[Paper Review] The Lambda-Fleming-Viot process and a connection with Wright-Fisher diffusion

Robert Griffiths|arXiv (Cornell University)|Jul 4, 2012

Stochastic processes and statistical mechanics3 citations

TL;DR

This paper establishes a novel connection between the $\Lambda$-Fleming-Viot process and Wright-Fisher diffusion by representing the former's generator as a modified Wright-Fisher generator with a random linear argument, enabling explicit computation of eigenvalues and eigenvectors via polynomial test functions. A key result is an analytic proof of fixation certainty under genic selection when the selection coefficient exceeds a critical threshold.

ABSTRACT

The d-dimensional Lambda-Fleming-Viot generator acting on functions g(x), with x being a vector of d allele frequencies, can be written as a Wright-Fisher generator acting on functions g with a modified random linear argument of x induced by partitioning occurring in the Lambda-Fleming-Viot process. The eigenvalues and right polynomial eigenvectors are easy to see from this representation. The two-dimensional process, which has a one-dimensional generator, is considered in detail. A non-linear equation is found for the Green's function. In a model with genic selection a proof is given that there is a critical selection value such that if the selection coefficient is greater or equal to the critical value then fixation, when the boundary 1 is hit, has probability 1 beginning from any non-zero frequency. This is an analytic proof different from proofs by Der, Epstein and Plotkin (2011) and Foucart (2013). When fixation is not certain the fixation probability can be computed from an algorithm in the paper. An application in the infinitely-many-alleles Lambda-Fleming-Viot process is finding an interesting identity for the frequency spectrum of alleles that is based on size-biassing. The moment dual process in the Fleming-Viot process is the usual Lambda-coalescent tree back in time. The Wright-Fisher representation using a different set of polynomials g_n(x) as test functions produces a dual death process which has a similarity to the Kingman coalescent and decreases by units of one. The eigenvalues of the process are analogous to the Jacobi polynomials when expressed in terms of g_n(x), playing the role of x^n. E[g_n(X)] under the stationary distribution when there is mutation is analogous to the n th moment in a Beta distribution. There is a version g_n(X), when X and n are d-dimensional, and even an intriguing Ewens' sampling formula analogy when d tends to infinity.

Motivation & Objective

• To establish a functional connection between the $\Lambda$-Fleming-Viot process and the Wright-Fisher diffusion through a modified generator representation.
• To derive explicit expressions for eigenvalues and polynomial eigenvectors of the $\Lambda$-Fleming-Viot generator using a Wright-Fisher-type framework.
• To analyze fixation probabilities under genic selection, proving that fixation occurs with probability one when the selection coefficient exceeds a critical value.
• To extend the moment duality between the $\Lambda$-Fleming-Viot process and the $\Lambda$-coalescent to higher-dimensional and infinite-alleles settings.
• To derive a generalized Ewens’ sampling formula analog in the infinitely-many-alleles $\Lambda$-Fleming-Viot process based on size-biased frequency spectra.

Proposed method

• Represents the $d$-dimensional $\Lambda$-Fleming-Viot generator as a Wright-Fisher generator acting on a transformed argument involving a random variable $W$ derived from the measure $\Lambda$.
• Uses polynomial test functions $g_n(x)$ and their multitype extensions $g_{\bm{n}}(\bm{x})$ to exploit moment duality with a death process dual to the $\Lambda$-coalescent.
• Applies the duality relation $\mathbb{E}_{\bm{X}(0)=\bm{x}}[h_{\bm{n}}(\bm{X}(t))] = \mathbb{E}_{\bm{N}(0)=\bm{n}}[h_{\bm{N}(t)}(\bm{x})]$ to compute expectations and eigenvalues.
• Derives the stationary distribution moments via a product formula involving $\mathbb{E}[(1-W)^{j-2}]$ and selection parameters $\theta_i$, generalizing the Beta distribution moment analog.
• Constructs a dual death process with transition rates proportional to $\frac{n_i}{n} \big( (n-1)\mathbb{E}[(1-W)^{n-2}] + \theta \big)$, resembling Kingman’s coalescent.
• Establishes a limit sampling formula as $d \to \infty$ that generalizes Ewens’ sampling formula by incorporating the $\Lambda$-coalescent’s size-biased structure.

Experimental results

Research questions

• RQ1 Can the $\Lambda$-Fleming-Viot generator be represented as a modified Wright-Fisher generator with a random linear argument?
• RQ2 What are the eigenvalues and polynomial eigenvectors of the $\Lambda$-Fleming-Viot process under this representation?
• RQ3 Under what conditions does genic selection lead to certain fixation in the $\Lambda$-Fleming-Viot process?
• RQ4 How does the moment duality with the $\Lambda$-coalescent extend to multitype and infinite-alleles models?
• RQ5 What is the limiting sampling formula for allele frequencies in the infinitely-many-alleles $\Lambda$-Fleming-Viot process?

Key findings

• The $\Lambda$-Fleming-Viot generator can be expressed as a Wright-Fisher generator acting on a random linear transformation of the allele frequency vector, enabling explicit computation of eigenvalues and eigenvectors.
• For the two-dimensional case, a non-linear equation for the Green’s function is derived, providing a path to potential explicit solutions.
• An analytic proof is given that fixation occurs with probability one when the selection coefficient is greater than or equal to a critical threshold, independent of initial non-zero frequency.
• The stationary distribution moment $\mathbb{E}[g_n(X)]$ is shown to be a product of terms involving $\mathbb{E}[(1-W)^{j-2}]$ and selection parameters $\theta_i$, generalizing the Beta distribution moment.
• In the multitype case, the moment generating function $\mathbb{E}[g_{\bm{n}}(\bm{X})]$ is given by a ratio of products over $j=1$ to $n_i$ and $j=1$ to $n$, with a structure analogous to the Ewens’ sampling formula.
• In the limit as $d \to \infty$, the sampling formula converges to a generalized form involving $\mathbb{E}[(1-W)^{j-2}]$, reducing to the classical Ewens’ formula when $W \equiv 0$.