Robert M. Burton

Seasonality and vertical structure of microbial communities in an ocean gyre.

Vertical, seasonal and geographical patterns in ocean microbial communities have been observed in many studies, but the resolution of community dynamics has been limited by the scope of data sets, which are seldom up to the task of illuminating the highly structured and rhythmic patterns of change found in ocean ecosystems. We studied vertical and temporal patterns in the microbial community composition in a set of 412 samples collected from the upper 300 m of the water column in the northwestern Sargasso Sea, on cruises between 1991 and 2004. The region sampled spans the extent of deep winter mixing and the transition between the euphotic and the upper mesopelagic zones, where most carbon fixation and reoxidation occurs. A bioinformatic pipeline was developed to de-noise, normalize and align terminal restriction fragment length polymorphism (T-RFLP) data from three restriction enzymes and link T-RFLP peaks to microbial clades. Non-metric multidimensional scaling statistics resolved three microbial communities with distinctive composition during seasonal stratification: a surface community in the region of lowest nutrients, a deep chlorophyll maximum community and an upper mesopelagic community. A fourth microbial community was associated with annual spring blooms of eukaryotic phytoplankton that occur in the northwestern Sargasso Sea as a consequence of winter convective mixing that entrains nutrients to the surface. Many bacterial clades bloomed in seasonal patterns that shifted with the progression of stratification. These richly detailed patterns of community change suggest that highly specialized adaptations and interactions govern the success of microbial populations in the oligotrophic ocean.

Limit theorems for functionals of mixing processes with applications to U-statistics and dimension estimation

In this paper we develop a general approach for investigating the asymptotic distribution of functional Xn = f((Zn+k)k∈z) of absolutely regular stochastic processes (Zn)n∈z. Such functional occur naturally as orbits of chaotic dynamical systems, and thus our results can be used to study probabilistic aspects of dynamical systems. We first prove some moment inequalities that are analogous to those for mixing sequences. With their help, several limit theorems can be proved in a rather straightforward manner. We illustrate this by re-proving a central limit theorem of Ibragimov and Linnik. Then we apply our techniques to U-statistics Matrix Equation with symmetric kernel h : R × R → R. We prove a law of large numbers, extending results of Aaronson, Burton, Dehling, Gilat, Hill and Weiss for absolutely regular processes. We also prove a central limit theorem under a different set of conditions than the known results of Denker and Keller. As our main application, we establish an invariance principle for U-processes (Un(h))h, indexed by some class of functions. We finally apply these results to study the asymptotic distribution of estimators of the fractal dimension of the attractor of a dynamical system.

Non-uniqueness of measures of maximal entropy for subshifts of finite type

It is known that in one dimension an irreducible subshift of finite type has a unique measure of maximal entropy, the so-called Parry measure. Here we give a counterexample to this in higher dimensions. For this example, we also describe the geometric structure of the measures of maximal entropy and show that there are exactly two extremal measures.

Large deviations for some weakly dependent random processes

In this paper we compute large deviation probabilities for two classes of weakly dependent processes, moving averages of i.i.d. random variables and Poisson center cluster random measures.

Phytoplankton distribution patterns in the northwestern Sargasso Sea revealed by small subunit rRNA genes from plastids.

Phytoplankton species vary in their physiological properties, and are expected to respond differently to seasonal changes in water column conditions. To assess these varying distribution patterns, we used 412 samples collected monthly over 12 years (1991–2004) at the Bermuda Atlantic Time-Series Study site, located in the northwestern Sargasso Sea. We measured plastid 16S ribosomal RNA gene abundances with a terminal restriction fragment length polymorphism approach and identified distribution patterns for members of the Prymnesiophyceae, Pelagophyceae, Chrysophyceae, Cryptophyceae, Bacillariophyceae and Prasinophyceae. The analysis revealed dynamic bloom patterns by these phytoplankton taxa that begin early in the year, when the mixed layer is deep. Previously, unreported open-ocean prasinophyte blooms dominated the plastid gene signal during convective mixing events. Quantitative PCR confirmed the blooms and transitions of Bathycoccus, Micromonas and Ostreococcus populations. In contrast, taxa belonging to the pelagophytes and chrysophytes, as well as cryptophytes, reached annual peaks during mixed layer shoaling, while Bacillariophyceae (diatoms) were observed only episodically in the 12-year record. Prymnesiophytes dominated the integrated plastid gene signal. They were abundant throughout the water column before mixing events, but persisted in the deep chlorophyll maximum during stratified conditions. Various models have been used to describe mechanisms that drive vernal phytoplankton blooms in temperate seas. The range of taxon-specific bloom patterns observed here indicates that different ‘spring bloom’ models can aptly describe the behavior of different phytoplankton taxa at a single geographical location. These findings provide insight into the subdivision of niche space by phytoplankton and may lead to improved predictions of phytoplankton responses to changes in ocean conditions.

Original Contribution: Event-dependent control of noise enhances learning in neural networks

We have devised noise-control algorithms, using biological adaptation as an analogy, for application to response optimization in adaptive systems generally. The present paper illustrates one of these algorithms by showing its effects on increasing the rate of learning in neural networks. Optimization procedures usually employ simulated annealing by which noise is systematically decreased at a constant rate. Our methods are time-invariant, and control the level of injected noise solely through the response of the system. Such time-invariant noise algorithms (TINA) may be more applicable than annealing to adaptive systems that must respond to unpredictable environments, and may find analogy in brain function. Both TINA and annealing have surprising properties of a new form of generalization in which networks that have been trained in the presence of noise are able to exhibit enhanced rates of learning in a subsequent learning task when no noise is present. We use special features of the geometry of error-surfaces, depicting the error as a function of changes in synaptic weights, to discuss the effect of noise in enhancing the rate of learning, and to compare learning strategies available to networks exposed to the different training procedures. The applicability of the findings to biological systems is discussed.

On the central limit theorem for dynamical systems

Etant donne un systeme dynamique aperiodique (X,T,μ), il y a un f∈L 2 (μ) avec ∫fdμ=0 satisfaisant le theoreme de la limite centrale

Strong laws for L- and U-statistics

Strong laws of large numbers are given for L-statistics (linear combinations of order statistics) and for U-statistics (averages of kernels of random samples) for ergodic stationary processes, extending classical theorems; of Hoeffding and of Helmers for lid sequences. Examples are given to show that strong and even weak convergence may fail if the given sufficient conditions are not satisfied, and an application is given to estimation of correlation dimension of invariant measures.

An invariance principle for weakly associated random vectors

The positive dependence notion of association for collections of random variables is generalized to that of weak association for collections of vector valued random elements in such a way as to allow negative dependencies in individual random elements. An invariance principle is stated and proven for a stationary, weakly associated sequence of d-valued or separable Hilbert space valued random elements which satisfy a covariance summability condition.

New results on measures of maximal entropy

It has recently been demonstrated that there are strongly irreducible subshifts of finite type with more than one measure of maximal entropy. Here we obtain a number of results concerning the uniqueness of the measure of maximal entropy. In addition, we construct for anyd≥2 andk a strongly irreducible subshift of finite type ind dimensions with exactlyk ergodic (extremal) measures of maximal entropy. Ford≥3, we construct a strongly irreducible subshift of finite type ind dimensions with a continuum of ergodic measures of maximal entropy.