This paper is devoted to the problem of determining the deviation bounds that are achievable in non-parametric regression. We consider the setting where features are supported on a bounded subset of R-d, the regression function is Lipschitz, and the noise is only assumed to have a finite second moment. We first specify the fundamental limits of the problem by establishing a general lower bound on deviation probabilities, and then construct explicit estimators that achieve this bound. These estimators are obtained by applying the median-of-means principle to classical local averaging rules in non-parametric regression, including nearest neighbors and kernel procedures.

The Biological Resource Center (BRC) ’Pome fruits and roses’ is hosted by INRAE near Angers, France. It includes collections of apple, pear, quince and rose. The apple, pear and quince collections are preserved in orchards on site and as DNA samples. Research accessions of the Rosa genus are preserved both in the field and as DNA samples, whereas traditional accessions are primarily maintained as DNA samples. Traditional roses accessions are preserved in the field by different private and public rose gardens in France. The main BRC’s mission is to preserve these biological resources while improving and optimizing its collections. It also aims to gather and enrich data associated with the conserved genetic accessions, in particular by recording and centralizing phenotypic and genetic characterization data. This ensures the provision of suitable biological material and data for research and selection purposes. Additionally, these data support genetic resources management, especially in the framework of the apple and pear French networks coordinated by BRC. Thanks to these data, pedigrees could be unravelled and association studies implemented.

The pear breeding programme at INRAE Angers-Beaucouze is partly devoted to screen for red-skinned fruit from mutants derived from European pear cultivars. These mutants were selected around 1950 in the USA from 'Doyenne du Comice', 'Bartlett' and 'Beurre Hardy'. Several combinations involving these red mutants and also cultivars such as 'Lombacad' and 'Red Satin' were studied for the inheritance of red leaves in the greenhouse and red-skinned fruit in the field. A single dominant gene (already named C/c) codes for red leaves and fruit skin in these different progenies. In the present study, the focus was evaluating vigour, tree habit and flowering time of field planted progenies. The traits were determined and compared between the segregants (C/C, C/c, c/c). The deep-red seedlings (C/C) have a columnar habit with some weak feathers and their flowering in the field is late compared to the other segregants (C/c and c/c). Our results highlight a pleiotropic effect of gene C on tree habit and flowering time. The other part of this breeding programme is related to red-fleshed pears. Progenies from the cultivar 'Sanguine d'Italie' were evaluated for this character. A single dominant gene (called F/f), independent from C/c, was found to control this character. In the greenhouse the red colour of the hypocotyls is a very accurate marker for the red-fleshed fruit hybrids screened in the field. An improved knowledge of the genetic basis of production and accumulation of red pigments in the fruit flesh will better support the pear breeding programmes aimed to select new pear cultivars carrying this attractive trait.

In this paper, we apply the median-of-means principle to derive robust versions of local averaging rules in non-parametric regression. For various estimates, including nearest neighbors and kernel procedures, we obtain non-asymptotic exponential inequalities, with only a second moment assumption on the noise. We then show that these bounds cannot be significantly improved by establishing a corresponding lower bound on tail probabilities.

We illustrate how the Hill relation and the notion of quasi-stationary distribution can be used to analyse the biasing error introduced by many numerical procedures that have been proposed in the literature, in particular in molecular dynamics, to compute mean reaction times between metastable states for Markov processes. The theoretical findings are illustrated on various examples demonstrating the sharpness of the biasing error analysis as well as the applicability of our study to elliptic diffusions.

Let g : $\Omega$ = [0, 1] d $\rightarrow$ R denote a Lipschitz function that can be evaluated at each point, but at the price of a heavy computational time. Let X stand for a random variable with values in $\Omega$ such that one is able to simulate, at least approximately, according to the restriction of the law of X to any subset of $\Omega$. For example, thanks to Markov chain Monte Carlo techniques, this is always possible when X admits a density that is known up to a normalizing constant. In this context, given a deterministic threshold T such that the failure probability p := P(g(X) > T) may be very low, our goal is to estimate the latter with a minimal number of calls to g. In this aim, building on Cohen et al. [9], we propose a recursive and optimal algorithm that selects on the fly areas of interest and estimate their respective probabilities.

Sequential Monte Carlo (SMC) methods represent a classical set of techniques to simulate a sequence of probability measures through a simple selection/mutation mechanism. However, the associated selection functions and mutation kernels usually depend on tuning parameters that are of first importance for the efficiency of the algorithm. A standard way to address this problem is to apply Adaptive Sequential Monte Carlo (ASMC) methods, which consist in exploiting the information given by the history of the sample to tune the parameters. This article is concerned with variance estimation in such ASMC methods. Specifically, we focus on the case where the asymptotic variance coincides with the one of the "limiting" Sequential Monte Carlo algorithm as defined by Beskos et al. (2016). We prove that, under natural assumptions, the estimator introduced by Lee and Whiteley (2018) in the nonadaptive case (i.e., SMC) is also a consistent estimator of the asymptotic variance for ASMC methods. To do this, we introduce a new estimator that is expressed in terms of coalescent tree-based measures, and explain its connection with the previous one. Our estimator is constructed by tracing the genealogy of the associated Interacting Particle System. The tools we use connect the study of Particle Markov Chain Monte Carlo methods and the variance estimation problem in SMC methods. As such, they may give some new insights when dealing with complex genealogy-involved problems of Interacting Particle Systems in more general scenarios.

Fleming-Viot type particle systems represent a classical way to approximate the distribution of a Markov process with killing, given that it is still alive at a final deterministic time. In this context, each particle evolves independently according to the law of the underlying Markov process until its killing, and then branches instantaneously on another randomly chosen particle. While the consistency of this algorithm in the large population limit has been recently studied in several articles, our purpose here is to prove Central Limit Theorems under very general assumptions. For this, we only suppose that the particle system does not explode in finite time, and that the jump and killing times have atomless distributions. In particular, this includes the case of elliptic diffusions with hard killing.

This article presents a variant of Fleming-Viot particle systems, which are a standard way to approximate the law of a Markov process with killing as well as related quantities. Classical Fleming-Viot particle systems proceed by simulating $N$ trajectories, or particles, according to the dynamics of the underlying process, until one of them is killed. At this killing time, the particle is instantaneously branched on one of the $(N-1)$ other ones, and so on until a fixed and finite final time $T$. In our variant, we propose to wait until $K$ particles are killed and then rebranch them independently on the $(N-K)$ alive ones. Specifically, we focus our attention on the large population limit and the regime where $K/N$ has a given limit when $N$ goes to infinity. In this context, we establish consistency and asymptotic normality results. The variant we propose is motivated by applications in rare event estimation problems.

This article first presents a short historical perpective of the importance splitting approach to simulate and estimate rare events, with a detailed description of several variants. We then give an account of recent theoretical results on these algorithms, including a central limit theorem for Adaptive Multilevel Splitting (AMS). Considering the asymptotic variance in the latter, the choice of the importance function, called the reaction coordinate in molecular dynamics, is also discussed. Finally, we briefly mention some worthwhile applications of AMS in various domains.

The French INRA Pip Fruit Biological Resource Center is in charge of the maintenance, management, characterization and promotion of the traditional and scientific genetic resources of apple, pear, quince and related species. In 2010, the first cryopreservation trials were performed on apple and pear using the dormant bud technique with the main aim of cryopreserving a major part of the INRA Malus and Pyrus collections. However, some accessions, mainly from the Pyrus genus, sometimes responded badly to the dormant bud technique. To optimize the ability to cryopreserve the largest number of accessions in our collections, the droplet-vitrification technique was tested on two cultivars of Pyrus: 'Williams' and 'Conference', used as controls since 2010. Our objective was to evaluate this method under our experimental conditions and to compare it with the dormant bud technique. Experiments were carried out using samples from a batch of budsticks collected in January 2015, in order to test the impact of several critical steps in the process. With both techniques, the immersion in liquid nitrogen step appeared to have the most important impact on the final results, much more than dehydration and slow-freezing for dormant bud technique or LS and PVS2 steps for droplet-vitrification technique. Regenerated plants were obtained after droplet-vitrification. With some improvements, this method could be a viable alternative to the dormant bud technique and opens up the possibility of new development paths for the optimization of long-term preservation systems of our genetic resources collections.

Adaptive Multilevel Splitting (AMS for short) is a generic Monte Carlo method for Markov processes that simulates rare events and estimates associated probabilities. Despite its practical efficiency, there are almost no theoretical results on the convergence of this algorithm. The purpose of this paper is to prove both consistency and asymptotic normality results in a general setting. This is done by associating to the original Markov process a level-indexed process, also called a stochastic wave, and by showing that AMS can then be seen as a Fleming-Viot type particle system. This being done, we can finally apply general results on Fleming-Viot particle systems that we have recently obtained.

The aim of Genome Wide Association Studies (GWAS) is to identify markers in tight linkage disequilibrium with loci controlling quantitative trait variation. These markers can then be used in marker-assisted selection (MAS) in fruit crops such as apple. The GWAS approach involves both phenotyping of a large population of mostly unrelated individuals for the traits of interest, and genotyping at high marker density. In the EU-FP7 project FruitBreedomics, almost 1,200 European diploid dessert apple accessions (old and/or local cultivars) from six germplasm collections were genotyped with the Affymetrix Axiom_Apple480K array (487,000 SNPs). Phenotypic data on a large number of traits have been gathered during the project. Here we focus on flowering period and harvesting date. Knowledge of the genetic control of these traits is necessary to develop cultivars that can face the challenges imposed by global climate change and to target cultivar development as a function of a prolonged vegetation period in the production regions. Different models were tested, including control for effects of population structure and relatedness between cultivars. The full model, controlling for both structure and relatedness, was shown to be the most appropriate to avoid spurious marker-trait associations. When analyzing data over all collections, one significant marker-trait association was obtained for each trait, on chromosomes 9 and 3, for flowering period and harvesting date, respectively. Thereby, genomic locations previously identified in bi-parental populations could now be confirmed for a genetically diverse germplasm.

Deciphering the genetic control of flowering and ripening periods in apple is essential for breeding cultivars adapted to their growing environments. We implemented a large Genome-Wide Association Study (GWAS) at the European level using an association panel of 1,168 different apple genotypes distributed over six locations and phenotyped for these phenological traits. The panel was genotyped at a high-density of SNPs using the Axiom®Apple 480 K SNP array. We ran GWAS with a multi-locus mixed model (MLMM), which handles the putatively confounding effect of significant SNPs elsewhere on the genome. Genomic regions were further investigated to reveal candidate genes responsible for the phenotypic variation. At the whole population level, GWAS retained two SNPs as cofactors on chromosome 9 for flowering period, and six for ripening period (four on chromosome 3, one on chromosome 10 and one on chromosome 16) which, together accounted for 8.9 and 17.2% of the phenotypic variance, respectively. For both traits, SNPs in weak linkage disequilibrium were detected nearby, thus suggesting the existence of allelic heterogeneity. The geographic origins and relationships of apple cultivars accounted for large parts of the phenotypic variation. Variation in genotypic frequency of the SNPs associated with the two traits was connected to the geographic origin of the genotypes (grouped as North+East, West and South Europe), and indicated differential selection in different growing environments. Genes encoding transcription factors containing either NAC or MADS domains were identified as major candidates within the small confidence intervals computed for the associated genomic regions. A strong microsynteny between apple and peach was revealed in all the four confidence interval regions. This study shows how association genetics can unravel the genetic control of important horticultural traits in apple, as well as reduce the confidence intervals of the associated regions identified by linkage mapping approaches. Our findings can be used for the improvement of apple through marker-assisted breeding strategies that take advantage of the accumulating additive effects of the identified SNPs.

Background The amount and structure of genetic diversity in dessert apple germplasm conserved at a European level is mostly unknown, since all diversity studies conducted in Europe until now have been performed on regional or national collections. Here, we applied a common set of 16 SSR markers to genotype more than 2,400 accessions across 14 collections representing three broad European geographic regions (North + East, West and South) with the aim to analyze the extent, distribution and structure of variation in the apple genetic resources in Europe. Results A Bayesian model-based clustering approach showed that diversity was organized in three groups, although these were only moderately differentiated (F ST = 0.031). A nested Bayesian clustering approach allowed identification of subgroups which revealed internal patterns of substructure within the groups, allowing a finer delineation of the variation into eight subgroups (F ST = 0.044). The first level of stratification revealed an asymmetric division of the germplasm among the three groups, and a clear association was found with the geographical regions of origin of the cultivars. The substructure revealed clear partitioning of genetic groups among countries, but also interesting associations between subgroups and breeding purposes of recent cultivars or particular usage such as cider production. Additional parentage analyses allowed us to identify both putative parents of more than 40 old and/or local cultivars giving interesting insights in the pedigree of some emblematic cultivars. Conclusions The variation found at group and subgroup levels may reflect a combination of historical processes of migration/selection and adaptive factors to diverse agricultural environments that, together with genetic drift, have resulted in extensive genetic variation but limited population structure. The European dessert apple germplasm represents an important source of genetic diversity with a strong historical and patrimonial value. The present work thus constitutes a decisive step in the field of conservation genetics. Moreover, the obtained data can be used for defining a European apple core collection useful for further identification of genomic regions associated with commercially important horticultural traits in apple through genome-wide association studies.

Multilevel Splitting is a Sequential Monte Carlo method to simulate realisations of a rare event as well as to estimate its probability. This article is concerned with the convergence and the fluctuation analysis of Adaptive Multilevel Splitting techniques. In contrast to their fixed level version, adaptive techniques estimate the sequence of levels on the fly and in an optimal way, with only a low additional computational cost. However, very few convergence results are available for this class of adaptive branching models, mainly because the sequence of levels depends on the occupation measures of the particle systems. This article proves the consistency of these methods as well as a central limit theorem. In particular, we show that the precision of the adaptive version is the same as the one of the fixed-levels version where the levels would have been placed in an optimal manner.

In-depth characterization of apple genetic resources is a prerequisite for genetic improvement and for germplasm management. In this study, we fingerprinted a very large French collection of 2163 accessions with 24 SSR markers in order to evaluate its genetic diversity, population structure, and genetic relationships, to link these features with cultivar selection date or usage (old or modern, dessert or cider cultivars), and to construct core collections. Most markers were highly discriminating and powerful for varietal identification, with a probability of identity P (ID) over the 21 retained SSR loci close to 10 −28 . Pairwise comparisons revealed 34 % redundancy and 18.5 % putative triploids. The results showed that the germplasm is highly diverse with an expected heterozygosity H e of 0.82 and observed heterozygosity H o of 0.83. A Bayesian model-based clustering approach revealed a weak but significant structure in three subgroups (F ST = 0.014–0.048) corresponding, albeit approximately, to the three subpopulations defined beforehand (Old Dessert, Old Cider, and Modern Cultivars). Parentage analyses established already known and yet unknown relationships, notably between old cultivars, with the frequent occurrence of cultivars such as “King of Pippin” and “Calville Rouge d’Hiver” as founders. Finally, core collections based on allelic diversity were constructed. A large dessert core collection of 278 cultivars contained 90 % of the total dessert allelic diversity, whereas a dessert subcore collection of 48 cultivars contained 71 % of diversity. For cider apples, a 48-cultivar core collection contained 83 % of the total cider allelic diversity.

Adaptive Multilevel Splitting (AMS for short) is a general Monte-Carlo method to simulate and estimate rare events. In the framework of molecular dynamics, this technique can for example be used to generate reactive trajectories, namely equilibrium trajectories leaving a metastable state and ending in another one. In this talk, we will present a connection between AMS and stochastic waves, i.e. the transformation of a Markov process by a random time change. In particular, this connection allows us to analyze AMS as a Fleming-Viot type particle system. ∗Speaker sciencesconf.org:largedeviations:70351