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Support Vector Machine and String Kernels for

Protein Classification

- Christina Leslie

Department of Computer Science Columbia University

Learning Sequence-based Protein Classification

- Problem classification of protein sequence data

into families and superfamilies - Motivation Many proteins have been sequenced,

but often structure/function remains unknown - Motivation infer structure/function from

sequence-based classification

Sequence Data versus Structure and Function

Sequences for four chains of human hemoglobin

Tertiary Structure

gt1A3NA HEMOGLOBIN VLSPADKTNVKAAWGKVGAHAGEYGAEALE

RMFLSFPTTKTYFPHFDLSHGSAQVKGHGK KVADALTNAVAHVDDMPNA

LSALSDLHAHKLRVDPVNFKLLSHCLLVTLAAHLPAEFTPA VHASLDKF

LASVSTVLTSKYR gt1A3NB HEMOGLOBIN VHLTPEEKSAVTALWG

KVNVDEVGGEALGRLLVVYPWTQRFFESFGDLSTPDAVMGNPKV KAHGK

KVLGAFSDGLAHLDNLKGTFATLSELHCDKLHVDPENFRLLGNVLVCVLA

HHFGK EFTPPVQAAYQKVVAGVANALAHKYH gt1A3NC

HEMOGLOBIN VLSPADKTNVKAAWGKVGAHAGEYGAEALERMFLSFPTT

KTYFPHFDLSHGSAQVKGHGK KVADALTNAVAHVDDMPNALSALSDLHA

HKLRVDPVNFKLLSHCLLVTLAAHLPAEFTPA VHASLDKFLASVSTVLT

SKYR gt1A3ND HEMOGLOBIN VHLTPEEKSAVTALWGKVNVDEVGGE

ALGRLLVVYPWTQRFFESFGDLSTPDAVMGNPKV KAHGKKVLGAFSDGL

AHLDNLKGTFATLSELHCDKLHVDPENFRLLGNVLVCVLAHHFGK EFTP

PVQAAYQKVVAGVANALAHKYH

Function oxygen transport

Structural Hierarchy

- SCOP Structural Classification of Proteins
- Interested in superfamily-level homology remote

evolutionary relationship

Learning Problem

- Reduce to binary classification problem positive

() if example belongs to a family (e.g. G

proteins) or superfamily (e.g. nucleoside

triphosphate hydrolases), negative (-) otherwise - Focus on remote homology detection
- Use supervised learning approach to train a

classifier

Labeled Training Sequences

Classification Rule

Learning Algorithm

Two supervised learning approaches to

classification

- Generative model approach
- Build a generative model for a single protein

family classify each candidate sequence based on

its fit to the model - Only uses positive training sequences
- Discriminative approach
- Learning algorithm tries to learn decision

boundary between positive and negative examples - Uses both positive and negative training

sequences

Hidden Markov Models for Protein Families

- Standard generative model profile HMM
- Training data multiple alignment of examples

from family - Columns of alignment determine model topology

7LES_DROME LKLLRFLGSGAFGEVYEGQLKTE....DSEEPQRVAIKS

LRK....... ABL1_CAEEL IIMHNKLGGGQYGDVYEGYWK.....

...RHDCTIAVKALK........ BFR2_HUMAN

LTLGKPLGEGCFGQVVMAEAVGIDK.DKPKEAVTVAVKMLKDD.....A

TRKA_HUMAN IVLKWELGEGAFGKVFLAECHNLL...PEQDKMLVA

VKALK........

??

??

Profile HMMs for Protein Families

- Match, insert and delete states
- Observed variables symbol sequence, x1 .. xL
- Hidden variables state sequence, ?1 .. ?L
- Parameters transition and emission probabilities
- Joint probability P(x, ? ?)

HMMs Pros and Cons

- Ladies and gentlemen, boys and girls

- Let us leave something for next week

Discriminative Learning

- Discriminative approach
- Train on both positive and negative
- examples to learn classifier

- Modern computational learning theory
- Goal learn a classifier that generalizes well

- to new examples
- Do not use training data to estimate
- parameters of probability distribution
- curse of dimensionality

Learning Theoretic Formalism for Classification

Problem

- Training and test data drawn i.i.d. from fixed

but unknown probability distribution D on - X ? -1,1
- Labeled training set
- S (x1, y1), , (xm, ym)

Support Vector Machines (SVMs)

- We use SVM as discriminative learning algorithm

- Training examples mapped to
- (usually high-dimensional)
- feature space by a feature
- map F(x) (F1(x), , Fd(x))

_

_

_

- Learn linear decision boundary
- Trade-off between maximizing
- geometric margin of the training
- data and minimizing margin violations

_

_

_

SVM Classifiers

- Linear classifier defined in feature space by
- f(x) ltw,xgt b
- SVM solution gives
- w ? ?i xi
- as a linear combination of support vectors, a

subset of the training vectors

w

_

_

b

_

_

_

_

Advantages of SVMs

- Large margin classifier leads to good

generalization (performance on test sets) - Sparse classifier depends only on support

vectors, leads to fast classification, good

generalization - Kernel method as well see, we can introduce

sequence-based kernel functions for use with SVMs

Hard Margin SVM

- Assume training data linearly separable in

feature space - Space of linear classifiers
- fw,b(x) ?w, x? b
- giving decision rule
- hw,b(x) sign(fw,b(x))
- If w 1, geometric margin of training data for

hw,b - ?S MinS yi (?w, xi? b)

w

_

b

_

_

_

_

_

Hard Margin Optimization

- Hard margin SVM optimization given training data

S, find linear classifier hw,b with maximal

geometric margin ?S - Convex quadratic dual optimization problem
- Sparse classifier in term of support vectors

_

_

_

_

_

_

Hard Margin Generalization Error Bounds

- Theorem Cristianini, Shawe-Taylor Fix a real

value M gt 0. For any probability distribution D

on X ? -1,1 with support in a ball of radius R

around the origin, with probability 1-? over m

random samples S, any linear hypothesis h with

geometric margin - ?S ? M on S
- has error no more than
- ErrD(h) ? ?(m, ?, M, R)
- provided that m is big enough

SVMs for Protein Classification

- Want to define feature map from space of protein

sequences to vector space - Goals
- Computational efficiency
- Competitive performance with known methods
- No reliance on generative model general method

for sequence-based classification problems

Spectrum Feature Map for SVM Protein

Classification

- New feature map based on
- spectrum of a sequence

- C. Leslie, E. Eskin, and W. Noble, The Spectrum

Kernel - A String Kernel for SVM Protein Classification.

- Pacific Symposium on Biocomputing, 2002.
- C. Leslie, E. Eskin, J. Weston and W. Noble,
- Mismatch String Kernels for SVM Protein

Classification. - NIPS 2002.

The k-Spectrum of a Sequence

AKQDYYYYEI

- Feature map for SVM based on spectrum of a

sequence - The k-spectrum of a sequence is the set of all

k-length contiguous subsequences that it contains - Feature map is indexed by all possible k-length

subsequences - (k-mers) from the alphabet of
- amino acids
- Dimension of feature space 20k
- Generalizes to any sequence data

AKQ KQD QDY DYY YYY YYY YYE

YEI

k-Spectrum Feature Map

- Feature map for k-spectrum with no mismatches
- For sequence x, F(k)(x) (Ft (x))k-mers t,

where Ft (x) occurrences of t in x

AKQDYYYYEI

( 0 , 0 , , 1 , , 1 , , 2 ) AAA AAC

AKQ DYY YYY

(k,m)-Mismatch Feature Map

- Feature map for k-spectrum, allowing m

mismatches - if s is a k-mer, F(k,m)(s) (Ft(s))k-mers t,

where Ft(s) 1 if s is within m mismatches from

t, 0 otherwise - extend additively to longer sequences x by

summing over all k-mers s in x

AKQ

DKQ

AKY

EKQ

AAQ

The Kernel Trick

- To train an SVM, can use kernel rather than

explicit feature map - For sequences x, y, feature map F, kernel value

is inner product in feature space - K(x, y) ? F(x), F(y) ?
- Gives sequence similarity score
- Example of a string kernel
- Can be efficiently computed via traversal of trie

data structure

Computing the (k,m)-Spectrum Kernel

- Use trie (retrieval tree) to organize lexical

traversal of all instances of k-length patterns

(with mismatches) in the training data - Each path down to a leaf in the trie corresponds

to a coordinate in feature map - Kernel values for all training sequences updated

at each leaf node - If m0, traversal time for trie is linear in size

of training data - Traversal time grows exponentially with m, but

usually small values of m are useful - Depth-first traversal makes efficient use of

memory

Example Traversing the Mismatch Tree

- Traversal for input sequence AVLALKAVLL, k8, m1

Example Traversing the Mismatch Tree

- Traversal for input sequence AVLALKAVLL, k8, m1

Example Traversing the Mismatch Tree

- Traversal for input sequence AVLALKAVLL, k8, m1

Example Computing the Kernel for Pair of

Sequences

- Traversal of trie for k3 (m0)

A

EADLALGKAVF

S1

S2

ADLALGADQVFNG

Example Computing the Kernel for Pair of

Sequences

- Traversal of trie for k3 (m0)

A

EADLALGKAVF

S1

D

S2

ADLALGADQVFNG

Example Computing the Kernel for Pair of

Sequences

- Traversal of trie for k3 (m0)

A

EADLALGKAVF

s1

D

s2

ADLALGADQVFNG

L

Update kernel value for K(s1,s2) by adding

contribution for feature ADL

Fast prediction

- SVM training determines subset of training

sequences corresponding to support vectors and

their weights - (xi, ?i), i 1 .. r
- Prediction with no mismatches
- Represent SVM classifier by hash table mapping

support k-mers to weights - Test sequences can be classified in linear time

via look-up of k-mers - Prediction with mismatches
- Represent classifier as sparse trie traverse

k-mer paths occurring with mismatches in test

sequence

Experimental Design

- Tested with set of experiments on SCOP dataset
- Experiments designed to ask Could the method

discover a new family of a known superfamily?

Diagram from Jaakkola et al.

Experiments

- 160 experiments for 33 target families from 16

superfamilies - Compared results against
- SVM-Fisher
- SAM-T98 (HMM-based method)
- PSI-BLAST (heuristic alignment-based method)

Conclusions for SCOP Experiments

- Spectrum Kernel with SVM performs as well as the

best-known method for remote homology detection

problem - Efficient computation of string kernel
- Fast prediction
- Can precompute per k-mer scores and represent

classifier as a lookup table - Gives linear time prdiction for both spectrum

kernel, (unnormalized) mismatch kernel - General approach to classification problems for

sequence data

Feature Selection Strategies

- Explicit feature filtering
- Compute score for each k-mer, based on training

data statistics, during trie traversal and filter

as we compute kernel - Feature elimination as a wrapper for SVM training
- Eliminate features corresponding to small

components wi in vector w defining SVM classifier - Kernel principal component analysis
- Project to principal components prior to training

Ongoing and Future Work

- New families of string kernels, mismatching

schemes - Applications to other sequence-based

classification problems, e.g. splice site

prediction - Feature selection
- Explicit and implicit dimension reduction
- Other machine learning approaches to using sparse

string-based models for classification - Boosting with string-based classifiers

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