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Recall Flexibility From Kernel Embedding Idea HDLSS Asymptotics & Kernel Methods.

Published byJocelin Garrett Modified over 8 years ago

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Presentation on theme: "Recall Flexibility From Kernel Embedding Idea HDLSS Asymptotics & Kernel Methods."— Presentation transcript:

1 Recall Flexibility From Kernel Embedding Idea HDLSS Asymptotics & Kernel Methods

2 Recall Flexibility From Kernel Embedding Idea HDLSS Asymptotics & Kernel Methods

3 Recall Flexibility From Kernel Embedding Idea HDLSS Asymptotics & Kernel Methods

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5 Interesting Question: Behavior in Very High Dimension? Implications for DWD:  Recall Main Advantage is for High d  So Not Clear Embedding Helps  Thus Not Yet Implemented in DWD HDLSS Asymptotics & Kernel Methods

6 Batch and Source Adjustment Recall from Class Notes 1/26/16 For Stanford Breast Cancer Data (C. Perou) Analysis in Benito, et al (2004) https://genome.unc.edu/pubsup/dwd/ Adjust for Source Effects –Different sources of mRNA Adjust for Batch Effects –Arrays fabricated at different times

7 Source Batch Adj: Biological Class Col. & Symbols

8 Source Batch Adj: Source Colors

9 Source Batch Adj: PC 1-3 & DWD direction

10 Source Batch Adj: DWD Source Adjustment

11 Source Batch Adj: Source Adj ’ d, PCA view

12 Source Batch Adj: S. & B Adj ’ d, Adj ’ d PCA

13 13 UNC, Stat & OR Why not adjust using SVM? Major Problem: Proj’d Distrib’al Shape Triangular Dist’ns (opposite skewed) Does not allow sensible rigid shift

14 14 UNC, Stat & OR Why not adjust using SVM? Nicely Fixed by DWD Projected Dist’ns near Gaussian Sensible to shift

15 15 UNC, Stat & OR Why not adjust by means? DWD is complicated: value added?  Because it is “cool”  Recall Improves SVM for HDLSS  Good Empirical Success  Routinely Used in Perou Lab  Many Comparisons Done  Similar Lessons from Wistar  Proven Statistical Power

16 16 UNC, Stat & OR Why not adjust by means? But Why Not PAM (~Mean Difference)?  Simpler is Better  Why not means, i.e. point cloud centerpoints? Elegant Answer: Xuxin Liu, et al (2009)

17 17 UNC, Stat & OR Why not adjust by means? But Why Not PAM (~Mean Difference)?  Simpler is Better  Why not means, i.e. point cloud centerpoints? Drawback to PAM:  Poor Handling of Unbalanced Biological Subtypes  DWD more Resistant to Unbalance

18 18 UNC, Stat & OR Why not adjust by means? Toy Example: Gaussian Clusters Two batches (denoted: + o) Two subtypes (red and blue) Goal: bring together – +  o and also +  o Challenge: unequal biological ratios within batches

19 19 UNC, Stat & OR Twiddle ratios of subtypes 2-d Toy Example Balanced Mixture

20 20 UNC, Stat & OR Twiddle ratios of subtypes 2-d Toy Example Unbalanced Mixture (Through “decimation”)

21 21 UNC, Stat & OR Twiddle ratios of subtypes 2-d Toy Example Unbalanced Mixture (Diminishing Discriminatory Power)

22 22 UNC, Stat & OR Twiddle ratios of subtypes 2-d Toy Example Unbalanced Mixture

23 23 UNC, Stat & OR Twiddle ratios of subtypes 2-d Toy Example Unbalanced Mixture

24 24 UNC, Stat & OR Twiddle ratios of subtypes 2-d Toy Example Unbalanced Mixture Note: Losing Distinction To Be Studied

25 25 UNC, Stat & OR Twiddle ratios of subtypes 2-d Toy Example Unbalanced Mixture

26 26 UNC, Stat & OR Why not adjust by means? DWD robust against non-proportional subtypes… Mathematical Statistical Question: Are there mathematics behind this?

27 HDLSS Data Combo Mathematics

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30 Asymptotic Results (as ) Let denote ratio between subgroup sizes

31 HDLSS Data Combo Mathematics Asymptotic Results (as ):  For, PAM Inconsistent Angle(PAM,Truth)  For, PAM Strongly Inconsistent Angle(PAM,Truth)

32 HDLSS Data Combo Mathematics Asymptotic Results (as ):  For, DWD Inconsistent Angle(DWD,Truth)  For, DWD Strongly Inconsistent Angle(DWD,Truth)

33 HDLSS Data Combo Mathematics Value of and, for sample size ratio : , only when  Otherwise for, both are Inconsistent

34 HDLSS Data Combo Mathematics Comparison between PAM and DWD? I.e. between and ?

35 HDLSS Data Combo Mathematics Comparison between PAM and DWD?

36 HDLSS Data Combo Mathematics Comparison between PAM and DWD? I.e. between and ? Shows Strong Difference Explains Above Empirical Observation

37 SVM & DWD Tuning Parameter

38 SVM Tuning Parameter

39 SVM & DWD Tuning Parameter Possible Approaches: Visually Tuned (Can be Effective, But Takes Time, Requires Expertise)

40 SVM & DWD Tuning Parameter Possible Approaches: Visually Tuned Simple Defaults DWD: 100 / median pairwise distance (Surprisingly Useful, Simple Answer) SVM: 1000 (Works Well Sometimes, Not Others)

41 SVM & DWD Tuning Parameter Possible Approaches: Visually Tuned Simple Defaults (Works Well for DWD, Less Effective for SVM)

42 SVM & DWD Tuning Parameter

43 Possible Approaches: Visually Tuned Simple Defaults Cross Validation (Very Popular – Useful for SVM, But Comes at Computational Cost)

44 SVM & DWD Tuning Parameter Possible Approaches: Visually Tuned Simple Defaults Cross Validation Scale Space (Work with Full Range of Choices, Will Explore More Soon)

45 Participant Presentation Frank Teets Characterizing Protein Assembly Graphs

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