Multivariate analysis of genetic data exploring group diversity

Transcription

1 Multivariate analysis of genetic data exploring group diversity Thibaut Jombart, Marie-Pauline Beugin MRC Centre for Outbreak Analysis and Modelling Imperial College London Genetic data analysis with PR Statistics, Millport Field Station 18 Aug /30

2 Outline Introduction Identifying groups Hierarchical clustering K-means Exploring group diversity Aggregating data Optimizing group differences Discriminant Analysis of Principal Components 2/30

3 Outline Introduction Identifying groups Hierarchical clustering K-means Exploring group diversity Aggregating data Optimizing group differences Discriminant Analysis of Principal Components 3/30

4 Genetic data: introducing group data markers individual sum=1 alleles How to identify groups? How to explore group diversity? 4/30

5 Genetic data: introducing group data markers group 1 group 2 group 3 alleles How to identify groups? How to explore group diversity? 4/30

6 Outline Introduction Identifying groups Hierarchical clustering K-means Exploring group diversity Aggregating data Optimizing group differences Discriminant Analysis of Principal Components 5/30

7 Hierarchical clustering: a variety of algorithms single linkage complete linkage UPGMA Ward... 6/30

8 Rationale 1. compute pairwise genetic distances D (or similarities) 2. group the closest pair(s) together 3. (optional) update D 4. return to 2) until no new group can be made 7/30

9 Rationale 1. compute pairwise genetic distances D (or similarities) 2. group the closest pair(s) together 3. (optional) update D 4. return to 2) until no new group can be made 7/30

10 Rationale 1. compute pairwise genetic distances D (or similarities) 2. group the closest pair(s) together 3. (optional) update D 4. return to 2) until no new group can be made 7/30

11 Rationale 1. compute pairwise genetic distances D (or similarities) 2. group the closest pair(s) together 3. (optional) update D 4. return to 2) until no new group can be made 7/30

12 Rationale individuals a b c d e alleles a b c d e a b c d e D 8/30

13 Differences between algorithms k D =... k,g single linkage: D k,g = min(d k,i, D k,j ) complete linkage: D k,g = max(d k,i, D k,j ) g i D i,j j UPGMA: D k,g = D k,i+d k,j 2 9/30

14 Differences between algorithms k D =... k,g single linkage: D k,g = min(d k,i, D k,j ) complete linkage: D k,g = max(d k,i, D k,j ) g i D i,j j UPGMA: D k,g = D k,i+d k,j 2 9/30

15 Differences between algorithms k D =... k,g single linkage: D k,g = min(d k,i, D k,j ) complete linkage: D k,g = max(d k,i, D k,j ) g i D i,j j UPGMA: D k,g = D k,i+d k,j 2 9/30

16 Differences between algorithms k D =... k,g single linkage: D k,g = min(d k,i, D k,j ) complete linkage: D k,g = max(d k,i, D k,j ) g i D i,j j UPGMA: D k,g = D k,i+d k,j 2 9/30

17 Differences between algorithms Data Single linkage e d c c b a d a e b Variable Complete linkage UPGMA (average linkage) a b a b e e c d d c 10/30

18 ANOVA model: K-means underlying model total var. = (var. between groups) + (var. within groups) variatibility between groups individuals variability within groups 11/30

19 K-means rationale Find groups which minimize within group var. (equally: maximize between group var.). In other words: Identify a partition G = {g 1,..., g k } solving: arg min G={g 1,...,g k } k i g k x i µ k 2 with: x i R p : vector of allele frequencies of individual i µ k R p : vector of means allele frequencies of group k 12/30

20 K-means rationale Find groups which minimize within group var. (equally: maximize between group var.). In other words: Identify a partition G = {g 1,..., g k } solving: arg min G={g 1,...,g k } k i g k x i µ k 2 with: x i R p : vector of allele frequencies of individual i µ k R p : vector of means allele frequencies of group k 12/30

21 K-means rationale Find groups which minimize within group var. (equally: maximize between group var.). In other words: Identify a partition G = {g 1,..., g k } solving: arg min G={g 1,...,g k } k i g k x i µ k 2 with: x i R p : vector of allele frequencies of individual i µ k R p : vector of means allele frequencies of group k 12/30

22 K-means algorithm The K-mean problem is solved by the following algorithm: 1. select random group means (µ k, k = 1,..., K) 2. assign each individual x i to the closest group g k 3. update group means µ k 4. go back to 2) until convergence (groups no longer change) 13/30

23 K-means algorithm The K-mean problem is solved by the following algorithm: 1. select random group means (µ k, k = 1,..., K) 2. assign each individual x i to the closest group g k 3. update group means µ k 4. go back to 2) until convergence (groups no longer change) 13/30

24 K-means algorithm The K-mean problem is solved by the following algorithm: 1. select random group means (µ k, k = 1,..., K) 2. assign each individual x i to the closest group g k 3. update group means µ k 4. go back to 2) until convergence (groups no longer change) 13/30

25 K-means algorithm The K-mean problem is solved by the following algorithm: 1. select random group means (µ k, k = 1,..., K) 2. assign each individual x i to the closest group g k 3. update group means µ k 4. go back to 2) until convergence (groups no longer change) 13/30

26 K-means algorithm initialization allele individual group mean step 1 step 2 14/30

27 K-means: limitations and extensions Limitations slower for large numbers of alleles (e.g. 100,000) K-means does not identify the number of clusters (K) Extension run K-means after dimension reduction using PCA try increasing values of K use Bayesian Information Criterion (BIC) for model selection 15/30

28 K-means: limitations and extensions Limitations slower for large numbers of alleles (e.g. 100,000) K-means does not identify the number of clusters (K) Extension run K-means after dimension reduction using PCA try increasing values of K use Bayesian Information Criterion (BIC) for model selection 15/30

29 Genetic clustering using K-means & BIC (Jombart et al. 2010, BMC Genetics) Simulated data: island model with 6 populations Performances: K-means STRUCTURE on simulated data (various island and stepping stone models) orders of magnitude faster (seconds vs hours/days) 16/30

30 Genetic clustering using K-means & BIC (Jombart et al. 2010, BMC Genetics) Simulated data: island model with 6 populations Performances: K-means STRUCTURE on simulated data (various island and stepping stone models) orders of magnitude faster (seconds vs hours/days) 16/30

31 Outline Introduction Identifying groups Hierarchical clustering K-means Exploring group diversity Aggregating data Optimizing group differences Discriminant Analysis of Principal Components 17/30

32 Why identifying clusters is not the whole story Example of cattle breeds diversity (30 microsatellites, 704 individuals). Group membership probabilities: membership probability Borgou Somba BretPieNoire MaineAnjou Zebu Aubrac Charolais Montbeliard Lagunaire Bazadais Gascon Salers NDama BlondeAquitaine Limousin Important to assess the relationships between clusters. 18/30

33 Why identifying clusters is not the whole story Example of cattle breeds diversity (30 microsatellites, 704 individuals). Group membership probabilities: Multivariate analysis: membership probability MaineAnjou BretPieNoire Charolais Bazadais BlondeAquitaine Montbeliard Limousin Gascon Aubrac Salers DA eigenvalues Lagunaire NDama Somba Borgou Zebu Borgou Zebu Lagunaire NDama Somba Aubrac Bazadais BlondeAquitaine BretPieNoire Charolais Gascon Limousin MaineAnjou Montbeliard Salers Important to assess the relationships between clusters. 18/30

34 Why identifying clusters is not the whole story Example of cattle breeds diversity (30 microsatellites, 704 individuals). Group membership probabilities: Multivariate analysis: membership probability MaineAnjou BretPieNoire Charolais Bazadais BlondeAquitaine Montbeliard Limousin Gascon Aubrac Salers DA eigenvalues Lagunaire NDama Somba Borgou Zebu Borgou Zebu Lagunaire NDama Somba Aubrac Bazadais BlondeAquitaine BretPieNoire Charolais Gascon Limousin MaineAnjou Montbeliard Salers Important to assess the relationships between clusters. 18/30

35 Aggregating data by groups group 1 group 2 average group 3 group 4 group individual alleles alleles average multivariate analysis of group allele frequencies. 19/30

36 Analysing group data Available methods: Principal Component Analysis (PCA) of allele frequency table Genetic distance between populations Principal Coordinates Analysis (PCoA) Correspondance Analysis (CA) of allele counts Criticism: Lose individual information Neglect within-group diversity CA: possible artefactual outliers 20/30

37 Analysing group data Available methods: Principal Component Analysis (PCA) of allele frequency table Genetic distance between populations Principal Coordinates Analysis (PCoA) Correspondance Analysis (CA) of allele counts Criticism: Lose individual information Neglect within-group diversity CA: possible artefactual outliers 20/30

38 Multivariate analysis: reminder Find principal components with maximum total variance. 21/30

39 Multivariate analysis: reminder Find principal components with maximum total variance. 21/30

40 Multivariate analysis: reminder Find principal components with maximum total variance. 21/30

41 Multivariate analysis: reminder Find principal components with maximum total variance. 21/30

42 But total variance may not reflect group differences Need to optimize different criteria. 22/30

43 But total variance may not reflect group differences Need to optimize different criteria. 22/30

44 But total variance may not reflect group differences Need to optimize different criteria. 22/30

45 Optimizing different criteria Similar approaches to PCA can be used to optimize different quantities: PCA: total variance Between-group PCA: variance between groups Within-group PCA: variance within groups Discriminant Analysis: variance between groups / variance within groups 23/30

46 Optimizing different criteria Similar approaches to PCA can be used to optimize different quantities: PCA: total variance Between-group PCA: variance between groups Within-group PCA: variance within groups Discriminant Analysis: variance between groups / variance within groups 23/30

47 Optimizing different criteria Similar approaches to PCA can be used to optimize different quantities: PCA: total variance Between-group PCA: variance between groups Within-group PCA: variance within groups Discriminant Analysis: variance between groups / variance within groups 23/30

48 Optimizing different criteria Similar approaches to PCA can be used to optimize different quantities: PCA: total variance Between-group PCA: variance between groups Within-group PCA: variance within groups Discriminant Analysis: variance between groups / variance within groups 23/30

49 From PCA to DA: increasing group differentiation Density on axis 1 Axis 1 max max Density max Density Density min min Classical PCA Axis 1 Between-group PCA Axis 1 Axis 1 Discriminant Analysis Max. total diversity Max. diversity between groups Max. separation of groups 24/30

50 Discriminant Analysis: limitations and extensions Limitations: DA requires less variables (alleles) than observations (individuals) DA requires uncorrelated variables (no frequencies, no linkage disequilibrium) Discriminant Analysis of Principal Components (DAPC) 1 : data orthogonalisation/reduction using PCA before DA overcomes limitations of DA group membership probabilities, group prediction 1 Jombart et al. 2010, BMC Genetics 25/30

51 Discriminant Analysis: limitations and extensions Limitations: DA requires less variables (alleles) than observations (individuals) DA requires uncorrelated variables (no frequencies, no linkage disequilibrium) Discriminant Analysis of Principal Components (DAPC) 1 : data orthogonalisation/reduction using PCA before DA overcomes limitations of DA group membership probabilities, group prediction 1 Jombart et al. 2010, BMC Genetics 25/30

52 Rationale of DAPC PCA p r Principal components (orthogonal variables) n n s Discriminant Analysis r Principal axes (allele loadings) p n Discriminant Functions (synthetic variables) 26/30

53 PCA of seasonal influenza (A/H3N2) data Data: seasonal influenza (A/H3N2), 500 HA segments. Little temporal evolution, burst of diversity in 2002?? 27/30

54 PCA of seasonal influenza (A/H3N2) data Data: seasonal influenza (A/H3N2), 500 HA segments. Little temporal evolution, burst of diversity in 2002?? 27/30

55 DAPC of seasonal influenza (A/H3N2) data Strong temporal signal, originality of 2006 isolates (new alleles). 28/30

56 DAPC of seasonal influenza (A/H3N2) data Strong temporal signal, originality of 2006 isolates (new alleles). 28/30

57 Other features DAPC can be used to: provides group assignment probabilities can use supplementary individuals can predict group membership of new data can be used for variable selection 29/30

58 Time to get your hands dirty (again)! The pdf of the practical is online: or Google adegenet documents GDAR August /30