1. Identifying Ancestry Informative Markers via the Singular Value Decomposition
Petros Drineas
Rensselaer Polytechnic Institute
Computer Science Department
To access my web page:
drineas

2. Population genetics can help translate that historical message.
Human genetic history
Much of the biological and evolutionary history of our species is written in our DNA sequences.
Population genetics can help translate that historical message.

3. Population genetics can help translate that historical message.
Human genetic history
Much of the biological and evolutionary history of our species is written in our DNA sequences.
Population genetics can help translate that historical message.
The genetic variation among humans is a small portion of the human genome. All humans are almost than 99.9% identical.

4. Our objective
Develop unsupervised, efficient algorithms for the selection of a small set of genetic markers that can be used to
capture population structure, and
predict individual ancestry.

5. Our objective
Develop unsupervised, efficient algorithms for the selection of a small set of genetic markers that can be used to
capture population structure, and
predict individual ancestry.
To this end, we employ matrix algorithms and matrix decompositions such as
the Singular Value Decomposition (SVD), and
the CX decomposition.
We provide the first unsupervised algorithm for selecting of such markers.

6. Overview Basic genetics background
The Singular Value Decomposition (SVD)
The CX decomposition
Selecting Ancestry Informative Markers
The HapMap data
A worldwide set of populations
An admixed population

7. Single Nucleotide Polymorphisms (SNPs)
Single Nucleotide Polymorphisms: the most common type of genetic variation in the genome across different individuals.
They are known locations at the human genome where two alternate nucleotide bases (alleles) are observed (out of A, C, G, T).
SNPs
… AG CT GT GG CT CC CC CC CC AG AG AG AG AG AA CT AA GG GG CC GG AG CG AC CC AA CC AA GG TT AG CT CG CG CG AT CT CT AG CT AG GG GT GA AG …
… GG TT TT GG TT CC CC CC CC GG AA AG AG AG AA CT AA GG GG CC GG AA GG AA CC AA CC AA GG TT AA TT GG GG GG TT TT CC GG TT GG GG TT GG AA …
… GG TT TT GG TT CC CC CC CC GG AA AG AG AA AG CT AA GG GG CC AG AG CG AC CC AA CC AA GG TT AG CT CG CG CG AT CT CT AG CT AG GG GT GA AG …
… GG TT TT GG TT CC CC CC CC GG AA AG AG AG AA CC GG AA CC CC AG GG CC AC CC AA CG AA GG TT AG CT CG CG CG AT CT CT AG CT AG GT GT GA AG …
… GG TT TT GG TT CC CC CC CC GG AA GG GG GG AA CT AA GG GG CT GG AA CC AC CG AA CC AA GG TT GG CC CG CG CG AT CT CT AG CT AG GG TT GG AA …
… GG TT TT GG TT CC CC CG CC AG AG AG AG AG AA CT AA GG GG CT GG AG CC CC CG AA CC AA GT TT AG CT CG CG CG AT CT CT AG CT AG GG TT GG AA …
… GG TT TT GG TT CC CC CC CC GG AA AG AG AG AA TT AA GG GG CC AG AG CG AA CC AA CG AA GG TT AA TT GG GG GG TT TT CC GG TT GG GT TT GG AA …
individuals
There are ¼ 10 million SNPs in the human genome, so this table could have ~10 million columns.

8. Two copies of a chromosome (father, mother)
Focus at a specific locus and assay the observed nucleotide bases (alleles).
SNP: exactly two alternate alleles appear. T C

9. Two copies of a chromosome (father, mother)
Focus at a specific locus and assay the observed alleles.
SNP: exactly two alternate alleles appear. C T
An individual could be:
Heterozygotic (in our study, CT = TC)
Two copies of a chromosome (father, mother)
SNPs
… AG CT GT GG CT CC CC CC CC AG AG AG AG AG AA CT AA GG GG CC GG AG CG AC CC AA CC AA GG TT AG CT CG CG CG AT CT CT AG CT AG GG GT GA AG …
… GG TT TT GG TT CC CC CC CC GG AA AG AG AG AA CT AA GG GG CC GG AA GG AA CC AA CC AA GG TT AA TT GG GG GG TT TT CC GG TT GG GG TT GG AA …
… GG TT TT GG TT CC CC CC CC GG AA AG AG AA AG CT AA GG GG CC AG AG CG AC CC AA CC AA GG TT AG CT CG CG CG AT CT CT AG CT AG GG GT GA AG …
… GG TT TT GG TT CC CC CC CC GG AA AG AG AG AA CC GG AA CC CC AG GG CC AC CC AA CG AA GG TT AG CT CG CG CG AT CT CT AG CT AG GT GT GA AG …
… GG TT TT GG TT CC CC CC CC GG AA GG GG GG AA CT AA GG GG CT GG AA CC AC CG AA CC AA GG TT GG CC CG CG CG AT CT CT AG CT AG GG TT GG AA …
… GG TT TT GG TT CC CC CG CC AG AG AG AG AG AA CT AA GG GG CT GG AG CC CC CG AA CC AA GT TT AG CT CG CG CG AT CT CT AG CT AG GG TT GG AA …
… GG TT TT GG TT CC CC CC CC GG AA AG AG AG AA TT AA GG GG CC AG AG CG AA CC AA CG AA GG TT AA TT GG GG GG TT TT CC GG TT GG GT TT GG AA …
individuals

10. Two copies of a chromosome (father, mother)
Focus at a specific locus and assay the observed alleles.
SNP: exactly two alternate alleles appear. C C
Two copies of a chromosome (father, mother)
An individual could be:
Heterozygotic (in our study, CT = TC)
Homozygotic at the first allele, e.g., C
SNPs
… AG CT GT GG CT CC CC CC CC AG AG AG AG AG AA CT AA GG GG CC GG AG CG AC CC AA CC AA GG TT AG CT CG CG CG AT CT CT AG CT AG GG GT GA AG …
… GG TT TT GG TT CC CC CC CC GG AA AG AG AG AA CT AA GG GG CC GG AA GG AA CC AA CC AA GG TT AA TT GG GG GG TT TT CC GG TT GG GG TT GG AA …
… GG TT TT GG TT CC CC CC CC GG AA AG AG AA AG CT AA GG GG CC AG AG CG AC CC AA CC AA GG TT AG CT CG CG CG AT CT CT AG CT AG GG GT GA AG …
… GG TT TT GG TT CC CC CC CC GG AA AG AG AG AA CC GG AA CC CC AG GG CC AC CC AA CG AA GG TT AG CT CG CG CG AT CT CT AG CT AG GT GT GA AG …
… GG TT TT GG TT CC CC CC CC GG AA GG GG GG AA CT AA GG GG CT GG AA CC AC CG AA CC AA GG TT GG CC CG CG CG AT CT CT AG CT AG GG TT GG AA …
… GG TT TT GG TT CC CC CG CC AG AG AG AG AG AA CT AA GG GG CT GG AG CC CC CG AA CC AA GT TT AG CT CG CG CG AT CT CT AG CT AG GG TT GG AA …
… GG TT TT GG TT CC CC CC CC GG AA AG AG AG AA TT AA GG GG CC AG AG CG AA CC AA CG AA GG TT AA TT GG GG GG TT TT CC GG TT GG GT TT GG AA …
individuals

11. Two copies of a chromosome (father, mother)
Focus at a specific locus and assay the observed alleles.
SNP: exactly two alternate alleles appear. T T
Two copies of a chromosome (father, mother)
An individual could be:
Heterozygotic (in our study, CT = TC)
Homozygotic at the first allele, e.g., C
Homozygotic at the second allele, e.g., T
Encode as 0
Encode as +1
Encode as -1
SNPs
… AG CT GT GG CT CC CC CC CC AG AG AG AG AG AA CT AA GG GG CC GG AG CG AC CC AA CC AA GG TT AG CT CG CG CG AT CT CT AG CT AG GG GT GA AG …
… GG TT TT GG TT CC CC CC CC GG AA AG AG AG AA CT AA GG GG CC GG AA GG AA CC AA CC AA GG TT AA TT GG GG GG TT TT CC GG TT GG GG TT GG AA …
… GG TT TT GG TT CC CC CC CC GG AA AG AG AA AG CT AA GG GG CC AG AG CG AC CC AA CC AA GG TT AG CT CG CG CG AT CT CT AG CT AG GG GT GA AG …
… GG TT TT GG TT CC CC CC CC GG AA AG AG AG AA CC GG AA CC CC AG GG CC AC CC AA CG AA GG TT AG CT CG CG CG AT CT CT AG CT AG GT GT GA AG …
… GG TT TT GG TT CC CC CC CC GG AA GG GG GG AA CT AA GG GG CT GG AA CC AC CG AA CC AA GG TT GG CC CG CG CG AT CT CT AG CT AG GG TT GG AA …
… GG TT TT GG TT CC CC CG CC AG AG AG AG AG AA CT AA GG GG CT GG AG CC CC CG AA CC AA GT TT AG CT CG CG CG AT CT CT AG CT AG GG TT GG AA …
… GG TT TT GG TT CC CC CC CC GG AA AG AG AG AA TT AA GG GG CC AG AG CG AA CC AA CG AA GG TT AA TT GG GG GG TT TT CC GG TT GG GT TT GG AA …
individuals

12. Two copies of a chromosome (father, mother)
Focus at a specific locus and assay the observed alleles.
SNP: exactly two alternate alleles appear.
Two copies of a chromosome (father, mother)
An individual could be:
Heterozygotic (in our study, CT = TC)
Homozygotic at the first allele, e.g., C
Homozygotic at the second allele, e.g., T
Rare (or Minor) Allele Frequency, RAF (or MAF):
The frequency of the “less frequent” allele in a SNP
e.g., freq(C) = 5/14.
SNPs
… AG CT GT GG CT CC CC CC CC AG AG AG AG AG AA CT AA GG GG CC GG AG CG AC CC AA CC AA GG TT AG CT CG CG CG AT CT CT AG CT AG GG GT GA AG …
… GG TT TT GG TT CC CC CC CC GG AA AG AG AG AA CT AA GG GG CC GG AA GG AA CC AA CC AA GG TT AA TT GG GG GG TT TT CC GG TT GG GG TT GG AA …
… GG TT TT GG TT CC CC CC CC GG AA AG AG AA AG CT AA GG GG CC AG AG CG AC CC AA CC AA GG TT AG CT CG CG CG AT CT CT AG CT AG GG GT GA AG …
… GG TT TT GG TT CC CC CC CC GG AA AG AG AG AA CC GG AA CC CC AG GG CC AC CC AA CG AA GG TT AG CT CG CG CG AT CT CT AG CT AG GT GT GA AG …
… GG TT TT GG TT CC CC CC CC GG AA GG GG GG AA CT AA GG GG CT GG AA CC AC CG AA CC AA GG TT GG CC CG CG CG AT CT CT AG CT AG GG TT GG AA …
… GG TT TT GG TT CC CC CG CC AG AG AG AG AG AA CT AA GG GG CT GG AG CC CC CG AA CC AA GT TT AG CT CG CG CG AT CT CT AG CT AG GG TT GG AA …
… GG TT TT GG TT CC CC CC CC GG AA AG AG AG AA TT AA GG GG CC AG AG CG AA CC AA CG AA GG TT AA TT GG GG GG TT TT CC GG TT GG GT TT GG AA …
individuals

13. Population Genetics 101
Genetic diversity and population (sub)structure is caused by…
Mutation
Mutations are changes to the base pair sequence of the DNA.
Natural selection
Genotypes that correspond to favorable traits and are heritable become more common in successive generations of a population of reproducing organisms.

14. Population Genetics 101
Genetic diversity and population (sub)structure is caused by…
Mutation
Mutations are changes to the base pair sequence of the DNA.
Natural selection
Genotypes that correspond to favorable traits and are heritable become more common in successive generations of a population of reproducing organisms.
 Mutations increase genetic diversity.
Under natural selection, beneficial mutations increase in frequency, and vice versa.

15. Population Genetics 101 (cont’d)
Genetic drift
Sampling effects on evolution!
Example: say that the RAF of a SNP in a small population is p.
The offspring generation would (in expectation) have a RAF of p as well for the same SNP.
In reality, it will have a RAF of p’ (a drifted frequency) …
Gene flow
Transfer of alleles between populations.

16. Population Genetics 101 (cont’d)
Genetic drift
Sampling effects on evolution!
Example: say that the RAF of a SNP in a small population is p.
The offspring generation would (in expectation) have a RAF of p as well for the same SNP.
In reality, it will have a RAF of p’ (a drifted frequency) …
Gene flow
Transfer of alleles between populations.
Genetic drift are a stronger force than natural selection in small populations.

17. Population Genetics 101 (cont’d)
Examples:
Population bottlenecks
Wars, epidemics, natural disasters wipe off a part of the population and lead to genetic drift in offspring population.
Founders effect
A new population is established by a small number of individuals, carrying only a fraction of the original population's genetic variation.
Non-random mating
Reduces interaction between (sub)populations.
Other demographic events
Immigration, etc.

18. Images courtesy of Kenneth Kidd,
Early Homo sapiens sapiens in Africa
150,000 to 100,000 years ago
Images courtesy of Kenneth Kidd,

19. colonizing South West Asia,
Homo sapiens sapiens
colonizing South West Asia,
approx. 100,000 years ago.

20. Homo sapiens sapiens
approx. 40,000 years ago.

21. Why study population structure?
History of human populations
Genealogy
Forensics
Mapping causative genes for common complex disorders
(e.g. diabetes, heart conditions, obesity, etc.)

22. Why are SNPs really important?
Genome Wide Association Studies (GWAS):
Locating causative genes for common complex disorders is based on identifying association between affection status and known SNPs.
No prior knowledge about the function of the gene(s) or the etiology of the disorder is necessary.

23. Why are SNPs really important?
Genome Wide Association Studies (GWAS):
Locating causative genes for common complex disorders is based on identifying association between affection status and known SNPs.
No prior knowledge about the function of the gene(s) or the etiology of the disorder is necessary.
The subsequent investigation of candidate genes that are in physical proximity with the associated SNPs is the first step towards understanding the etiological pathway of a disorder and designing a drug.
Numerous such studies revealed (and will continue to reveal) correlations between genes and diseases.

24. Recall our objective
Develop unsupervised, efficient algorithms for the selection of a small set of SNPs that can be used to
capture population structure, and
predict individual ancestry.
Why? cost efficiency, identification of regions of natural selection.
Let’s discuss (briefly) prior work …

25. Inferring population structure
Oceania
Africa
Europe
Middle East
Central Asia
East Asia
America
377 STRPs, Rosenberg et al. Science ’02
Developed a software package called STRUCTURE.
Works well, however:
It is based on explicit assumptions that may not always hold.
It cannot handle large genome-wide datasets (computationally expensive).

26. Selecting ancestry informative markers
Existing methods (Fst, Informativeness, δ)
Rosenberg et al. Am J Hum Genet ’03
Allele frequency based.
Require prior knowledge of individual ancestry (supervised).
Such knowledge may not be available.
(E.g., populations of complex ancestry, large multi-centered studies of anonymous samples, etc.)

27. Overview Basic genetics background
The Singular Value Decomposition (SVD)
The CX decomposition
Selecting Ancestry Informative Markers
The HapMap data
A worldwide set of populations
An admixed population

28. The Singular Value Decomposition (SVD)
Let A be a matrix with m rows (one for each subject) and n columns (one for each SNP).
Matrix rows: points (vectors) in a Euclidean space,
e.g., given 2 objects (x & d), each described with respect to two features, we get a 2-by-2 matrix.
Two objects are “close” if the angle between their corresponding vectors is small.

29. SVD, intuition
Let the blue circles represent m data points in a 2-D Euclidean space.
Then, the SVD of the m-by-2 matrix of the data will return …
2nd (right) singular vector
2nd (right) singular vector:
direction of maximal variance, after removing the projection of the data along the first singular vector.
1st (right) singular vector
1st (right) singular vector:
direction of maximal variance,

30. Singular values
2nd (right) singular vector 2
1st (right) singular vector
1: measures how much of the data variance is explained by the first singular vector.
2: measures how much of the data variance is explained by the second singular vector. 1

31. SVD: formal definition
Let 1 ¸ 2 ¸ … ¸  be the entries of .
Exact computation of the SVD takes O(min{mn2 , m2n}) time.
The top k left/right singular vectors/values can be computed faster using Lanczos/Arnoldi methods.
: rank of A
U (V): orthogonal matrix containing the left (right) singular vectors of A.
S: diagonal matrix containing the singular values of A.

32. Rank-k approximations via the SVD= U  VT
features
sig.
significant
noise
noise =
significant
noise
objects

33. Rank-k approximations (Ak)
Uk (Vk): orthogonal matrix containing the top k left (right) singular vectors of A.
k: diagonal matrix containing the top k singular values of A.
Also,

34. PCA and SVD
Principal Components Analysis (PCA) essentially amounts to the computation of the Singular Value Decomposition (SVD) of a covariance matrix.
SVD is the algorithmic tool behind MultiDimensional Scaling (MDS) and Factor Analysis.

35. Back to data: the HapMap project
(~$130,000,000 funding from NIH)
Map approx. 3.1 million SNPs for 270 individuals from 4 populations (YRI, CEU, CHB, JPT), to create a “genetic map” for researchers.
Let A be the 90£2.7 million matrix of the CHB and JPT population in HapMap.
Run SVD on A, keep the two (left) singular vectors (eigenSNPs).
Run a (naïve, e.g., k-means) clustering algorithm to split the data in two clusters.

36. PCA for analyzing population structure was introduced by L
PCA for analyzing population structure was introduced by L. Cavalli-Sforza in the ’70s.

37. (Mathematically: spanning the same subspace.)
PCA for analyzing population structure was introduced by L. Cavalli-Sforza in the ’70s.
Not altogether satisfactory: the (top two left) singular vectors (eigenSNPs) are linear combinations of all SNPs, and can not be genotyped!
Can we find actual SNPs that capture the “information” in the eigenSNPs?
(Mathematically: spanning the same subspace.)

38. Overview Basic genetics background
The Singular Value Decomposition (SVD)
The CX decomposition
Selecting Ancestry Informative Markers
The HapMap data
A worldwide set of populations
An admixed population

39. CX decomposition
Carefully chosen X
Goal: make (some norm) of A-CX small.
c columns of A
Why?
If A is an subject-SNP matrix, then selecting representative columns is equivalent to selecting representative SNPs to capture the same structure as the top eigenSNPs.
We want c as small as possible!

40. CX decomposition Goal: make (some norm) of A-CX small.
Carefully chosen X
Goal: make (some norm) of A-CX small.
c columns of A
Theory: for any matrix A, we can find C such that
is almost equal to the norm of A-Ak with c ¼ k.

41. CX decomposition
c columns of A
Easy to prove that optimal X = C+A. (C+ is the Moore-Penrose pseudoinverse of C.)
Thus, the challenging part is to find good columns (SNPs) of A to include in C.

42. CX decomposition
c columns of A
Easy to prove that optimal X = C+A. (C+ is the Moore-Penrose pseudoinverse of C.)
Thus, the challenging part is to find good columns (SNPs) of A to include in C.
From a mathematical perspective, this is a hard combinatorial problem, closely related to the so-called Column Subset Selection Problem (CSSP).
The CSSP has been heavily studied in Numerical Linear Algebra.

43. A theorem (Drineas, Mahoney, and Muthukrishnan ’06, ’07)
Given an m-by-n matrix A, there exists an algorithm that picks, in expectation,
at most O( k log k / 2 ) columns of A
runs in O(mn2) time (m · n), and with probability at least

44. The CX algorithm Input: m-by-n matrix A, integer k
Output: C, the matrix consisting of the selected columns
CX algorithm
Compute probabilities pj summing to 1
Let c = O(k log k / 2)
For each j = 1,2,…,n, pick the j-th column of A with probability min{1,cpj}
Let C be the matrix consisting of the chosen columns
(C has – in expectation – at most c columns)

45. Subspace sampling (Frobenius norm)
Vk: orthogonal matrix containing the top k right singular vectors of A.
S k: diagonal matrix containing the top k singular values of A.
Remark: The rows of VkT are orthonormal vectors, but its columns (VkT)(i) are not.

46. Subspace sampling (Frobenius norm)
Vk: orthogonal matrix containing the top k right singular vectors of A.
S k: diagonal matrix containing the top k singular values of A.
Remark: The rows of VkT are orthonormal vectors, but its columns (VkT)(i) are not.
Subspace sampling in O(mn2) time
Normalization s.t. the pj sum up to 1

47. Deterministic variant of CX
Input: m-by-n matrix A,
integer k, and
c (number of SNPs to pick)
Output: the selected SNPs
CX algorithm
Compute the scores pj
Pick the columns (SNPs) corresponding to the top c scores.

48. Deterministic variant of CX
Input: m-by-n matrix A,
integer k, and
c (number of SNPs to pick)
Output: the selected SNPs: we will call them PCA Informative Markers or PCAIMs
CX algorithm
Compute the scores pj
Pick the columns (SNPs) corresponding to the top c scores.
In order to estimate k for SNP data, we developed a permutation-based test to determine whether a certain principal component is significant or not.
(A similar test was presented in Patterson et al. PLoS Genet ’06.)

49. Number of SNPs Misclassifications50 6
100+ 1
As good as the best existing metric (informativeness).
However, our metric is unsupervised!

50. Overview Basic genetics background
The Singular Value Decomposition (SVD)
The CX decomposition
Selecting Ancestry Informative Markers
The HapMap data
A worldwide set of populations
An admixed population

51. Worldwide data
European Americans
South Altaians - -
Spanish
African Americans
Japanese
Chinese
Puerto Rico
Mende
Nahua
Mbuti
Mala
Burunge
Quechua
Africa
Europe
E Asia
America
274 individuals, 12 populations, ~10,000 SNPs using the Affymetrix array (Shriver et al. Hum Genom ‘05)

52. (Africa, Europe, Asia, America)
Selecting PCA-correlated SNPs for individual assignment to four continents
(Africa, Europe, Asia, America)
Afr
Eur
Asi
Ame
Africa
Europe
Asia
America
* top 30 PCA-correlated SNPs
PCA-scores
SNPs by chromosomal order

53. (Use a subset of SNPs, cluster the individuals using k-means.)
Correlation coefficient between true and predicted membership of an individual to a particular geographic continent.
(Use a subset of SNPs, cluster the individuals using k-means.)

54. Cross-validation on HapMap data

55. All SNPs and 10 principal components: perfect clustering! B
9 indigenous populations from four different continents (Africa, Europe, Asia, Americas)
All SNPs and 10 principal components: perfect clustering!
50 PCAIMs SNPs, almost perfect clustering; 200 PCAIMs SNPs, perfect clustering.
Informativeness performed poorly (maybe an artifact of k-means clustering…)

56. Admixed populations
Two (independent) Puerto Rican datasets, two major axes of variation:
Dataset A: 192 individuals, ~7,000 SNPs
Dataset B: 30 individuals, same ~7,000 SNP
European - West African axis of variation
Native American axis of variation

57. West Africa
America
Europe

58. Europe-Africa axis of variation
Ancestry coefficient: location of the individual in the Europe - Africa axis.
Africa

59. Predicting ancestry using PCAIMs

60. Predicting ancestry using PCAIMs

61. Cross-validation on Puerto-Rican dataset B

62. Conclusions
Using linear algebraic techniques (e.g., matrix decompositions) we selected markers that capture population structure.
Our technique requires no prior assumptions and builds upon the power of SVD to identify population structure in various settings, including admixed populations.
Prior theoretical work and mathematical understanding of the underlying problem was fundamental in designing our algorithm!

63. Acknowledgements Collaborators Students
P. Paschou, Democritus University, Greece
E. Ziv, UCSF
E. Burchard, UCSF
M.W. Mahoney, Yahoo! Research
K. K. Kidd, Yale University
M. Shriver, Penn State
R. Krauss, Oakland Research Institute
Students
Asif Javed, RPI
Jamey Lewis, RPI
Funding
NSF CAREER award to Petros Drineas
NIH U19 AG23122 and K22CA Breast Cancer Research Program grant to E. Ziv
Hellenic Endocrine Society Research grant award to P. Paschou
Ref: Paschou, Elad, Burchard,…, Mahoney, and Drineas (2007) PLoS Genetics, 3: e160