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- 1. Application of Bayesian and Sparse Network Models for Assessing Linkage Disequilibrium in Animals and Plants Master’s Defense Gota Morota Dec 5, 2011 1 / 44
- 2. Outline Overview of standard LD metrics Bayesian Network L1 regularized Markov Network Exome sequence analysis 2 / 44
- 3. Outline Overview of standard LD metrics Bayesian Network L1 regularized Markov Network Exome sequence analysis 3 / 44
- 4. Outline Overview of standard LD metrics Bayesian Network L1 regularized Markov Network Exome sequence analysis 4 / 44
- 5. Outline Overview of standard LD metrics Bayesian Network L1 regularized Markov Network Exome sequence analysis 5 / 44
- 6. Linkage Disequilibrium (LD) Deﬁnition non-random association of alleles at different loci • also known as gametic phase disequilibrium • ﬁrst used in 1960 (Lewontin and Kojima) • has been used extensively in many area 1. genome enabled selection 2. genome-wide association study 3. understand past evolutionary and demographic events 6 / 44
- 7. Systems Genetics Figure 1: A multi-dimensional gene feature/knowledge network Purpose of the thesis • take the view that loci associate and interact together as a network • evaluate LD reﬂecting the biological nature that loci interact as a complex system 7 / 44
- 8. Graphical Models • provide a way to visualize the structure of a model • a graph is comprised of 1. nodes (vertices) → random variables 2. edges (links, arcs) → probabilistic relationship 8 / 44
- 9. Graphical Models • provide a way to visualize the structure of a model • a graph is comprised of 1. nodes (vertices) → random variables 2. edges (links, arcs) → probabilistic relationship Figure 2: A soccer ball 8 / 44
- 10. Bayesian Networks (BN) • directed acyclic graph (DAG) with graphical structure G = (V , E ) • Application of BN to genomic 1. DNA microarrays (Friedman et al. 2000) 2. protein signaling networks (Sachs et al. 2005) 3. GWAS (Sebastiani et al. 2005) 4. genome-enabled prediction and classiﬁcation in livestock (Long et al. 2009) Objective I Apply BN to uncover associations among a set of marker loci found to have the strongest effects on milk protein yield in Holstein cattle. 9 / 44
- 11. Factorization Properties p (a , b , c ) = p (a )p (b |a )p (c |a , b ) (1) Figure 3: The joint distribution of x1 , · · · , x3 10 / 44
- 12. Factorization Properties (cont.) p P (X1 , ..., Xp ) = P (Xj |Pa (Xj )) (2) j =1 • Pa (Xj ) is a set of parent nodes of Xj • a node is conditionally independent of its non-descendants given its parents One restriction Must be no directed cycles (need to be DAG). 11 / 44
- 13. Structure Learning • Local score-based algorithms • Scoring metrics • • • • Akaike Information Criterion (AIC) Minimum Description Length (MDL) K2 Bayesian Dirichlet Equivalent (BDe) • Constraint-based algorithms (Causal inference algorithms) • Koller-Sahami (KS) • Grow Shrink (GS) • Incremental Association Markov Blanket (IAMB) 12 / 44
- 14. IAMB algorithm Incremental Association Markov Blanket (Tsamardinos et al. 2003) 1. Compute Markov Blankets (MB) 2. Compute Graph Structure 3. Orient Edges Figure 4: The Markov Blanket of a node xi 13 / 44
- 15. Identifying the MB of a node • Growing phase • heuristic funtion: f (X ; T |CMB ) = MI(X ; T |CMB ) = cmb ∈CMB P (CMB ) P (X , T |CMB ) P (X , T |CMB ) log P (X |CMB )P (T |CMB ) x ∈X t ∈T • conditional independence tests (pearson’s χ2 test): H0 : P (X , T |CMB ) = P (X |CMB ) · P (T |CMB ) (do not add X ) HA : P (X , T |CMB ) P (X |CMB ) · P (T |CMB ) (add X to the CMB) • Shrinking phase • conditional independence tests (pearson’s χ2 test): H0 : P (X , T |CMB − X ) P (X |CMB − X ) · P (T |CMB − X ) (keep X ) HA : P (X , T |CMB − X ) = P (X |CMB − X ) · P (T |CMB − X ) (remove X 14 / 44
- 16. Identifying the MB of a node (example) Suppose we have target variable T , and a set of nodes X = (A , B , C , D , E , F ) • Growing phase 1. MI(T , X |CMB = (∅)) → X = (C , E , A , D , B , F ) 2. CI(T , C |CMB =(∅)) → CMB = (C ) 3. MI(T , X |CMB = (C)) → X = (A , E , D , F , B ) 4. CI(T , A |CMB =(C)) → CMB = (C , A ) • Shrinking phase 1. CI(T , C |CMB = (C , A ) − C ) 2. CI(T , A |CMB = (C , A ) − A ) 15 / 44
- 17. Network Structure Algorithm Suppose Y ∈ MB (T ). Then T and Y are connected if they are conditionaly dependent given all subsets of the smaller of MB (T ) − (Y ) and MB (Y ) − (T ). Example: • MB (T ) = (A , B , Y ), MB (Y ) = (C , D , E , F , T ) • since MB (T ) < MB (Y ), independence tests are conditional on all subsets of MB (T ) − (Y ) = (A , B ). • if any of the CI(T , Y |{}), CI(T , Y |{A }), CI(T , Y |{B }), andCI(T , Y |{A , B }) imply conditional independence, ↓ • T and Y are considered separate (spouses) • repeat for T ∈ S and Y ∈ MB (T ), 16 / 44
- 18. Data, Missing Genotype Imputation, and Subset selection 1. Data • 4,898 progeny tested Holstein bulls (USDA-ARS AIPL) • 37,217 SNP markers (MAF > 0.025) • Predicted Transmitting Ability (PTA) for milk protein yield 2. Missing genotypes imputation • fastPHASE (Scheet and Stephens, 2006) 3. Select 15-30 SNPs • Bayesian LASSO (BLR R pakcage, Perez et al, 2010) ´ 4. SNPs ranking strategies ˆ • |βj | ˆ • |βj |/ Var (βj ) ˆ • 2pj (1 − pj )β2 j 17 / 44
- 19. Results – Top 15 SNPs from Strategy 1 IAMB algorithm Pairwise LD among SNPs (r2) J d A c b a Z L Y X M N W V U F T S B K G R Q P H O N O E M L K J I I H G F E C D C B R2 Color Key A 0 Figure 5: r 2 0.2 0.4 0.6 0.8 1 D Figure 6: IAMB 18 / 44
- 20. Conclusion The result conﬁrms that LD relationships are of a multivariate nature, and that r 2 gives an incomplete description and understanding of LD. • capture association among SNPs as a network • no limitation with respect to the type of loci 19 / 44
- 21. Possible Improvements • associations among loci are assumed bi-directional • LD is expected to decline rapidly as the physical distance between two loci increases, and that pairs of loci on different chromosomes rarely show high LD • conditional independence property 20 / 44
- 22. Possible Improvements • associations among loci are assumed bi-directional • LD is expected to decline rapidly as the physical distance between two loci increases, and that pairs of loci on different chromosomes rarely show high LD • conditional independence property ⇓ therefore • undirected networks • sparisty • conditional independence property 20 / 44
- 23. Undirected Graphical Models Figure 7: An undirected graph • Markov networks (Markov random ﬁelds) • G = (V , E ) • express an afﬁnity instead of a causal relationship 21 / 44
- 24. Pairwise Conditional Independence Property pairwise conditional independence property • an absence of edge between two nodes, xj and xk , implies conditional independence, given all other nodes p (xj , xk |x−j ,−k ) = p (xj |x−j ,−k )p (xj |x−j ,−k ) (3) In Figure (8), (a ⊥ d |b , c) and (b ⊥ d |a , c). Figure 8: Example 1 22 / 44
- 25. Cliques A clique is a subset of nodes in a graph such that every pair of nodes are connected by edges • (a) {X , Y }, {Y , Z } • (b) {X , Y , W }, {Z } • (c) {X , Y }, {Y , Z }, {Z , W }, {X , W } • (d) {X , Y }, {Y , Z }, {Z , W } Maximum cliques Figure 9: Example 3 a maximum clique is deﬁned as the clique having the largest size 23 / 44
- 26. The Factorization of Markov Networks the Hammersley-Clifford theorem for any positive distributions, the distribution factorizes according to the Markov network structure deﬁned by cliques. Consider X = xi , · · · , xn , p (X ) = 1 Z φc (Xc ) (4) C ∈G where Z is a normalizing constant deﬁned by φc (Xc ) Z= (5) x C ∈G and φ is called a potential function or a clique potential. 24 / 44
- 27. The Factorization of Markov Networks (cont.) • the sets of two node cliques (a , b ), (a , d ), (b , d ) and (b , c ), (c , d ), (b , d ) • maximum cliques (a , b , d ) and (b , c , d ) respectively. Figure 10: Example 5 1 φ1 (a , b , d ) · φ2 (b , c , d ) (6) Z 1 (7) P (a , b , c , d ) = φ1 (a , b ) · φ2 (a , d ) · φ3 (b , d ) · φ4 (b , c , d ) Z 1 P (a , b , c , d ) = φ1 (a , b ) · φ2 (a , d ) · φ3 (b , d ) · φ4 (a , b , d ) · φ5 (b , c , d ) Z (8) 25 / 44 P (a , b , c , d ) =
- 28. Log-Linear Models k 1 θq φq (Xq ) p (X ) = exp Z q =1 (9) where • (X1 , ..., Xk ) are cliques in the MN • (φ1 (X1 ), ..., φk (Xk )) are sets of clique potentials asoociated with k th clique • (θ1 , ..., θk ) are parameters of the log-linear models as weights 26 / 44
- 29. Pairwise Binary Markov Networks We estimate the Markov network parameters Θp ×p by maximizing a log-likelihood. f (x1 , ..., xp ) = exp Ψ(Θ) p 1 θj ,j xj + j =1 1 ≤j <k ≤p θj ,k xj xk (10) where xj ∈ {0, 1} Ψ(Θ) = x ∈0 , 1 (11) exp p θj ,j xj + j =1 1 ≤j <k ≤p θj ,k xj xk (12) • the ﬁrst term is a main effect of binary marker xj (node potential) • the second term corresponds to an“interaction effect” between binary markers xj and xk (link potential) • Ψ(Θ) is the normalization constant (partition function) 27 / 44
- 30. Ravikumar et al. (2010) The pseudo-likelihood based on the local conditional likelihood associated with each binary marker can be represented as n p x φi ,ij,j (1 − φi ,j )1−xi,j l (Θ) = (13) i =1 j =1 where φi ,j is the conditional probability of xi ,j = 1 given all other variables. Using a logistic link function, φi ,j = P(xi ,j = 1|xi ,k , k j ; θj ,k , 1 ≤ k ≤ p ) exp(θj ,j + k j θj ,k xi ,k ) = 1 + exp(θj ,j + k j θj ,k xi ,k ) (14) (15) 28 / 44
- 31. Ravikumar et al. (2010) (cont.) • L1 regularized logistic regressions problem • regressing each marker on the rest of the markers • the network structure is recovered from the sparsity pattern of the regression coefﬁcients 0 ˆ−2 β 1 ˆ . . Θ= . −(p −1) ˆ β 1 −p ˆ β1 ˆ β −1 , 2 0 ··· , ··· , ··· , 0 ˆ−(p −1) · · · , β p −2 ˆ p · · · , β−−2 p ˜ Θ= ˆ ˆ Θ • ΘT ˆ 1 β−−1 p ˆ 2 β−−1 p ˆp β −1 ˆp β −2 . . . ··· , −(p −1) ˆp 0 β −p ˆ β p −1 0 (16) (17) 29 / 44
- 32. L1 Regularization n log L = i =1 [xi ,j p k j θk xi ,k − log(1 + e p θ x k j k i ,k )] + λ(θ) (18) • Cyclic Coordinate Descent (CCD) algorithm (Friedman, 2010) • the smallest λmax that shrinks every coefﬁcient to zero • λmin = λmax = 0.01λmax • Tuning the LASSO: • AIC • BIC • CV • GCV • Ten fold cross validation • goodness of ﬁt → deviance 30 / 44
- 33. Summary of Ravikumar et al. (2010) • computation of the partition function is not needed • p different regularization parameters • leads to asymptotically consistent estimates of MN parameters as well as to model selection. ⇓ Implementation Implemented in R with glmnet and with igraph packages. 31 / 44
- 34. ¨ Hoﬂing and Tibshirani’s method (2009) Aims to optimize jointly over Θ f (x1 , ..., xp ) = exp Ψ(Θ) 1 p θj ,j xj + j =1 1 ≤j <k ≤p θj ,k xj xk (19) The log likelihood for all n observations is given by n l (Θ) = i =1 p θj ,j xij + j =1 1 ≤j <k ≤p θj ,k xij xik − log(nΨ(Θ)) (20) Now, adding the L1 penalty to equation (20) yields n log f (x1 , ..., xp ) − n||S • Θ||1 (21) i =1 where S = 2R − diag (R ); R is a p × p lower triangular matrix of containing the penalty parameter 32 / 44
- 35. ¨ Hoﬂing and Tibshirani’s method (2009) Consider a local quadratic Taylor expansion of the log-likelihood around Θ(m) fΘ( m) (Θ) = C + j ≥k ∂l 1 ∂2 l (m ) (m) (θjk − θjk ) + (θ − θjk )2 − n||S • Θ||1 2 jk ∂θjk 2 (∂θjk ) (22) the solution is soft thresholding because the Hessian is diagonal sjk ˆ ˜ ˜ θjk = sign(θjk ) |θjk | − 2 ∂ l (∂θjk ) ∂ l (∂θjk )2 2 (m ) ˜ θjk = θjk − −1 2 jk ∂ ˜ |θjk | − sjk / (∂θ l 2 jk ) ∂ ˜ if |θjk | > sjk / (∂θ 2 l 2 jk ) + ∂l ∂θjk ∂2 ˜ The soft thresholding operator |θjk | − sjk / (∂θ l)2 2 (23) (24) returns + , and zero otherwise. 33 / 44
- 36. Reconstruction of the network Since weak associations are shrunk toward zero, • no need to conduct a series of multiple testings Reconstruction of the LD network ˆ • if Θj ,k = 0, then (xj , ⊥ xk )|else ˆ • if Θj ,k 0, then (xj , not ⊥ xk )|else The matrix entries can be considered as edge weights 34 / 44
- 37. Data, Subset selection and the reference models • 599 inbred wheat lines with 1447 Diversity Array Technology (DArT) binary markers (CIMMYT) • grain yields • Bayesian LASSO • IAMB (Incremental Association Markov Blanket) algorithm for learning BN • r 2 metric 35 / 44
- 38. 10th lambda 15th lambda 8 7 6 9 qqq 10 5 qq q4 11 q q 3 12 q q 13 2 q q 14 1 q q 0 15 q q 29 16 q q 28 17 q q 18 27 q q 19 26 q 20 q 25 q q 22 23 q q 21 q q 24 8 7 6 9 qqq 10 5 qq q4 11 q q 3 12 q q 13 2 q q 14 1 q q 0 15 q q 29 16 q q 28 17 q q 18 27 q q 19 26 q 20 q 25 q q 22 23 q q 21 q q 24 25th lambda 40th lambda 8 7 6 9 qqq 10 5 qq q4 11 q q 3 12 q q 13 2 q q 14 1 q q 0 15 q q 29 16 q q 28 17 q q 18 27 q q 19 26 q 20 q 25 q q 22 23 q q 21 q q 24 8 7 6 9 qqq 10 5 qq q4 11 q q 3 12 q q 13 2 q q 14 1 q q 0 15 q q 29 16 q q 28 17 q q 18 27 q q 19 26 q 20 q 25 q q 22 23 q q 21 q q 24 50th lambda 55th lambda 8 7 6 9 qqq 10 5 qq q4 11 q q 3 12 q q 13 2 q q 14 1 q q 0 15 q q 29 16 q q 28 17 q q 18 27 q q 19 26 q 20 q 25 q q 22 23 q q 21 24 qq 8 7 6 9 qqq 10 5 qq q4 11 q q 3 12 q q 13 2 q q 14 1 q q 0 15 q q 29 16 q q 28 17 q q 18 27 q q 19 26 q 20 q 25 q q 22 23 q q 21 24 qq Figure 11: LD networks with 6 different λ values 36 / 44
- 39. lambda = CV 9 8 7 6 10 5 11 4 3 12 13 2 14 1 15 0 16 29 28 17 18 27 19 26 20 25 21 22 23 24 Figure 12: L1 regularized LD network learned by the method of Ravikumar et al. with chosen by CV. Nodes denote 30 marker loci. 37 / 44
- 40. lambda = sqrt(log(p)/n) 9 8 7 6 10 5 11 4 3 12 13 2 14 1 15 0 16 29 28 17 18 27 19 26 20 25 21 22 23 24 ¨ Figure 13: L1 regularized LD network learned by Hoﬂing’s method with λ chosen as log (p )/n = 0.075, where p = 30, n = 599. Each node denotes a marker locus. 38 / 44
- 41. lambda = sqrt(log(p)/n) lambda = CV 9 8 7 9 6 10 8 7 6 10 5 11 5 11 4 4 3 12 3 12 13 13 2 2 14 14 1 1 15 0 16 29 0 15 29 16 28 17 28 17 18 18 27 27 19 19 26 20 25 21 22 23 24 Figure 14: Ravikumar et al. 26 20 25 21 22 23 24 ¨ Figure 15: Hoﬂing and Tibshirani’s method 39 / 44
- 42. lambda = CV 9 8 7 Bayesian Network 9 6 10 8 7 6 10 5 11 4 3 12 5 11 13 4 3 12 13 2 2 14 1 14 15 0 15 0 16 29 16 29 28 17 18 27 19 26 20 25 21 22 23 24 Figure 16: Ravikumar et al. 1 28 17 18 27 19 26 20 25 21 22 23 24 Figure 17: IAMB 40 / 44
- 43. Summary interactions and associations among the cells and genes form a complex biological system ⇓ r 2 only capture superﬁcial marginal correlations ⇓ explored the possibility of employing graphical models as an alternative approach • r 2 → association(m1, m2)|∅ (emtpyset) • L1 regularized MN → association(m1, m2) | else 41 / 44
- 44. Summary (cont.) • higher-order associations → Reproducing Kernel Hilbert Spaces methods • suitable for binary-valued variables only A ﬁnal remark selecting tag SNPs unconditionally, as well as conditionally, on other markers when the dimension of the data is high, → data generated from next generation sequence technologies. 42 / 44
- 45. GAW17 GAW 17 = Genetic Analysis Workshop 17 • common disease common variant hypothesis vs. common disease rare variant hypothesis • exome sequence from the 1000 Genomes project • 119/166 papers have been accepted for publication • Bayesian hierarchical mixture model GAW18 Scheduled for October 14-17, 2012. 43 / 44
- 46. Acknowledgments University of Wisconsin-Madison • Daniel Gianola • Guilherme Rosa • Kent Weigel • Bruno Valente University College London • Marco Scutari Unversity of Freiburg • Holger Hoﬂing ¨ • fellow graduate students in the 4th and 6th ﬂoors 44 / 44