J. Andrés Christen

A Bayesian framework for age modeling of radiocarbon-dated peat deposits: Case studies from the Netherlands

Recently, Bayesian statistical software has been developed for age-depth modeling (wiggle-match dating) of sequences of densely spaced radiocarbon dates from peat cores. The method is described in non-statistical terms, and is compared with an alternative method of chronological ordering of 14C dates. Case studies include the dating of the start of agriculture in the northeastern part of the Netherlands, and of a possible Hekla-3 tephra layer in the same country. We discuss future enhancements in Bayesian age modeling.

A New Robust Statistical Model for Radiocarbon Data

The general method currently used to analyze radiocarbon data (y) is conditional on the standard deviation (σ), reported by 14C laboratories, which reflects the uncertainty in the dating process. This uncertainty is measured through a series of empirical as well as theoretical considerations about the dating process, chemical preprocessing, etc. Nevertheless, σ is assumed as known in the statistical model for 14C data used since the dawn of the discipline. This paper proposes a method for the analysis of 14C data where the associated variance is taken as the product of an unknown constant α with the sum of the variance reported by the laboratory σ2 and the variance of the calibration curve σ2(θ) (that is, an unknown error multiplier). Using this approach, assuming that the 14C determination y arises from a Normal population and that, a priori, α has an inverse gamma distribution InvGa(a, b), the resulting dating model is a t distribution with 2a degrees of freedom. The introduction of parameters a and b allows a robust analysis in the presence of atypical data and at the same time incorporates the uncertainty associated with the intra- and interlaboratory error assessment processes. Comparisons with the common Normal model show that the proposed t model produces smoother posterior distributions and seem to be far more robust to atypical data, presenting a simpler alternative to the standard 14C outlier analysis. Moreover, this new model might be a step forward in understanding and explaining the otherwise elusive scatter in 14C data seen in interlaboratory studies.

Species Diversity and Distribution in Presence-Absence Matrices: Mathematical Relationships and Biological Implications

The diversity of sites and the distribution of species are fundamental pieces in the analysis of biogeographic and macroecological questions. A link between these two variables is the correlation between the species diversity of sites and the mean range size of species occurring there. Alternatively, one could correlate the range sizes of species and the mean species diversity within those ranges. Here we show that both approaches are mirror images of the same patterns, reflecting fundamental mathematical and biological relationships. We develop a theory and analyze data for North American mammals to interpret range‐diversity plots in which the species diversity of sites and the geographic range of species can be depicted simultaneously. We show that such plots contain much more information than traditional correlative approaches do, and we demonstrate that the positions of points in the plots are determined to a large extent by the average, minimum, and maximum values of range and diversity but that the dispersion of points depends on the association among species and the similitude among sites. These generalizations can be applied to biogeographic studies of diversity and distribution and in the identification of hotspots of diversity and endemism.

Random walk simulations of fossil proxy data

A wealth of palaeoecological studies (e.g. pollen, diatoms, chironomids and macrofossils from deposits such as lakes or bogs) have revealed major as well as more subtle ecosystem changes over decadal to multimillennial timescales. Such ecosystem changes are usually assumed to have been forced by specific environmental changes. Here, we test if the observed changes in palaeoecological records may be reproduced by random simulations, and we find that simple procedures generate abrupt events, long-term trends, quasi-cyclic behaviour, extinctions and immigrations. Our results highlight the importance of replicated and multiproxy data for reliable reconstructions of past climate and environmental changes.

Sequential stopping rules for species accumulation

Identifying and counting the total number of biological species observed to date, and plotting versusa measure of the effort used to record them, gives rise to a species accumulation curve. Interest typically is concerned with estimating the total number of species in the area of study, having observed only the accumulation curve, having no information on species frequencies. This article considers the problem of optimally stopping the sampling process. We use a sequential procedure with a fixed maximum horizon for accumulation. A utility function based on the number of new species to be observed and the effort saved from the maximum horizon is adopted, and a workable algorithm based on backward induction is obtained. An example in accumulation of bat species is also presented.

Prediction of potential areas of species distributions based on presence-only data

We introduce a methodology to infer zones of high potential for the habitat of a species, useful for management of biodiversity, conservation, biogeography, ecology, or sustainable use. Inference is based on a set of sites where the presence of the species has been reported. Each site is associated with covariate values, measured on discrete scales. We compute the predictive probability that the species is present at each node of a regular grid. Possible spatial bias for sites of presence is accounted for. Since the resulting posterior distribution does not have a closed form, a Markov chain Monte Carlo (MCMC) algorithm is implemented. However, we also describe an approximation to the posterior distribution, which avoids MCMC. Relevant features of the approach are that specific notions of data acquisition such as sampling intensity and detectability are accounted for, and that available a priori information regarding areas of distribution of the species is incorporated in a clear-cut way. These concepts, arising in the presence-only context, are not addressed in alternative methods. We also consider an uncertainty map, which measures the variability for the predictive probability at each node on the grid. A simulation study is carried out to test and compare our approach with other standard methods. Two case studies are also presented.

Bayesian Analysis of ODEs: Solver Optimal Accuracy and Bayes Factors

In most cases in the Bayesian analysis of ODE inverse problems, a numerical solver needs to be used. Therefore, we cannot work with the exact theoretical posterior distribution but only with an approximate posterior derived from the error in the numerical solver. To compare an approximate posterior distribution with the theoretical one, we propose using Bayes factors (BFs), considering both of them as models for the data at hand. From a theoretical point of view, we prove that the theoretical vs. numerical posterior BF tends to 1, in the same order as the numerical solver used. In practice, we illustrate the fact that for higher order solvers (e.g., Runge--Kutta) the BF is already nearly 1 for step sizes that would take far less computational effort. Considerable CPU time may be saved by using coarser solvers that nevertheless produce practically error-free posteriors. Two examples are presented where nearly 90% CPU time is saved, with all inference results being identical to those obtained using a solver...

Advances in the Sequential Design of Computer Experiments Based on Active Learning

We present some advances in the design of computer experiments. A Gaussian Process (GP) model is fitted to the computer experiment data as a surrogate model. We investigate using the Active Learning (AL) strategy of finding design points that maximize reduction on predictive variance. Using a series of approximations based on standard results from linear algebra (Weyls inequalities), we establish a score that approximates the AL utility. Our method is illustrated with a simulated example as well as with an intermediate climate computer model.

The comparison of 14C wiggle-matching results for the 'floating' tree-ring chronology of the Ulandryk-4 burial ground (Altai mountains, Siberia)

Two independent (super 14) C data sets of 10 tree-ring samples from the longest master chronology of the Pazyryk cultural complex were obtained and wiggle-matched to the absolute timescale. The results show very good agreement, within 10-15 calendar yr. The Ulandryk-4 burial ground (mound 1) was dated to about 320-310 cal BC, and this is consistent with wiggle-matching of the Pazyryk burial ground date series.

Bayesian sequential analysis for multiple-arm clinical trials

Use of full Bayesian decision-theoretic approaches to obtain optimal stopping rules for clinical trial designs typically requires the use of Backward Induction. However, the implementation of Backward Induction, apart from simple trial designs, is generally impossible due to analytical and computational difficulties. In this paper we present a numerical approximation of Backward Induction in a multiple-arm clinical trial design comparing k experimental treatments with a standard treatment where patient response is binary. We propose a novel stopping rule, denoted by τp, as an approximation of the optimal stopping rule, using the optimal stopping rule of a single-arm clinical trial obtained by Backward Induction. We then present an example of a double-arm (k=2) clinical trial where we use a simulation-based algorithm together with τp to estimate the expected utility of continuing and compare our estimates with exact values obtained by an implementation of Backward Induction. For trials with more than two treatment arms, we evaluate τp by studying its operating characteristics in a three-arm trial example. Results from these examples show that our approximate trial design has attractive properties and hence offers a relevant solution to the problem posed by Backward Induction.