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A continuous-time Markov process X can be conditioned to be in a given state at a fixed time T>0 using Doob's h-transform. This transform requires the typically intractable transition density of X. The effect of the h-transform can be described as introducing a guiding force on the process. Replacing this force with an approximation defines the wider class of guided processes. For certain approximations the law of a guided process approximates–and is equivalent to–the actual conditional distribution, with tractable likelihood-ratio. The main contribution of this paper is to prove that the principle of a guided process, introduced in [M. Schauer, F. van der Meulen, and H. van Zanten, Guided proposals for simulating multi-dimensional diffusion bridges, Bernoulli 23 (2017a), pp. 2917–2950. doi:10.3150/16-BEJ833] for stochastic differential equations, can be extended to a more general class of Markov processes. In particular we apply the guiding technique to jump processes in discrete state spaces. The Markov process perspective enables us to improve upon existing results for hypo-elliptic diffusions.

Suppose X is a multidimensional diffusion process. Assume that at time zero the state of X is fully observed, but at time $T>0$ only linear combinations of its components are observed. That is, one only observes the vector $L X_T$ for a given matrix L. In this paper we show how samples from the conditioned process can be generated. The main contribution of this paper is to prove that guided proposals, introduced in [35], can be used in a unified way for both uniformly elliptic and hypo-elliptic diffusions, even when L is not the identity matrix. This is illustrated by excellent performance in two challenging cases: a partially observed twice-integrated diffusion with multiple wells and the partially observed FitzHugh–Nagumo model.

Suppose that a compound Poisson process is observed discretely in time and assume that its jump distribution is supported on the set of natural numbers. In this paper we propose a non-parametric Bayesian approach to estimate the intensity of the underlying Poisson process and the distribution of the jumps. We provide a MCMC scheme for obtaining samples from the posterior. We apply our method on both simulated and real data examples, and compare its performance with the frequentist plug-in estimator proposed by Buchmann and Gr\"ubel. On a theoretical side, we study the posterior from the frequentist point of view and prove that as the sample size $n\rightarrow\infty$, it contracts around the `true', data-generating parameters at rate $1/\sqrt{n}$, up to a $\log n$ factor., Comment: 27 pages, 7 figures

We introduce the use of the Zig-Zag sampler to the problem of sampling conditional diffusion processes (diffusion bridges). The Zig-Zag sampler is a rejection-free sampling scheme based on a non-reversible continuous piecewise deterministic Markov process. Similar to the Lévy–Ciesielski construction of a Brownian motion, we expand the diffusion path in a truncated Faber–Schauder basis. The coefficients within the basis are sampled using a Zig-Zag sampler. A key innovation is the use of thefully localalgorithm for the Zig-Zag sampler that allows to exploit the sparsity structure implied by the dependency graph of the coefficients and by thesubsamplingtechnique to reduce the complexity of the algorithm. We illustrate the performance of the proposed methods in a number of examples.

Suppose X is a multivariate diffusion process that is observed discretely in time. At each observation time, a transformation of the state of the process is observed with noise. The smoothing problem consists of recovering the path of the process, consistent with the observations. We derive a novel Markov Chain Monte Carlo algorithm to sample from the exact smoothing distribution. The resulting algorithm is called the Backward Filtering Forward Guiding (BFFG) algorithm. We extend the algorithm to include parameter estimation. The proposed method relies on guided proposals introduced in Schauer et al. (2017). We illustrate its efficiency in a number of challenging problems., Revised article with additional author Marcin Mider. Article contains an animated figure

We present a general framework for Bayesian estimation of incompletely observed multivariate diffusion processes. Observations are assumed to be discrete in time, noisy and incomplete. We assume the drift and diffusion coefficient depend on an unknown parameter. A data-augmentation algorithm for drawing from the posterior distribution is presented which is based on simulating diffusion bridges conditional on a noisy incomplete observation at an intermediate time. The dynamics of such filtered bridges are derived and it is shown how these can be simulated using a generalised version of the guided proposals introduced in Schauer, Van der Meulen and Van Zanten (2017, Bernoulli 23(4A)).

Stochastically evolving geometric systems are studied in shape analysis and computational anatomy for modelling random evolutions of human organ shapes. The notion of geodesic paths between shapes is central to shape analysis and has a natural generalisation as diffusion bridges in a stochastic setting. Simulation of such bridges is key to solve inference and registration problems in shape analysis. We demonstrate how to apply state-of-the-art diffusion bridge simulation methods to recently introduced stochastic shape deformation models thereby substantially expanding the applicability of such models. We exemplify these methods by estimating template shapes from observed shape configurations while simultaneously learning model parameters.

In 1990, livestock grazing was introduced in Meijendel, a 1800 ha lime-rich coastal dune area, at a density of 0.06–0.07 LLU.ha-1.year−1 (1:12–18 ha) to counteract encroachment of tall grasses and shrubland on dune grassland and increase the bare sand area. Monitoring was based on four digital orthophotos (1975–1990–2001-2009) with a high spatial resolution (pixel size 25 × 25 cm). The changes were tested using Generalized Estimating Equations. Habitat changes occurred, but contradicting our hypothesis, there was no significant impact from the grazing on bare sand, grassland or shrubland within 11 and 19 years post livestock introduction. (1) After several decennia of decreasing bare sand, there was a significant increase between 2001 and 2009, irrespective of livestock presence. (2) The changes in grasslands and shrublands are independent of the livestock, but dependent on distance to the coast. (3) Bare sand and shrub cover determine the space left for the dune grasslands. It appears other factors than livestock grazing must have induced the changes. Changes in climate conditions and nitrogen load might have stimulated bare sand. An interaction with the end of Marram planting in 1990 cannot be concluded from available data. The disease-led reduction of rabbit grazing from the mid-1950s led to an expansion of the dominant shrub Hippophae rhamnoides. However, Hippophae shrubland typically regresses to grasslands on its collapse after 25–40 years. Tree species like Crataegus, Betula and Quercus will gradually dominate the landscape for far longer. Active removal of these indigenous species is necessary to prevent future loss of dune grasslands.

Available climate change projections, which can be used for quantifying future changes in marine and coastal ecosystems, usually consist of a few scenarios. Studies addressing ecological impacts of climate change often make use of a low- (RCP2.6), moderate- (RCP4.5) or high climate scenario (RCP8.5), without taking into account further uncertainties in these scenarios. In this research a methodology is proposed to generate further synthetic scenarios, based on existing datasets, for a better representation of climate change induced uncertainties. The methodology builds on Regional Climate Model scenarios provided by the EURO-CORDEX experiment. In order to generate new realizations of climate variables, such as radiation or temperature, a hierarchical Bayesian model is developed. In addition, a parameterized time series model is introduced, which includes a linear trend component, a seasonal shape with varying amplitude and time shift, and an additive residual term. The seasonal shape is derived with the non-parametric locally weighted scatterplot smoothing, and the residual term includes the smoothed variance of residuals and independent and identically distributed noise. The distributions of the time series model parameters are estimated through Bayesian parameter inference with Markov chain Monte Carlo sampling (Gibbs sampler). By sampling from the predictive distribution numerous new statistically representative synthetic scenarios can be generated including uncertainty estimates. As a demonstration case, utilizing these generated synthetic scenarios and a physically based ecological model (Delft3D-WAQ) that relates climate variables to ecosystem variables, a probabilistic simulation is conducted to further propagate the climate change induced uncertainties to marine and coastal ecosystem indicators.

We consider a nonparametric Bayesian approach to estimate the diffusion coefficient of a stochastic differential equation given discrete time observations over a fixed time interval. As a prior on the diffusion coefficient, we employ a histogram-type prior with piecewise constant realisations on bins forming a partition of the time interval. Specifically, these constants are realizations of independent inverse Gamma distributed randoma variables. We justify our approach by deriving the rate at which the corresponding posterior distribution asymptotically concentrates around the data-generating diffusion coefficient. This posterior contraction rate turns out to be optimal for estimation of a H\"older-continuous diffusion coefficient with smoothness parameter $0

Estimation of parameters of a diffusion based on discrete time observations poses a difficult problem due to the lack of a closed form expression for the likelihood. From a Bayesian computational perspective it can be casted as a missing data problem where the diffusion bridges in between discrete-time observations are missing. The computational problem can then be dealt with using a Markov-chain Monte-Carlo method known as data-augmentation. If unknown parameters appear in the diffusion coefficient, direct implementation of data-augmentation results in a Markov chain that is reducible. Furthermore, data-augmentation requires efficient sampling of diffusion bridges, which can be difficult, especially in the multidimensional case. We present a general framework to deal with with these problems that does not rely on discretisation. The construction generalises previous approaches and sheds light on the assumptions necessary to make these approaches work. We define a random-walk type Metropolis-Hastings sampler for updating diffusion bridges. Our methods are illustrated using guided proposals for sampling diffusion bridges. These are Markov processes obtained by adding a guiding term to the drift of the diffusion. We give general guidelines on the construction of these proposals and introduce a time change and scaling of the guided proposal that reduces discretisation error. Numerical examples demonstrate the performance of our methods.

According to both domain expert knowledge and empirical evidence, wavelet coefficients of real signals tend to exhibit clustering patterns, in that they contain connected regions of coefficients of similar magnitude (large or small). A wavelet de-noising approach that takes into account such a feature of the signal may in practice outperform other, more vanilla methods, both in terms of the estimation error and visual appearance of the estimates. Motivated by this observation, we present a Bayesian approach to wavelet de-noising, where dependencies between neighbouring wavelet coefficients are a priori modelled via a Markov chain-based prior, that we term the caravan prior. Posterior computations in our method are performed via the Gibbs sampler. Using representative synthetic and real data examples, we conduct a detailed comparison of our approach with a benchmark empirical Bayes de-noising method (due to Johnstone and Silverman). We show that the caravan prior fares well and is therefore a useful addition to the wavelet de-noising toolbox., 32 pages, 15 figures, 4 tables

Assume we observe a finite number of inspection times together with information on whether a specific event has occurred before each of these times. Suppose replicated measurements are available on multiple event times. The set of inspection times, including the number of inspections, may be different for each event. This is known as mixed case interval censored data. We consider Bayesian estimation of the distribution function of the event time while assuming it is concave. We provide sufficient conditions on the prior such that the resulting procedure is consistent from the Bayesian point of view. We also provide computational methods for drawing from the posterior and illustrate the performance of the Bayesian method in both a simulation study and two real datasets.

Given a sample from a discretely observed multidimensional compound Poisson process, we study the problem of nonparametric estimation of its jump size density $r_0$ and intensity $\lambda_0$. We take a nonparametric Bayesian approach to the problem and determine posterior contraction rates in this context, which, under some assumptions, we argue to be optimal posterior contraction rates. In particular, our results imply the existence of Bayesian point estimates that converge to the true parameter pair $(r_0,\lambda_0)$ at these rates. To the best of our knowledge, construction of nonparametric density estimators for inference in the class of discretely observed multidimensional L\'{e}vy processes, and the study of their rates of convergence is a new contribution to the literature., Comment: Published at http://dx.doi.org/10.15559/15-VMSTA20 in the Modern Stochastics: Theory and Applications (https://www.i-journals.org/vtxpp/VMSTA) by VTeX (http://www.vtex.lt/)

Suppose X1,…,Xn is a random sample from a bounded and decreasing density f0 on [0,∞). We are interested in estimating such f0, with special interest in f0(0). This problem is encountered in various statistical applications and has gained quite some attention in the statistical literature. It is well known that the maximum likelihood estimator is inconsistent at zero. This has led several authors to propose alternative estimators which are consistent. As any decreasing density can be represented as a scale mixture of uniform densities, a Bayesian estimator is obtained by endowing the mixture distribution with the Dirichlet process prior. Assuming this prior, we derive contraction rates of the posterior density at zero by carefully revising arguments presented in Salomond (Electronic Journal of Statistics 8 (2014) 1380–1404). Several choices of base measure are numerically evaluated and compared. In a simulation various frequentist methods and a Bayesian estimator are compared. Finally, the Bayesian procedure is applied to current durations data described in Slama et al. (Human Reproduction 27 (2012) 1489–1498).

Given a sample from a discretely observed compound Poisson process, we consider non-parametric estimation of the density $$f_0$$ of its jump sizes, as well as of its intensity $$\lambda _0.$$ We take a Bayesian approach to the problem and specify the prior on $$f_0$$ as the Dirichlet location mixture of normal densities. An independent prior for $$\lambda _0$$ is assumed to be compactly supported and to possess a positive density with respect to the Lebesgue measure. We show that under suitable assumptions the posterior contracts around the pair $$(\lambda _0,\,f_0)$$ at essentially (up to a logarithmic factor) the $$\sqrt{n\Delta }$$ -rate, where n is the number of observations and $$\Delta $$ is the mesh size at which the process is sampled. The emphasis is on high frequency data, $$\Delta \rightarrow 0,$$ but the obtained results are also valid for fixed $$\Delta .$$ In either case we assume that $$n\Delta \rightarrow \infty .$$ Our main result implies existence of Bayesian point estimates converging (in the frequentist sense, in probability) to $$(\lambda _0,\,f_0)$$ at the same rate. We also discuss a practical implementation of our approach. The computational problem is dealt with by inclusion of auxiliary variables and we develop a Markov chain Monte Carlo algorithm that samples from the joint distribution of the unknown parameters in the mixture density and the introduced auxiliary variables. Numerical examples illustrate the feasibility of this approach.

Given discrete time observations over a fixed time interval, we study a nonparametric Bayesian approach to estimation of the volatility coefficient of a stochastic differential equation. We postulate a histogram-type prior on the volatility with piecewise constant realisations on bins forming a partition of the time interval. The values on the bins are assigned an inverse Gamma Markov chain (IGMC) prior. Posterior inference is straightforward to implement via Gibbs sampling, as the full conditional distributions are available explicitly and turn out to be inverse Gamma. We also discuss in detail the hyperparameter selection for our method. Our nonparametric Bayesian approach leads to good practical results in representative simulation examples. Finally, we apply it on a classical data set in change-point analysis: weekly closings of the Dow-Jones industrial averages.

A new dune area was constructed using beach and foreshore nourishment along the Dutch dune coast in the delta area. The new area is intended as a compensation for losses of existing high quality dune habitat as a result of the use of Maasvlakte 2 (MV-2), the new extension of Rotterdam harbour. Compensation was required under international EU Community regulations because the existing dune areas that could potentially be affected are protected under the EU Natura2000. The paper discusses the political and management aspects at EU- and regional levels. In The Netherlands, the construction of a compensation area along the coast, due to the extension of an international harbour, has been a procedure without precedence. The knowledge-based design and construction of the area and the set up of monitoring and evaluation programs in the context of policy and management are described. We believe that they provide innovative ideas for coastal management worldwide. Aspects of construction and landscape ecological developments of the same area have been reported in an earlier paper (van der Meulen et al. 2014).

A Monte Carlo method for simulating a multi-dimensional diffusion process conditioned on hitting a fixed point at a fixed future time is developed. Proposals for such diffusion bridges are obtained by superimposing an additional guiding term to the drift of the process under consideration. The guiding term is derived via approximation of the target process by a simpler diffusion processes with known transition densities. Acceptance of a proposal can be determined by computing the likelihood ratio between the proposal and the target bridge, which is derived in closed form. We show under general conditions that the likelihood ratio is well defined and show that a class of proposals with guiding term obtained from linear approximations fall under these conditions.

A new dune area was constructed by beach and foreshore nourishments along the Delfland Coast in the southwestern Netherlands. The new area is intended to be a compensation for losses of existing high quality dunes in nearby areas, due to the use/construction of Maasvlakte 2, the new harbour extension of Rotterdam opening in 2013. The target habitats for compensation include dry grey dunes and moist dune valley grassland. Due to the habitat type, the nourished material had to have a special character. The nourishment was carried out according to a special design and initial development of the new area had to follow certain pathways. In order to meet the compensation requirements in time. Careful monitoring is done to assess whether development of the new habitat is taking place in the direction of the desired target habitats. This paper reports on the construction of the new area and on the first years of development and monitoring. The monitoring mainly concerns development of the abiotic landscape elements, geomorphology, groundwater and soils. The lessons learned from this area are discussed in the light of the development of other recent young nature areas in the sandy Dutch delta coast. This gives new insights for coastal management in general in the context of building with nature.

We consider a nonparametric Bayesian approach to estimate the diffusion coefficient of a stochastic differential equation given discrete time observations over a fixed time interval. As a prior on the diffusion coefficient, we employ a histogram-type prior with piecewise constant realisations on bins forming a partition of the time interval. Specifically, these constants are realizations of independent inverse Gamma distributed randoma variables. We justify our approach by deriving the rate at which the corresponding posterior distribution asymptotically concentrates around the data-generating diffusion coefficient. This posterior contraction rate turns out to be optimal for estimation of a Holder-continuous diffusion coefficient with smoothness parameter 0

One of Risso’s dolphin’s distinctive characteristics is the tendency to “lighten” with age due to the accumulation of unpigmented scars. These accumulated scars may provide an indication of age. Photographic skin recaptures gathered from 61 free-ranging animals over a period of 15 years were analyzed to develop a skin classification model in 6 skin stages. Classification of photographic skin captures following this model was tested by 15 experts and 13 nonexpert rankers, with a general probability of agreement of 79%. The duration of each skin stage was estimated using a statistical model based on the recorded dates in which individual animals were known to have entered and/or exited a given stage. A Bayesian approach was used to combine available photographic skin recapture data using expert knowledge as prior to predict the duration for each skin stage and thus the mean age at each stage. Results suggest that animals may live more than 45 years, which is in agreement with published information based on dental layers. The proposed skin stages can be correlated with reproduction, with the transition to stage 3 linked to the onset of maturity. Adult females are less scarified than males and were not observed in the whiter skin classes. The proposed skin stage model is noninvasive and easy to apply and could be a valuable tool in further studies of population structure and dynamics of Risso’s dolphins.

The standard method to assess a measurement system’s precision is a gauge repeatability and reproducibility (gauge R&R) study. It exploits replications to estimate variance components that are interpreted as measurement spread. For nonrepeatable measurements, it is not feasible to obtain replications because objects are destroyed when they are measured or because the object changes over time. Possible solutions are to replace replications with measurements of multiple objects or with the measurement of one object at multiple times. Subsequently, these measurements are modeled by a fixed pattern (over time or over positions). We show that the experimental design used in this type of nonrepeatable gauge R&R studies is best constructed in a way that is similar to a Latin square design. These designs have a great flexibility, can be applied in many situations encountered in practice, and have nice mathematical properties as well. We consider several examples in which this approach is applied and worked out. For the examples given, we provide the analysis and the results following the worked-out approach. Analysis of the envisaged experimental set-up is done with linear and nonlinear mixed models in which variance components are estimated by restricted maximum-likelihood estimators.

Key words and Phrases: cumulant, empirical characteristic function,Lévy process, self-decomposable distribution, stationary process.Consider a stationary sequence of random variables with infinitely divisiblemarginal law, characterized by its Lévy density. We analyze the behaviorof a so-called cumulant M-estimator, in case this Lévy density ischaracterized by a Euclidean (finite-dimensional) parameter. Under mildconditions, we prove consistency and asymptotic normality of the estimator.The estimator is considered in the situation where the data are incrementsof a subordinator as well as the situation where the data consist of adiscretely sampled Ornstein Uhlenbeck process induced by the subordinator.We illustrate our results for the Gamma-process and theInverse-Gaussian-OU-process. For these processes we also explain how theestimator can be computed numerically.

Main coastal characteristics The Netherlands occupy a small low lying country (35000 km) in the Delta of main European rivers, Rhine, Meuse and Scheldt. This has put coastal management always high on the national policy agenda. The coast (ca 330 km long) consists mainly of sandy beaches and dunes and estuaries. There are a few hard parts, where dikes, storm surge barriers and sluices are found (Fig. 1). In the past 6000 years sea level has been rising and flooding and erosion ware prevalent. Dunes were also formed. The present prominent, so called Younger Dunes, date between 1000-1600 AD. Since 1900 to 2000 average sea level rise, based on measurements, measures appr. 18 cm/100 yr. Due to climate change the sea level rise is expected to increase again. Predictions differ from 30-80 cm for the coming 100 year. The coast fulfills a number of important functions for the population of the Netherlands. They are: coastal defense, recreation, nature conservation and drinking water production. Coastal defense has always been primary, but in the course of time other functions gained importance.

A five-year experimental study was carried out to examine the combined effects of grazing and atmospheric nitrogen deposition on the vegetation of three dry dune grasslands: one short species-rich, one short species-poor, and one predominated by tall graminoids. Additional fertilization with nitrogen had no significant effect, neither in grazed nor in non-grazed plots. Exclusion of grazing by rabbits resulted in an increase in the frequency of perennial graminoids and a decrease in the frequency of annual graminoids and herbs. Nevertheless, species diversity remained the same in the species-rich grassland. During the experiment, the above-ground biomass increased in all nongrazed plots and the amount of bare soil and mosses decreased. The vegetation changes occurred mainly within one year after the exclusion of grazing. An exception is the grass-dominated site where the amount of Calamagrostis epigejos increased gradually from ca. 20 % in the first two years to about 50 % in the fourth and fifth year. Grazing by rabbits seems essential to prevent graminoids to become predominant in the dry dunes. If graminoids are dominant, grazing by horses can be an appropriate method to restore the original grassland vegetation. After six months of grazing by horses the grass-dominated site showed a decrease of the frequency of perennial graminoids, from 95 % to 80 %, and an increase of the frequency of perennial herbs, from 2.5 % to between 13 and 20 %.

A Belgium hospital's cardiology department advises discharged patients to participate in a rehabilitation program. The article models the probability that a patient joins the program as a function of various numerical and categorical influence factors.

We consider two nonparametric procedures for estimating a concave distribution function based on data corrupted with additive noise generated by a bounded decreasing density on $(0,\infty)$. For the maximum likelihood (ML) estimator and least squares (LS) estimator, we state qualitative properties, prove consistency and propose a computational algorithm. For the LS estimator and its derivative, we also derive the pointwise asymptotic distribution. Moreover, the rate $n^{-2/5}$ achieved by the LS estimator is shown to be minimax for estimating the distribution function at a fixed point., Published in at http://dx.doi.org/10.1214/07-AOS579 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We consider nonparametric estimation of the Levy measure of a hidden Levy process driving a stationary Omstein-Uhlenbeck process which is observed at discrete time points. This Levy measure can be expressed in terms of the canonical function of the stationary distribution of the Omstein-Uhlenbeck process, which is known to be self-decomposable. We propose an estimator for this canonical function based on a preliminary estimator of the characteristic function of the stationary distribution. We provide a suppport-reduction algorithm for the numerical computation of the estimator, and show that the estimator is asymptotically consistent under various sampling schemes. We also define a simple consistent estimator of the intensity parameter of the process. Along the way, a nonparametric procedure for estimating a self-decomposable density function is constructed, and it is shown that the Oenstein-Uhlenbeck process is β-mixing. Some general results on uniform convergence of random characteristic functions are included. © 2005 ISI/BS.