Parametric Inference for Diffusion Processes Observed at Discrete Points in Time: a Survey

Sørensen, Helle

doi:10.1111/j.1751-5823.2004.tb00241.x

Cited by 159 publications

(109 citation statements)

References 59 publications

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“…Since diffusion processes also arise in other disciplines (e.g., physics and mathematical finance), several approaches have been proposed to obtain efficient approximations of the transition densities for diffusions more general than the Wright-Fisher diffusion (see Srensen 2004;Aït-Sahalia 2008, for example). It would be interesting to investigate whether one could borrow techniques from those fields to the population genetics applications mentioned above.…”

Section: Discussionmentioning

confidence: 99%

A Simple Method for Finding Explicit Analytic Transition Densities of Diffusion Processes with General Diploid Selection

2012

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The transition density function of the Wright-Fisher diffusion describes the evolution of population-wide allele frequencies over time. This function has important practical applications in population genetics, but finding an explicit formula under a general diploid selection model has remained a difficult open problem. In this article, we develop a new computational method to tackle this classic problem. Specifically, our method explicitly finds the eigenvalues and eigenfunctions of the diffusion generator associated with the Wright-Fisher diffusion with recurrent mutation and arbitrary diploid selection, thus allowing one to obtain an accurate spectral representation of the transition density function. Simplicity is one of the appealing features of our approach. Although our derivation involves somewhat advanced mathematical concepts, the resulting algorithm is quite simple and efficient, only involving standard linear algebra. Furthermore, unlike previous approaches based on perturbation, which is applicable only when the population-scaled selection coefficient is small, our method is nonperturbative and is valid for a broad range of parameter values. As a by-product of our work, we obtain the rate of convergence to the stationary distribution under mutation-selection balance. DIFFUSION processes, which are continuous-time Markov processes with almost surely continuous sample paths, have been successfully applied in various population genetic analyses in the past. Examples include finding the stationary distribution of allele frequencies and approximating fixation times and probabilities (see Karlin and Taylor 1981;Ewens 2004;Durrett 2008 for other applications of diffusion processes). This success is largely due to the fact that the diffusion approximation captures the key features of an evolutionary model while ignoring unimportant details, thereby arriving at a simpler process that facilitates computation. However, when a reasonably complex model of evolution is considered, one is faced with unwieldy equations even under the diffusion approximation. In particular, for Wright-Fisher diffusions with general diploid selection, finding an explicit analytic transition density function, which characterizes the evolution of population-wide allele frequencies over time, has remained a challenging open problem. The diffusion theory allows one to write down a partial differential equation (PDE) satisfied by the transition density, but solving the PDE analytically has proved to be difficult.The transition density has several practical applications, including the following: Recently, there has been growing interest in analyzing samples taken from the same or related populations at different time points. For example, such data arise from experimental evolution of model organisms in the laboratory (e.g., bacteria, see Lenski 2011), from viral/ phage populations (Shankarappa et al. 1999;Wichman et al. 1999), or from ancient DNA (Hummel et al. 2005); see also Bollback et al. (2008) and references therein. In part...

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Section: Discussionmentioning

confidence: 99%

A Simple Method for Finding Explicit Analytic Transition Densities of Diffusion Processes with General Diploid Selection

2012

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“…As a result of this, the literature contains various methodologies that may or not be based on the likelihood; see Sørensen (2004) for an extensive review. Likelihood based approaches are either analytical (Aït-Sahalia (2002), Aït-Sahalia (2005)), or they use simulations (Pedersen (1995), Durham and Gallant (2002)).…”

Section: Introductionmentioning

confidence: 99%

Likelihood-based inference for a class of multivariate diffusions with unobserved paths

Καλογερόπουλος

2007

Journal of Statistical Planning and Inference

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This document is the author's final manuscript accepted version of the journal article, incorporating any revisions agreed during the peer review process. Some differences between this version and the published version may remain. You are advised to consult the publisher's version if you wish to cite from it. Likelihood-based Inference for a Class of Multivariate Diffusions with Unobserved Paths Konstantinos Kalogeropoulos May 18, 2007Abstract This paper presents a Markov chain Monte Carlo algorithm for a class of multivariate diffusion models with unobserved paths. This class is of high practical interest as it includes most diffusion driven stochastic volatility models. The algorithm is based on a data augmentation scheme where the paths are treated as missing data. However, unless these paths are transformed so that the dominating measure is independent of any parameters, the algorithm becomes reducible. The methodology developed in Roberts and Stramer (2001 Biometrika 88(3):603-621) circumvents the problem for scalar diffusions. We extend this framework to the class of models of this paper by introducing an appropriate reparametrisation of the likelihood that can be used to construct an irreducible data augmentation scheme. Practical implementation issues are considered and the methodology is applied to simulated data from the Heston model.

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“…Alternative approaches include the use of estimating functions [10,27,11] and spectral methods [23,22,14]. An overview of different approaches can be found in [38]; the difficulties of estimation from low-frequency data are highlighted in [22].…”

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Diffusion Estimation from Multiscale Data by Operator Eigenpairs

Crommelin¹,

Vanden‐Eijnden²

2011

Multiscale Model. Simul.

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Abstract. In this paper we present a new procedure for the estimation of diffusion processes from discretely sampled data. It is based on the close relation between eigenpairs of the diffusion operator L and those of the conditional expectation operator Pt, a relation stemming from the semigroup structure Pt = exp(tL ) for t ≥ 0. It allows for estimation without making time discretization errors, an aspect that is particularly advantageous in case of data with low sampling frequency. After estimating eigenpairs of L via eigenpairs of Pt, we infer the drift and diffusion functions that determine L by fitting L to the estimated eigenpairs using a convex optimization procedure. We present numerical examples where we apply the procedure to one-and two-dimensional diffusions, reversible as well as nonreversible.In the second part of the paper we consider estimation of coarse-grained (homogenized) diffusion processes from multiscale data. We show that eigenpairs of the homogenized diffusion operator are asymptotically close to eigenpairs of the underlying multiscale diffusion operator. This implies that we can infer the correct homogenized process from data of the multiscale process, using the estimation procedure discussed in the first part of the paper. This is illustrated with numerical examples.

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Parametric Inference for Diffusion Processes Observed at Discrete Points in Time: a Survey

Cited by 159 publications

References 59 publications

A Simple Method for Finding Explicit Analytic Transition Densities of Diffusion Processes with General Diploid Selection

A Simple Method for Finding Explicit Analytic Transition Densities of Diffusion Processes with General Diploid Selection

Likelihood-based inference for a class of multivariate diffusions with unobserved paths

Diffusion Estimation from Multiscale Data by Operator Eigenpairs

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