Object Orie’d Data Analysis, Last Time

Object Orie’d Data Analysis, Last Time

• Kernel Embedding– Embed data in higher dimensional

manifold

– Gives greater flexibility to linear methods

• Support Vector Machines– Aimed at very non-Gaussian Data

– E.g. from Kernel Embedding

• Distance Weighted Discrimination– HDLSS Improvement of SVM

Support Vector MachinesGraphical View, using Toy Example:

• Find separating plane

• To maximize distances from data to plane

• In particular smallest distance

• Data points closest are called

support vectors

• Gap between is called margin

Support Vector MachinesGraphical View, using Toy Example:

Support Vector MachinesForgotten last time,

Important Extension:

Multi-Class SVMs

Hsu & Lin (2002)

Lee, Lin, & Wahba (2002)

• Defined for “implicit” version

• “Direction Based” variation???

Support Vector MachinesAlso forgotten last time,

Toy examples illustrating

Explicit vs. Implicit

Kernel Embedding

As well as effect of window width, σ

on Gaussian kernel embedding

SVMs, Comput’n & Embedding

For an “Embedding Map”,

e.g.

Explicit Embedding:

Maximize:

Get classification function:

• Straightforward application of embedding

• But loses inner product advantage

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SVMs, Comput’n & EmbeddingImplicit Embedding:

Maximize:

Get classification function:

• Still defined only via inner products• Retains optimization advantage• Thus used very commonly• Comparison to explicit embedding?• Which is “better”???

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Support Vector MachinesTarget Toy Data set:

Support Vector MachinesExplicit Embedding, window σ = 0.1:

Support Vector MachinesExplicit Embedding, window σ = 1:



Support Vector MachinesNotes on Explicit Embedding:

• Too small Poor generalizability

• Too big miss important regions

• Classical lessons from kernel smoothing

• Surprisingly large “reasonable region”

• I.e. parameter less critical (sometimes?)

Also explore projections (in kernel space)

Support Vector MachinesKernel space projection, window σ =

0.1:

Support Vector MachinesKernel space projection, window σ = 1:


10:


100:


100:

Support Vector MachinesNotes on Kernel space projection:

• Too small – Great separation

– But recall, poor generalizability

• Too big no longer separable

• As above:– Classical lessons from kernel smoothing

– Surprisingly large “reasonable region”

– I.e. parameter less critical (sometimes?)

Also explore projections (in kernel space)

Support Vector MachinesImplicit Embedding, window σ = 0.1:

Support Vector MachinesImplicit Embedding, window σ = 0.5:

Support Vector MachinesImplicit Embedding, window σ = 1:

Support Vector MachinesImplicit Embedding, window σ = 10:

Support Vector MachinesNotes on Implicit Embedding:

• Similar Large vs. Small lessons

• Range of “reasonable results”

Seems to be smaller

(note different range of windows)

• Much different “edge” behavior

Interesting topic for future work…

Distance Weighted Discrim’n 2-d Visualization:

Pushes PlaneAway FromData

All PointsHave SomeInfluence

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Distance Weighted Discrim’n References for more on DWD:

• Current paper:Marron, Todd and Ahn (2007)

• Links to more papers:Ahn (2007)

• JAVA Implementation of DWD:caBIG (2006)

• SDPT3 Software:Toh (2007)

Batch and Source Adjustment

Recall from Class Notes 8/28/07• For Stanford Breast Cancer Data (C.

Perou)• Analysis in Benito, et al (2004)

Bioinformatics, 20, 105-114. https://genome.unc.edu/pubsup/dwd/

• Adjust for Source Effects– Different sources of mRNA

• Adjust for Batch Effects– Arrays fabricated at different times

Source Batch Adj: Biological Class Col. &

Symbols

Source Batch Adj: Source Colors

Source Batch Adj: PC 1-3 & DWD direction

Source Batch Adj: DWD Source Adjustment

Source Batch Adj: Source Adj’d, PCA view

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

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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

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Why not adjust using SVM?

Nicely Fixed by DWD

Projected Dist’ns near Gaussian

Sensible to shift

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Why not adjust by means?

DWD is complicated: value added?

Xuxin Liu example…

Key is sizes of biological subtypes

Differing ratio trips up mean

But DWD more robust

(although still not perfect)

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Next time:

Work in before and after, slides like

138-141 from DWDnormPreso.ppt

In Research/Bioinf/caBIG

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Twiddle ratios of subtypes

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DWD robust against non-

proportional subtypes…

Mathematical Statistical Question:

Are there mathematics behind this?

(will answer next time…)

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DWD in Face Recognition

Face Images as Data

(with M. Benito & D. Peña)

Male – Female Difference?

Discrimination Rule?

Represented as long

vector of pixel gray levels

Registration is critical

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UNC, Stat & OR

DWD in Face Recognition, (cont.)

Registered Data

Shifts and scale

Manually chosen

To align eyes and mouth

Still large variation

See males vs. females???

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DWD in Face Recognition , (cont.)

DWD Direction

Good separation

Images “make

sense”

Garbage at ends?

(extrapolation

effects?)

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Unregistered Version

Much blurrier

Since features don’t

properly line up

Nonlinear Variation

But DWD still works

Can see M-F

differ’ce?

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Interesting summary:

Jump between

means

(in DWD direction)

Clear separation of

Maleness vs.

Femaleness

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Fun Comparison:

Jump between means

(in SVM direction)

Also distinguishes

Maleness vs.

Femaleness

But not as well as

DWD

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Analysis of difference: Project onto normals SVM has “small gap” (feels noise artifacts?) DWD “more informative” (feels real structure?)

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DWD in Face Recognition, (cont.)

Current Work:

Focus on “drivers”:

(regions of interest)

Relation to Discr’n?

Which is “best”?

Lessons for human

perception?

Outcomes DataBreast Cancer Study (C. M. Perou):

• Outcome of interest = death or survival

• Connection with gene expression?

Approach:

• Treat death vs. survival during study as “classes”

• Find “direction that best separates the classes”

Outcomes DataFind “direction that best separates the classes”

Outcomes DataFind “direction that best separates the classes”

Outcomes DataFind “direction that best separates classes”• DWD Projection• SVM Projection

Notes:• SVM is “better separated”?

(recall “data piling” problems….)• DWD gives “more spread between sub-

populations”???(perhaps “more stable”?)

Outcomes DataWhich is “better”?

Approach:

• Find “genes of interest”

• To maximize loadings of direction vectors

(reflects pointing in gene direction)

• Show intensity plot (of gene expression)

• Using top 20 genes in each direction


• Study with gene intensity plot

• Order cases by DWD score (proj’n)

• Order genes by DWD loading (vec. entry)

• Reduce to top & bottom 20

• Color map shows gene expression

• Shows genes that drive classification

• Gene names also available

Outcomes DataWhich is “better”? DWD direction

Outcomes DataWhich is “better”? SVM direction


• DWD finds genes showing better separation

• SVM genes are less informative

Outcomes DataHow about Centroid (Mean Diff’nce) Method?


Outcomes DataCompare to DWD direction


• Best yet, in terms of red – green plot?

• Projections unacceptably mixed?

• These are two different goals…

• Try for trade-off?

Scale space approach???

• Interesting philosophical point:

Very simple things often “best”

Outcomes DataWeakness of above analysis:

• Some with “genes prone to disease” have not died yet

• Perhaps can see in DWD expression plot?

Better analysis:

• More sophisticated survival methods

• Work in progress w/ Brent Johnson, Danyu Li, Helen Zhang

Distance Weighted Discrim’n 2=d Visualization:

Pushes PlaneAway FromData

All PointsHave SomeInfluence

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Distance Weighted Discrim’n Maximal Data Piling

HDLSS Discrim’n Simulations

Main idea:

Comparison of

• SVM (Support Vector Machine)

• DWD (Distance Weighted Discrimination)

• MD (Mean Difference, a.k.a. Centroid)

Linear versions, across dimensions


Overall Approach:

• Study different known phenomena– Spherical Gaussians

– Outliers

– Polynomial Embedding

• Common Sample Sizes

• But wide range of dimensions

25 nn

1600,400,100,40,10d


Spherical Gaussians:


Spherical Gaussians:• Same setup as before• Means shifted in dim 1 only,• All methods pretty good• Harder problem for higher dimension• SVM noticeably worse• MD best (Likelihood method)• DWD very close to MD• Methods converge for higher

dimension??

2.21


Outlier Mixture:


Outlier Mixture:80% dim. 1 , other dims 020% dim. 1 ±100, dim. 2 ±500, others 0• MD is a disaster, driven by outliers• SVM & DWD are both very robust• SVM is best• DWD very close to SVM (insig’t

difference)• Methods converge for higher dimension??

Ignore RLR (a mistake)

2.21


Wobble Mixture:


Wobble Mixture:80% dim. 1 , other dims 020% dim. 1 ±0.1, rand dim ±100, others

0• MD still very bad, driven by outliers• SVM & DWD are both very robust• SVM loses (affected by margin push)• DWD slightly better (by w’ted influence)• Methods converge for higher dimension??


2.21


Nested Spheres:

HDLSS Discrim’n SimulationsNested Spheres:

1st d/2 dim’s, Gaussian with var 1 or C2nd d/2 dim’s, the squares of the 1st dim’s(as for 2nd degree polynomial embedding)

• Each method best somewhere• MD best in highest d (data non-Gaussian)• Methods not comparable (realistic)• Methods converge for higher

dimension??• HDLSS space is a strange place


HDLSS Discrim’n SimulationsConclusions:

• Everything (sensible) is best sometimes• DWD often very near best• MD weak beyond Gaussian

Caution about simulations (and examples):• Very easy to cherry pick best ones• Good practice in Machine Learning

– “Ignore method proposed, but read paper for useful comparison of

others”


Caution: There are additional players

E.g. Regularized Logistic Regression

looks also very competitive

Interesting Phenomenon:

All methods come together

in very high dimensions???


Can we say more about:

All methods come together

in very high dimensions???

Mathematical Statistical Question:

Mathematics behind this???

(will answer next time)

Object Orie’d Data Analysis, Last Time

Documents

support vector machinesnotes

support vector machinesalso

explicit embedding

implicit embedding

embedding map

support vectors gap

effect of window width

gaussian kernel embeddingsvms