| 1. |
- Favero, Martina, et al.
(författare)
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A dual process for the coupled Wright-Fisher diffusion
- 2021
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Ingår i: Journal of Mathematical Biology. - : Springer Nature. - 0303-6812 .- 1432-1416. ; 82:1-2
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Tidskriftsartikel (refereegranskat)abstract
- The coupled Wright-Fisher diffusion is a multi-dimensional Wright-Fisher diffusion for multi-locus and multi-allelic genetic frequencies, expressed as the strong solution to a system of stochastic differential equations that are coupled in the drift, where the pairwise interaction among loci is modelled by an inter-locus selection. In this paper, an ancestral process, which is dual to the coupled Wright-Fisher diffusion, is derived. The dual process corresponds to the block counting process of coupled ancestral selection graphs, one for each locus. Jumps of the dual process arise from coalescence, mutation, single-branching, which occur at one locus at the time, and double-branching, which occur simultaneously at two loci. The coalescence and mutation rates have the typical structure of the transition rates of the Kingman coalescent process. The single-branching rate not only contains the one-locus selection parameters in a form that generalises the rates of an ancestral selection graph, but it also contains the two-locus selection parameters to include the effect of the pairwise interaction on the single loci. The double-branching rate reflects the particular structure of pairwise selection interactions of the coupled Wright-Fisher diffusion. Moreover, in the special case of two loci, two alleles, with selection and parent independent mutation, the stationary density for the coupled Wright-Fisher diffusion and the transition rates of the dual process are obtained in an explicit form.
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| 2. |
- Garcia-Pareja, Celia, et al.
(författare)
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EXACT SIMULATION OF COUPLED WRIGHT-FISHER DIFFUSIONS
- 2021
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Ingår i: Advances in Applied Probability. - : Cambridge University Press (CUP). - 0001-8678 .- 1475-6064. ; 53:4, s. 923-950
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Tidskriftsartikel (refereegranskat)abstract
- In this paper an exact rejection algorithm for simulating paths of the coupled Wright- Fisher diffusion is introduced. The coupled Wright-Fisher diffusion is a family of multivariate Wright-Fisher diffusions that have drifts depending on each other through a coupling term and that find applications in the study of networks of interacting genes. The proposed rejection algorithm uses independent neutral Wright-Fisher diffusions as candidate proposals, which are only needed at a finite number of points. Once a candidate is accepted, the remainder of the path can be recovered by sampling from neutral multivariate Wright-Fisher bridges, for which an exact sampling strategy is also provided. Finally, the algorithm's complexity is derived and its performance demonstrated in a simulation study.
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| 3. |
- Saize, Stefane, 1984-
(författare)
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Contributions to reciprocal processes
- 2023
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Reciprocal processes are stochastic processes such that the current state only depends on the nearest past and future, and they have found many applications in various fields such as Euclidean quantum physics. This thesis focuses on the study of some classes of reciprocal processes in both discrete-time and continuous-time frameworks. The main contributions of this thesis can be splitted into two parts. The first and major part of this thesis consists of the study of several discrete-time reciprocal processes: reciprocal chains, hidden Markov models (HMMs) and hidden reciprocal models (HRMs). More specifically, we (i) formally define reciprocal chains and explore their properties and similarities/differences to Markov chains; (ii) point out that one of the three most commonly used definitions of HMMs has fatal flaws and list some key properties of HMMs; (iii) present the connections between undirected graphical models and HMMs/reciprocal chains; (iv) learn and compare HMMs and HRMs parameters based on the expectation-maximization algorithm. The second part of the thesis focuses on the study of Brownian bridges with pre-scribed terminal densities which are special continuous-time reciprocal processes. We derive large deviations for the family of Brownian bridges with the help of Girsanov transformation.
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| 4. |
- Zickert, Gustav
(författare)
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Analytic and data-driven methods for 3D electron microscopy
- 2020
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- The central theme of this thesis is theoretical and algorithmic aspects of 3D electron microscopy (3D-EM). In particular, the thesis explores three parts of this theme. The first part concerns analysis of forward operators that, compared to those traditionally used, better account for the wave properties of the imaging electron. The second part concerns the adoption of data-driven methods in 3D-EM. The third part concerns the use of Gaussian dictionaries in image decomposition and image reconstruction.The thesis consists primarily of five papers, which are preceded by two introductory chapters. The first chapter provides a background for the thesis and the second one constitutes a summary of the papers.In paper A we propose a fast non-linear reconstruction method for joint phase-retrieval and image reconstruction in cryo electron tomography. We evaluate the method on simulated and real data. In paper B we train a deep convolutional neural network on a database of previously determined molecular structures. This network is used to model a prior distribution in single particle analysis (SPA) within a maximum-a-posteriori framework. We show in a simulation study that the proposed method is able to significantly improve on one of the current state-of-the-art methods.In paper C we propose a greedy method for decomposing a signal as a mixture of Gaussians. We also derive an upper bound for the distance from any local maximum of a Gaussian mixture to the set of mean vectors.In paper D we generalise the method in paper C and introduce an algorithm for reconstructing a mixture of Gaussians from its ray-transform projection images. We also derive exact and approximate expressions for the Riesz potential of isotropic and anisotropic Gaussians, respectively.In paper E we prove a uniqueness theorem for an Ewald sphere corrected model for SPA. The theorem shows that accounting for a non-zero curvature of the Ewald sphere renders the noise-free SPA problem uniquely solvable, including the hand of the structure.
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