Fitting Effective Diffusion Models to Data Associated with a "Glassy Potential": Estimation, Classical Inference Procedures and Some Heuristics

A variety of researchers have successfully obtained the parameters of low dimensional diffusion models using the data that comes out of atomistic simulations. This naturally raises a variety of questions about efficient estimation, goodness-of-fit tests, and confidence interval estimation. The first part of this article uses maximum likelihood estimation to obtain the parameters of a diffusion model from a scalar time series. I address numerical issues associated with attempting to realize asymptotic statistics results with moderate sample sizes in the presence of exact and approximated transition densities. Approximate transition densities are used because the analytic solution of a transition density associated with a parametric diffusion model is often unknown.I am primarily interested in how well the deterministic transition density expansions of Ait-Sahalia capture the curvature of the transition density in (idealized) situations that occur when one carries out simulations in the presence of a "glassy" interaction potential. Accurate approximation of the curvature of the transition density is desirable because it can be used to quantify the goodness-of-fit of the model and to calculate asymptotic confidence intervals of the estimated parameters. The second part of this paper contributes a heuristic estimation technique for approximating a nonlinear diffusion model. A "global" nonlinear model is obtained by taking a batch of time series and applying simple local models to portions of the data. I demonstrate the technique on a diffusion model with a known transition density and on data generated by the Stochastic Simulation Algorithm.Comment: 30 pages 10 figures Submitted to SIAM MMS (typos removed and slightly shortened

Minimax Estimation of Nonregular Parameters and Discontinuity in Minimax Risk

When a parameter of interest is nondifferentiable in the probability, the existing theory of semiparametric efficient estimation is not applicable, as it does not have an influence function. Song (2014) recently developed a local asymptotic minimax estimation theory for a parameter that is a nondifferentiable transform of a regular parameter, where the nondifferentiable transform is a composite map of a continuous piecewise linear map with a single kink point and a translation-scale equivariant map. The contribution of this paper is two fold. First, this paper extends the local asymptotic minimax theory to nondifferentiable transforms that are a composite map of a Lipschitz continuous map having a finite set of nondifferentiability points and a translation-scale equivariant map. Second, this paper investigates the discontinuity of the local asymptotic minimax risk in the true probability and shows that the proposed estimator remains to be optimal even when the risk is locally robustified not only over the scores at the true probability, but also over the true probability itself. However, the local robustification does not resolve the issue of discontinuity in the local asymptotic minimax risk

Maximum likelihood drift estimation for a threshold diffusion

We study the maximum likelihood estimator of the drift parameters of a stochastic differential equation, with both drift and diffusion coefficients constant on the positive and negative axis, yet discontinuous at zero. This threshold diffusion is called drifted Oscillating Brownian motion.For this continuously observed diffusion, the maximum likelihood estimator coincide with a quasi-likelihood estimator with constant diffusion term. We show that this estimator is the limit, as observations become dense in time, of the (quasi)-maximum likelihood estimator based on discrete observations. In long time, the asymptotic behaviors of the positive and negative occupation times rule the ones of the estimators. Differently from most known results in the literature, we do not restrict ourselves to the ergodic framework: indeed, depending on the signs of the drift, the process may be ergodic, transient or null recurrent. For each regime, we establish whether or not the estimators are consistent; if they are, we prove the convergence in long time of the properly rescaled difference of the estimators towards a normal or mixed normal distribution. These theoretical results are backed by numerical simulations

Magic traits drive the emergence of pathogens

An important branch of evolutionary biology strives to understand how divergent selection for an ecologically important trait can foster the emergence of new species specialized on different niches. Such ecological speciation is usually difficult to achieve because recombination between different subsets of a population that are adapting to different environments counteracts selection for locally adapted gene combinations. Traits pleiotropically controlling adaptation to different environments and reproductive isolation are therefore the most favourable for ecological speciation, and are thus called “magic traits”. We used genetic markers and cross-inoculations to show that pathogenicity-related loci are responsible for both host adaptation and reproductive isolation in emerging populations of Venturia inaequalis, the fungus causing apple scab disease. Because the fungus mates within its host and because the pathogenicity-related loci prevent infection of the non-host trees, host adaptation pleiotropically maintains genetic differentiation and adaptive allelic combinations between sympatric populations specific to different apple varieties. Such “magic traits” are likely frequent in fungal pathogens, and likely drive the emergence of new diseases.

Differential selection pressures exerted by host resistance quantitative trait loci on a pathogen population: a case study in an apple × Venturia inaequalis pathosystem

Understanding how pathogens evolve according to pressures exerted by their plant hosts is essential for the derivation of strategies aimed at the durable management of resistant cultivars. The spectrum of action of the resistance factors in the partially resistant cultivars is thought to be an important determinant of resistance durability. However, it has not yet been demonstrated whether the pressures exerted by quantitative resistance are different according to their spectrum of action.To investigate selection pressures exerted by apple genotypes harbouring various resistance quantitative trait loci (QTLs) on a mixed inoculum of the scab disease agent, Venturia inaequalis, we monitored V. inaequalis isolate proportions on diseased apple leaves of an F1 progeny using quantitative pyrosequencing technology and QTL mapping. Broad-spectrum resistances did not exert any differential selection pressures on the mixed inoculum, whereas narrow-spectrum resistances decreased the frequencies of some isolates in the mixture relative to the susceptible host genotypes. Our results suggest that the management of resistant cultivars should be different according to the spectrum of action of their resistance factors. The pyramiding of broad-spectrum factors or the use of a mixture of apple genotypes that carry narrow-spectrum resistance factors are two possible strategies for the minimization of resistance erosion

Asymptotically optimal quantum channel reversal for qudit ensembles and multimode Gaussian states

We investigate the problem of optimally reversing the action of an arbitrary quantum channel C which acts independently on each component of an ensemble of n identically prepared d-dimensional quantum systems. In the limit of large ensembles, we construct the optimal reversing channel R* which has to be applied at the output ensemble state, to retrieve a smaller ensemble of m systems prepared in the input state, with the highest possible rate m/n. The solution is found by mapping the problem into the optimal reversal of Gaussian channels on quantum-classical continuous variable systems, which is here solved as well. Our general results can be readily applied to improve the implementation of robust long-distance quantum communication. As an example, we investigate the optimal reversal rate of phase flip channels acting on a multi-qubit register.Comment: 17 pages, 3 figure

Emergence of novel fungal pathogens by ecological speciation: importance of the reduced viability of immigrants

Expanding global trade and the domestication of ecosystems have greatly accelerated the rate of emerging infectious fungal diseases, and host-shift speciation appears to be a major route for disease emergence. There is therefore an increased interest in identifying the factors that drive the evolution of reproductive isolation between populations adapting to different hosts. Here, we used genetic markers and cross-inoculations to assess the level of gene flow and investigate barriers responsible for reproductive isolation between two sympatric populations of Venturia inaequalis, the fungal pathogen causing apple scab disease, one of the fungal populations causing a recent emerging disease on resistant varieties. Our results showed the maintenance over several years of strong and stable differentiation between the two populations in the same orchards, suggesting ongoing ecological divergence following a host shift. We identified strong selection against immigrants (i.e. host specificity) from different host varieties as the strongest and likely most efficient barrier to gene flow between local and emerging populations. Cross-variety disease transmission events were indeed rare in the field and cross-inoculation tests confirmed high host specificity. Because the fungus mates within its host after successful infection and because pathogenicity-related loci prevent infection of nonhost trees, adaptation to specific hosts may alone maintain both genetic differentiation between and adaptive allelic combinations within sympatric populations parasitizing different apple varieties, thus acting as a ‘magic trait’. Additional intrinsic and extrinsic postzygotic barriers might complete reproductive isolation and explain why the rare migrants and F1 hybrids detected do not lead to pervasive gene flow across years

Host-specific differentiation among populations of Venturia inaequalis causing scab on apple, pyracantha and loquat

Patterns of multilocus DNA sequence variation within and between closely related taxa can provide insights into the history of divergence. Here, we report on DNA polymorphism and divergence at six nuclear loci in globally distributed samples of the ascomycete Venturia inaequalis, responsible for scab on apple, loquat, and pyracantha. Isolates from different hosts were differentiated but did not form diagnosable distinct phylogenetic species. Parameters of an Isolation-with-Migration model estimated from the data suggested that the large amount of variation shared among groups more likely resulted from recent splitting than from extensive genetic exchanges. Inferred levels of gene flow among groups were low and more concentrated toward recent times, and we identified two potentially recent one-off shifters from apple and pyracantha to loquat. These findings support a scenario of recent divergence in allopatry followed by introgression through secondary contact, with groups from loquat and pyracantha being the most recently differentiated

The singular continuous diffraction measure of the Thue-Morse chain

The paradigm for singular continuous spectra in symbolic dynamics and in mathematical diffraction is provided by the Thue-Morse chain, in its realisation as a binary sequence with values in $\{\pm 1\}$. We revisit this example and derive a functional equation together with an explicit form of the corresponding singular continuous diffraction measure, which is related to the known representation as a Riesz product.Comment: 6 pages, 1 figure; revised and improved versio