Idea Transcript
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 o Inference in Coupled Wright-Fisher Models.
My Advisors
Timo Koski Main advisor KTH
Boualem Djehiche Assistant advisor KTH
Yasin
Juma Kasozi Assistant advisor Makerere Univ.
J. Y. T Mugisha Assistant advisor Makerere Univ.
Department of Mathematics, Makerere University Department o Inference in Coupled Wright-Fisher Models.
Coupled Wright-Fisher model for two alleles and two loci
Yasin
Department of Mathematics, Makerere University Department o Inference in Coupled Wright-Fisher Models.
Outline Coupled Wright-Fisher Model
Yasin
Department of Mathematics, Makerere University Department o Inference in Coupled Wright-Fisher Models.
Outline Coupled Wright-Fisher Model Parameter Estimation
Yasin
Department of Mathematics, Makerere University Department o Inference in Coupled Wright-Fisher Models.
Outline Coupled Wright-Fisher Model Parameter Estimation Results
Yasin
Department of Mathematics, Makerere University Department o 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 o Inference in Coupled Wright-Fisher Models.
Coupled Wright-Fisher Model It is a system of Stochastic differential equations of the form dXt = β11 − (β11 + β12 )Xt + θ(1 − Xt )Xt Yt +
q
dYt = β21 − (β21 + β22 )Yt + θ(1 − Yt )Yt Xt +
Xt (1 − Xt )dW1
q
Yt (1 − Yt )dW2 (1)
with initial conditions X(0) = X0 , Y (0) = Y0 and 0 ≤ Xt ≤ 1, 0 ≤ Yt ≤ 1 This is a special case of the multi locus - multiallele model by Aurell, Ekeberg, Koski (2017)
Yasin
Department of Mathematics, Makerere University Department o 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 o Inference in Coupled Wright-Fisher Models.
Simulation
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Yasin Figure: Simulation of coupled Wright-Fisher model for various θ.
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Department of Mathematics, Makerere University Department o Inference in Coupled Wright-Fisher Models.
Simulation
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Yasin Figure: Simulation of coupled Wright-Fisher model for various θ.
Department of Mathematics, Makerere University Department o Inference in Coupled Wright-Fisher Models.
Stationary distribution This has been found by Aurell, Ekeberg, Koski (2017). The Stationary density is P (x, y) =
1 π(x)π(y)e2θxy Z
(2)
where
Z=
π(x) =
x(2β11 −1) (1 − x)2β12 −1
π(y) =
y (2β21 −1) (1 − y)2β22 −1
∞ Γ(2(β21 + β22 )) X (2β21 )n (2θ)n Γ(2β12 ) Γ(2β11 + n) Γ(2β21 )Γ(2β22 ) n=0 (2β¯2 )n n! Γ(2β¯2 + n)
(3) Γ(z) is the Euler gamma function. Yasin
Department of Mathematics, Makerere University Department o Inference in Coupled Wright-Fisher Models.
Estimation of θ eqn(1) is of the form √ dXt = C(Xt , θ)dt +
A dW
(4)
where !
C(Xt , θ) =
!
(1 − Xt )Xt Yt β11 − (β11 + β12 )Yt (5) +θ (1 − Yt )Yt Xt β21 − (β21 + β22 )Yt
C(Xt , θ) = a(Xt ) + θg(Xt )
(6) !
A(Xt ) =
Xt (1 − Xt ) 0 0 Yt (1 − Yt )
Yasin
Department of Mathematics, Makerere University Department o Inference in Coupled Wright-Fisher Models.
Likelihood function of θ By Girsanov theorem, Z 1
n dPθ = L(θ) = exp dPθ0
1 − 2
Z 1
A−1 (C(Xt , θ) − C(Xt , θ0 )) · dXt
0
o
[A−1 C(Xt , θ) · C(Xt , θ) − A−1 C(Xt , θ0 ) · C(Xt , θ0 )]dt
(7)
0
Pθ and Pθ0 are probability measures induced by solutions of eqn(4). Particularly, θ0 = 0. A · B denotes the dot product between A and B.
Yasin
Department of Mathematics, Makerere University Department o Inference in Coupled Wright-Fisher Models.
Derivative of log-likelihood function of θ Thus, dlogL(θ) d n 1 −1 = A C(Xt , θ) · dXt dθ dθ 0 Z o 1 1 −1 − A C(Xt , θ) · C(Xt , θ)dt 2 0 Z
(8)
Subsituting for C(Xt , θ),
Yasin
Department of Mathematics, Makerere University Department o Inference in Coupled Wright-Fisher Models.
Derivative of log-likelihood function of θ Hence, dlogL(θ) dn = θ dθ dθ 1 − 2
Z 1
2
[θ A
−1
g(Xt ) · g(Xt ) + 2θA
Z 1
−1
A−1 g(Xt ) · dXt
0
o
g(Xt ) · a(Xt )]dt
(9)
0
Yasin
Department of Mathematics, Makerere University Department o Inference in Coupled Wright-Fisher Models.
MamixumLikelihood estimate of θ For MLE, θˆ =
dlogL(θ) dθ
R1 0
=0
A−1 g(Xt ) · dXt − 01 A−1 g(Xt ) · a(Xt )dt R1 −1 0 A g(Xt ) · g(Xt )dt
Yasin
R
(10)
Department of Mathematics, Makerere University Department o Inference in Coupled Wright-Fisher Models.
But, Z 1
A−1 g(X
−1
A
t)
=
Yt Xt
g(Xt ) · dXt =
0
!
⇒ By Ito’s formula, Z 1
Yt dXt + Xt dYt = X(1)Y (1) − X(0)Y (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 o Inference in Coupled Wright-Fisher Models.
Simulation Results
(a)
Yasin
(b)
Department of Mathematics, Makerere University Department o Inference in Coupled Wright-Fisher Models.
Simulation Results
(e)
Yasin
(f)
Department of Mathematics, Makerere University Department o Inference in Coupled Wright-Fisher Models.
Future work In the future we intend to explore the following;
Yasin
Department of Mathematics, Makerere University Department o 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 o 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 o 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 o 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 o Inference in Coupled Wright-Fisher Models.
Tack så mycket! Thank you!
Yasin
Department of Mathematics, Makerere University Department o Inference in Coupled Wright-Fisher Models.