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 dXt = β11 − (β11 + β12)Xt + θ(1 − Xt)XtYt +

• Xt(1 − Xt)dW1

dYt = β21 − (β21 + β22)Yt + θ(1 − Yt)YtXt +

• 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 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

0.7 0.75 0.8 0.85 0.9 0.95 1

X1

0.4 0.6 0.8 1

X2

=0 0.75 0.8 0.85 0.9 0.95 1 X1 0.85 0.9 0.95 1

X2

=0.02

(a)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

X1

0.2 0.4 0.6 0.8 1

X2

=0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 X1 0.2 0.4 0.6 0.8 1

X2

=0.02

(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

0.88 0.9 0.92 0.94 0.96 0.98 1

X1

0.5 0.6 0.7 0.8 0.9 1

X2

=0.04 0.9 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 X1 0.92 0.94 0.96 0.98 1

X2

=0.06

(a)

0.75 0.8 0.85 0.9 0.95 1

X1

0.8 0.85 0.9 0.95 1

X2

=0.04 0.75 0.8 0.85 0.9 0.95 1 X1 0.9 0.92 0.94 0.96 0.98 1

X2

=0.06

(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 P(x, y) = 1 Z π(x)π(y)e2θxy (2) where π(x) = x(2β11−1)(1 − x)2β12−1 π(y) = y(2β21−1)(1 − y)2β22−1 Z = Γ(2(β21 + β22)) Γ(2β21)Γ(2β22)

∞

• n=0

(2β21)n(2θ)n (2 ¯ β2)nn! Γ(2β12) Γ(2 ¯ β2 + n)Γ(2β11 + n) (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 dXt = C(Xt, θ)dt + √ A dW (4) where C(Xt, θ) =

• β11 − (β11 + β12)Yt

β21 − (β21 + β22)Yt

• + θ
• (1 − Xt)XtYt

(1 − Yt)YtXt

• (5)

C(Xt, θ) = a(Xt) + θg(Xt) (6) A(Xt) =

• Xt(1 − Xt)

Yt(1 − Yt)

• Yasin

Department of Mathematics, Makerere University Department of

Inference in Coupled Wright-Fisher Models.

Likelihood function of θ

By Girsanov theorem, dPθ dPθ0 = L(θ) = exp

1

A−1(C(Xt, θ) − C(Xt, θ0)) · dXt −1 2

1

[A−1C(Xt, θ) · C(Xt, θ) − A−1C(Xt, θ0) · C(Xt, θ0)]dt

• (7)

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 of

Inference in Coupled Wright-Fisher Models.

Derivative of log-likelihood function of θ

Thus, dlogL(θ) dθ = d dθ

1

A−1C(Xt, θ) · dXt −1 2

1

A−1C(Xt, θ) · C(Xt, θ)dt

• (8)

Subsituting for C(Xt, θ),

Yasin

Department of Mathematics, Makerere University Department of

Inference in Coupled Wright-Fisher Models.

Derivative of log-likelihood function of θ

Hence, dlogL(θ) dθ = d dθ

• θ

1

A−1g(Xt) · dXt −1 2

1

[θ2A−1g(Xt) · g(Xt) + 2θA−1g(Xt) · a(Xt)]dt

• (9)

Yasin

Department of Mathematics, Makerere University Department of

Inference in Coupled Wright-Fisher Models.

MamixumLikelihood estimate of θ

For MLE, dlogL(θ)

dθ

= 0 ˆ θ =

1

0 A−1g(Xt) · dXt −

1

0 A−1g(Xt) · a(Xt)dt

1

0 A−1g(Xt) · g(Xt)dt

(10)

Yasin

Department of Mathematics, Makerere University Department of

Inference in Coupled Wright-Fisher Models.

But, A−1g(Xt) =

• Yt

Xt

• ⇒ By Ito’s formula,

1

A−1g(Xt) · dXt =

1

YtdXt + XtdYt = X(1)Y (1) − X(0)Y (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.