Inference in Coupled Wright-Fisher Models. Yasin Department of - PowerPoint PPT Presentation

Inference in Coupled Wright-Fisher Models. Yasin Department of Mathematics, Makerere University Department of Mathematics, KTH Royal Institute of Technology First Network Meeting for Sida- and ISP-funded PhD Students in Mathematics Stockholm 7–8 March 2017 Yasin Department of Mathematics, Makerere University Department of Inference in Coupled Wright-Fisher Models.

My Advisors Timo Koski Boualem Djehiche Juma Kasozi J. Y. T Mugisha Main advisor Assistant advisor Assistant advisor Assistant advisor KTH KTH Makerere Univ. Makerere Univ. Yasin Department of Mathematics, Makerere University Department of Inference in Coupled Wright-Fisher Models.

Coupled Wright-Fisher model for two alleles and two loci Yasin Department of Mathematics, Makerere University Department of Inference in Coupled Wright-Fisher Models.

Outline Coupled Wright-Fisher Model Yasin Department of Mathematics, Makerere University Department of Inference in Coupled Wright-Fisher Models.

Outline Coupled Wright-Fisher Model Parameter Estimation Yasin Department of Mathematics, Makerere University Department of Inference in Coupled Wright-Fisher Models.

Outline Coupled Wright-Fisher Model Parameter Estimation Results Yasin Department of Mathematics, Makerere University Department of Inference in Coupled Wright-Fisher Models.

Outline Coupled Wright-Fisher Model Parameter Estimation Results Conclusion and Future work Yasin Department of Mathematics, Makerere University Department of Inference in Coupled Wright-Fisher Models.

Coupled Wright-Fisher Model It is a system of Stochastic differential equations of the form � dX t = β 11 − ( β 11 + β 12 ) X t + θ (1 − X t ) X t Y t + X t (1 − X t ) dW 1 � dY t = β 21 − ( β 21 + β 22 ) Y t + θ (1 − Y t ) Y t X t + Y t (1 − Y t ) dW 2 (1) with initial conditions X (0) = X 0 , Y (0) = Y 0 and 0 ≤ X t ≤ 1 , 0 ≤ Y t ≤ 1 This is a special case of the multi locus - multiallele model by Aurell, Ekeberg, Koski (2017) Yasin Department of Mathematics, Makerere University Department of Inference in Coupled Wright-Fisher Models.

the parameters are : βij assumed to be nuisance parameters θ the interaction between the two loci. Purpose: To infer the interaction from data. Yasin Department of Mathematics, Makerere University Department of Inference in Coupled Wright-Fisher Models.

Simulation =0 =0 1 1 0.8 0.8 X 2 X 2 0.6 0.6 0.4 0.4 0.2 0.7 0.75 0.8 0.85 0.9 0.95 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 X 1 X 1 =0.02 =0.02 1 1 0.8 0.95 X 2 X 2 0.6 0.9 0.4 0.85 0.2 0.75 0.8 0.85 0.9 0.95 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 X 1 X 1 (a) (b) Figure: Simulation of coupled Wright-Fisher model for various θ. Yasin Department of Mathematics, Makerere University Department of Inference in Coupled Wright-Fisher Models.

Simulation =0.04 =0.04 1 1 0.9 0.95 X 2 X 2 0.8 0.9 0.7 0.85 0.6 0.5 0.8 0.88 0.9 0.92 0.94 0.96 0.98 1 0.75 0.8 0.85 0.9 0.95 1 X 1 X 1 =0.06 =0.06 1 1 0.98 0.98 X 2 X 2 0.96 0.96 0.94 0.94 0.92 0.92 0.9 0.9 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 0.75 0.8 0.85 0.9 0.95 1 X 1 X 1 (a) (b) Figure: Simulation of coupled Wright-Fisher model for various θ. Yasin Department of Mathematics, Makerere University Department of Inference in Coupled Wright-Fisher Models.

Stationary distribution This has been found by Aurell, Ekeberg, Koski (2017). The Stationary density is 1 Z π ( x ) π ( y ) e 2 θxy P ( x, y ) = (2) where x (2 β 11 − 1) (1 − x ) 2 β 12 − 1 π ( x ) = y (2 β 21 − 1) (1 − y ) 2 β 22 − 1 π ( y ) = (2 β 21 ) n (2 θ ) n ∞ Γ(2( β 21 + β 22 )) Γ(2 β 12 ) � Z = β 2 + n )Γ(2 β 11 + n ) (2 ¯ Γ(2 ¯ Γ(2 β 21 )Γ(2 β 22 ) β 2 ) n n ! n =0 (3) Γ( z ) is the Euler gamma function. Yasin Department of Mathematics, Makerere University Department of Inference in Coupled Wright-Fisher Models.

Estimation of θ eqn(1) is of the form √ d X t = C ( X t , θ ) dt + A d W (4) where � � � � β 11 − ( β 11 + β 12 ) Y t (1 − X t ) X t Y t C ( X t , θ ) = + θ (5) β 21 − ( β 21 + β 22 ) Y t (1 − Y t ) Y t X t C ( X t , θ ) = a ( X t ) + θ g ( X t ) (6) � � X t (1 − X t ) 0 A ( X t ) = 0 Y t (1 − Y t ) Yasin Department of Mathematics, Makerere University Department of Inference in Coupled Wright-Fisher Models.

Likelihood function of θ By Girsanov theorem, �� 1 d P θ A − 1 ( C ( X t , θ ) − C ( X t , θ 0 )) · d X t = L ( θ ) = exp d P θ 0 0 � 1 − 1 � [ A − 1 C ( X t , θ ) · C ( X t , θ ) − A − 1 C ( X t , θ 0 ) · C ( X t , θ 0 )] dt (7) 2 0 P θ and P θ 0 are probability measures induced by solutions of eqn(4). Particularly, θ 0 = 0 . A · B denotes the dot product between A and B . Yasin Department of Mathematics, Makerere University Department of Inference in Coupled Wright-Fisher Models.

Derivative of log-likelihood function of θ Thus, �� 1 dlogL ( θ ) = d A − 1 C ( X t , θ ) · d X t dθ dθ 0 � 1 − 1 � A − 1 C ( X t , θ ) · C ( X t , θ ) dt (8) 2 0 Subsituting for C ( X t , θ ) , Yasin Department of Mathematics, Makerere University Department of Inference in Coupled Wright-Fisher Models.

Derivative of log-likelihood function of θ Hence, � 1 dlogL ( θ ) = d � A − 1 g ( X t ) · d X t θ dθ dθ 0 � 1 − 1 � [ θ 2 A − 1 g ( X t ) · g ( X t ) + 2 θ A − 1 g ( X t ) · a ( X t )] dt (9) 2 0 Yasin Department of Mathematics, Makerere University Department of Inference in Coupled Wright-Fisher Models.

MamixumLikelihood estimate of θ For MLE, dlogL ( θ ) = 0 dθ � 1 � 1 0 A − 1 g ( X t ) · d X t − 0 A − 1 g ( X t ) · a ( X t ) dt ˆ θ = (10) � 1 0 A − 1 g ( X t ) · g ( X t ) dt Yasin Department of Mathematics, Makerere University Department of Inference in Coupled Wright-Fisher Models.

� � Y t But, A − 1 g ( X t ) = ⇒ By Ito’s formula, X t � 1 � 1 A − 1 g ( X t ) · d X t = Y t dX t + X t dY t = X (1) Y (1) − X (0) Y (0) 0 0 The other integrals can be discretised by Classical methods for instance, Trapezoidal rule (Lucus M., 2008) Yasin Department of Mathematics, Makerere University Department of Inference in Coupled Wright-Fisher Models.

Simulation Results (a) (b) Yasin Department of Mathematics, Makerere University Department of Inference in Coupled Wright-Fisher Models.

Simulation Results (e) (f) Yasin Department of Mathematics, Makerere University Department of Inference in Coupled Wright-Fisher Models.

Future work In the future we intend to explore the following; Yasin Department of Mathematics, Makerere University Department of Inference in Coupled Wright-Fisher Models.

Future work In the future we intend to explore the following; extension our work to L loci and d alleles Yasin Department of Mathematics, Makerere University Department of Inference in Coupled Wright-Fisher Models.

Future work In the future we intend to explore the following; extension our work to L loci and d alleles consider a more general interaction function Yasin Department of Mathematics, Makerere University Department of Inference in Coupled Wright-Fisher Models.

Future work In the future we intend to explore the following; extension our work to L loci and d alleles consider a more general interaction function consider inference from real allele frequency time series data Yasin Department of Mathematics, Makerere University Department of Inference in Coupled Wright-Fisher Models.

Future work In the future we intend to explore the following; extension our work to L loci and d alleles consider a more general interaction function consider inference from real allele frequency time series data use bayesian inference Yasin Department of Mathematics, Makerere University Department of Inference in Coupled Wright-Fisher Models.

Tack så mycket! Thank you! Yasin Department of Mathematics, Makerere University Department of Inference in Coupled Wright-Fisher Models.