Lacunarity of the zero crossings of Gaussian processes.

A lacunarity analysis of the zero crossings derived from Gaussian stochastic processes with oscillatory autocorrelation functions is evaluated and reveals distinct multiscaling signatures depending on the smoothness and degree of anticorrelation of the process. These bear qualitative similarities and quantitative distinctions from an oscillatory deterministic signal and a Poisson random process both possessing the same mean interval size between crossings. At very small and large scales compared with the correlation length of the random processes, the lacunarity is similar to the Poisson but exhibits significant departures from Poisson behavior if there is a zero-frequency component to the process's power spectrum. A comparison of exact results with the gliding box technique that is frequently used to determine lacunarity demonstrates its inherent bias.

Quantifying age and model uncertainties in palaeoclimate data and dynamical climate models with a joint inferential analysis.

The study of palaeoclimates relies on information sampled in natural archives such as deep sea cores. Scientific investigations often use such information in multi-stage analyses, typically with an age model being fitted to a core to convert depths into ages at stage one. These age estimates are then used as inputs to develop, calibrate or select climate models in a second stage of analysis. Here, we show that such multi-stage approaches can lead to misleading conclusions, and develop a joint inferential approach for climate reconstruction, model calibration and age estimation. As an illustration, we investigate the glacial-interglacial cycle, fitting both an age model and dynamical climate model to two benthic sediment cores spanning the past 780 kyr. To show the danger of a multi-stage analysis, we sample ages from the posterior distribution, then perform model selection conditional on the sampled age estimates, mimicking standard practice. Doing so repeatedly for different samples leads to model selection conclusions that are substantially different from each other, and from the joint inferential analysis. We conclude that multi-stage analyses are insufficient when dealing with uncertainty, and that to draw sound conclusions the full joint inferential analysis should be performed.

Piecewise Approximate Bayesian Computation: fast inference for discretely observed Markov models using a factorised posterior distribution.

Many modern statistical applications involve inference for complicated stochastic models for which the likelihood function is difficult or even impossible to calculate, and hence conventional likelihood-based inferential techniques cannot be used. In such settings, Bayesian inference can be performed using Approximate Bayesian Computation (ABC). However, in spite of many recent developments to ABC methodology, in many applications the computational cost of ABC necessitates the choice of summary statistics and tolerances that can potentially severely bias the estimate of the posterior. We propose a new "piecewise" ABC approach suitable for discretely observed Markov models that involves writing the posterior density of the parameters as a product of factors, each a function of only a subset of the data, and then using ABC within each factor. The approach has the advantage of side-stepping the need to choose a summary statistic and it enables a stringent tolerance to be set, making the posterior "less approximate". We investigate two methods for estimating the posterior density based on ABC samples for each of the factors: the first is to use a Gaussian approximation for each factor, and the second is to use a kernel density estimate. Both methods have their merits. The Gaussian approximation is simple, fast, and probably adequate for many applications. On the other hand, using instead a kernel density estimate has the benefit of consistently estimating the true piecewise ABC posterior as the number of ABC samples tends to infinity. We illustrate the piecewise ABC approach with four examples; in each case, the approach offers fast and accurate inference.

T-cell motility in the early stages of the immune response modeled as a random walk amongst targets.

The transport process by which a T cell makes high-frequency encounters with antigen-presenting cells following infection is an important element of adaptive immunity. Recent experimental work has allowed in vivo cell motility to be characterized in detail. On the basis of experimental data we develop a quantitative model for encounters between T cells and antigen-presenting cells. We model this as a transport-limited chemical reaction with the dynamics dependent on physical contact between randomly moving reactants. We use asymptotic methods to calculate a time distribution which characterizes the delay before a T cell is activated and use Monte Carlo simulations to verify the analysis. We find that the density of antigen-primed dendritic cells within the lymph node paracortex must be greater than 35 cells/mm3 for a T cell to have a more than 50% chance of encountering a dendritic cell within 24 h. This density is much larger than existing estimates based on calculations which neglect the transport process. We also use simulations to compare a T cell which re-orients isotropically with a T cell which turns according to an experimentally observed distribution and find that the effects of anisotropy on the solution are small.