not observed variables Stat 541, Multivariate Analysis Oleg Melnikov, Rice University, Fall 2013 RW 12/17/2021 14
Kernel trick • Stat 541, Multivariate Analysis Oleg Melnikov, Rice University, Fall 2013 RW 12/17/2021 15
Kernel trick • Source: Kernel Methods for Pattern Analysis, John Shawe, 2004 Stat 541, Multivariate Analysis Oleg Melnikov, Rice University, Fall 2013 12/17/2021 RW 16
Mercer condition • Stat 541, Multivariate Analysis Oleg Melnikov, Rice University, Fall 2013 RW 12/17/2021 17
Mercer theorem • Stat 541, Multivariate Analysis Oleg Melnikov, Rice University, Fall 2013 RW 12/17/2021 18
PCA vs k. PCA • Source: Learning with. Kernels SVM, … by A. J. Smola, 2001 Stat 541, Multivariate Analysis Oleg Melnikov, Rice University, Fall 2013 RW 12/17/2021 19
k. PCA example Fig 1 • Fig 3 Fig 2 Source: Wikipedia, Author: Petter Strandmark Stat 541, Multivariate Analysis Oleg Melnikov, Rice University, Fall 2013 RW 12/17/2021 20
PCA: Review • Stat 541, Multivariate Analysis Oleg Melnikov, Rice University, Fall 2013 RW 12/17/2021 21
Kernel PCA (k. PCA) • Source: Wikipedia, Author: Petter Strandmark Stat 541, Multivariate Analysis Oleg Melnikov, Rice University, Fall 2013 RW 12/17/2021 22
- Cis419
- Aldi popcorn kernels
- Walnut kernels meaning
- Protein function prediction via graph kernels
- Parallel research kernels
- Iterative deepening search
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- Comp541
- Roger crawfis osu
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- Unc comp 541
- Comp 541
- Comp 541
- Formation of complex ions
- 29 cfr part 541
- Cse 541
- S-o-r
- Roger crawfis
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- What is multivariate analysis?
- Anova meaning
- Nature of multivariate analysis
- Multivariate analysis of variance and covariance