Promotion carryover as a missing data problem

An important feature of agribusiness promotion programs is their lagged impact on consumption. Efficient investment in advertising requires reliable estimates of these lagged responses and it is desirable from both applied and theoretical standpoints to have a flexible method for estimating them. This note derives an alternative Bayesian methodology for estimating lagged responses when investments occur intermittently within a time series. The method exploits a latent-variable extension of the natural-conjugate, normal-linear model, Gibbs sampling and data augmentation. It is applied to a monthly time series on Turkish pasta consumption (1993:5-1998:3) and three, nonconsecutive promotion campaigns (1996:3, 1997:3, 1997:10). The results suggest that responses were greatest to the second campaign, which allocated its entire budget to television media; that its impact peaked in the sixth month following expenditure; and that the rate of return (measured in metric tons additional consumption per thousand dollars expended) was around a factor of 20.

Data assimilation for state and parameter estimation: application to morphodynamic modelling

Data assimilation is predominantly used for state estimation; combining observational data with model predictions to produce an updated model state that most accurately approximates the true system state whilst keeping the model parameters fixed. This updated model state is then used to initiate the next model forecast. Even with perfect initial data, inaccurate model parameters will lead to the growth of prediction errors. To generate reliable forecasts we need good estimates of both the current system state and the model parameters. This paper presents research into data assimilation methods for morphodynamic model state and parameter estimation. First, we focus on state estimation and describe implementation of a three dimensional variational(3D-Var) data assimilation scheme in a simple 2D morphodynamic model of Morecambe Bay, UK. The assimilation of observations of bathymetry derived from SAR satellite imagery and a ship-borne survey is shown to significantly improve the predictive capability of the model over a 2 year run. Here, the model parameters are set by manual calibration; this is laborious and is found to produce different parameter values depending on the type and coverage of the validation dataset. The second part of this paper considers the problem of model parameter estimation in more detail. We explain how, by employing the technique of state augmentation, it is possible to use data assimilation to estimate uncertain model parameters concurrently with the model state. This approach removes inefficiencies associated with manual calibration and enables more effective use of observational data. We outline the development of a novel hybrid sequential 3D-Var data assimilation algorithm for joint state-parameter estimation and demonstrate its efficacy using an idealised 1D sediment transport model. The results of this study are extremely positive and suggest that there is great potential for the use of data assimilation-based state-parameter estimation in coastal morphodynamic modelling.

Four-dimensional data assimilation

Numerical weather prediction can be regarded as an initial value problem whereby the governing atmospheric equations are integrated forward from fully determined initial values of the meteorological parameters. However, in spite of the considerable improvements of the observing systems in recent years, the initial values are known only incompletely and inaccurately and one of the major tasks of any forecasting centre is to determine the best possible initial state from available observations.

Analytical solutions of a minimal model of species migration in a bounded domain

A minimal model of species migration is presented which takes the form of a parabolic equation with boundary conditions and initial data. Solutions to the differential problem are obtained that can be used to describe the small- and large-time evolution of a species distribution within a bounded domain. These expressions are compared with the results of numerical simulations and are found to be satisfactory within appropriate temporal regimes. The solutions presented can be used to describe existing observations of nematode distributions, can be used as the basis for further work on nematode migration, and may also be interpreted more generally.

On the impact of humidity observations in numerical weather prediction

The impact of humidity observations on forecast skill is explored by producing a series of global forecasts using initial data derived from the ERA-40 reanalyses system, in which all humidity data have been removed during the data assimilation. The new forecasts have been compared with the original ERA-40 analyses and forecasts made from them. Both sets of forecasts show virtually identical prediction skill in the extratropics and the tropics. Differences between the forecasts are small and undergo characteristic amplification rate. There are larger differences in temperature and geopotential in the tropics but the differences are small-scale and unstructured and have no noticeable effect on the skill of the wind forecasts. The results highlight the current very limited impact of the humidity observations, used to produce the initial state, on the forecasts.

Small- and waiting-time behaviour of the thin-film-equation

We consider the small-time behavior of interfaces of zero contact angle solutions to the thin-film equation. For a certain class of initial data, through asymptotic analyses, we deduce a wide variety of behavior for the free boundary point. These are supported by extensive numerical simulations. © 2007 Society for Industrial and Applied Mathematics

The Klein-Gordon equation on the half line: a Riemann-Hilbert approach

We solve an initial-boundary problem for the Klein-Gordon equation on the half line using the Riemann-Hilbert approach to solving linear boundary value problems advocated by Fokas. The approach we present can be also used to solve more complicated boundary value problems for this equation, such as problems posed on time-dependent domains. Furthermore, it can be extended to treat integrable nonlinearisations of the Klein-Gordon equation. In this respect, we briefly discuss how our results could motivate a novel treatment of the sine-Gordon equation.

A simple theoretical model for the intensification of tropical cyclones and polar lows

A simple theoretical model for the intensification of tropical cyclones and polar lows is developed using a minimal set of physical assumptions. These disturbances are assumed to be balanced systems intensifying through the WISHE (Wind-Induced Surface Heat Exchange) intensification mechanism, driven by surface fluxes of heat and moisture into an atmosphere which is neutral to moist convection. The equation set is linearized about a resting basic state and solved as an initial-value problem. A system is predicted to intensify with an exponential perturbation growth rate scaled by the radial gradient of an efficiency parameter which crudely represents the effects of unsaturated processes. The form of this efficiency parameter is assumed to be defined by initial conditions, dependent on the nature of a pre-existing vortex required to precondition the atmosphere to a state in which the vortex can intensify. Evaluation of the simple model using a primitive-equation, nonlinear numerical model provides support for the prediction of exponential perturbation growth. Good agreement is found between the simple and numerical models for the sensitivities of the measured growth rate to various parameters, including surface roughness, the rate of transfer of heat and moisture from the ocean surface, and the scale for the growing vortex.

Virtual reality Post Traumatic Stress Disorder (PTSD) exposure therapy results with active duty Iraq war combatants

Post Traumatic Stress Disorder (PTSD) is reported to be caused by traumatic events that are outside the range of usual human experience including (but not limited to) military combat, violent personal assault, being kidnapped or taken hostage and terrorist attacks. Initial data suggests that at least 1 out of 6 Iraq War veterans are exhibiting symptoms of depression, anxiety and PTSD. Virtual Reality (VR) delivered exposure therapy for PTSD has been used with reports of positive outcomes. The aim of the current paper is to present the rationale and brief description of a Virtual Iraq PTSD VR therapy application and present initial findings from its use with PTSD patients. Thus far, Virtual Iraq consists of a series of customizable virtual scenarios designed to represent relevant Middle Eastern VR contexts for exposure therapy, including a city and desert road convoy environment. User-centered design feedback needed to iteratively evolve the system was gathered from returning Iraq War veterans in the USA and from a system deployed in Iraq and tested by an Army Combat Stress Control Team. Results from an open clinical trial at San Diego Naval Medical Center of the first 18 treatment completers indicate that 14 no longer meet PTSD diagnostic criteria at post-treatment, with only one not maintaining treatment gains at 3 month follow-up. Clinical tests are also currently underway at Ft. Lewis, Emory University, Weill Cornell Medical College, Walter Reed Army Medical Center and 10 other sites. Other sites are preparing to use the application for a variety of PTSD and VR research purposes.

Virtual reality Post Traumatic Stress Disorder (PTSD) exposure therapy results with active duty OIF/OEF service members

Posttraumatic stress disorder (PTSD) is reported to be caused by exposure to traumatic events including (but not limited to) military combat, violent personal assault, being kidnapped or taken hostage and terrorist attacks. Initial data suggest that at least 1 out of 6 Iraq War veterans are exhibiting symptoms of depression, anxiety and PTSD. Virtual reality (VR) delivered exposure therapy for PTSD has been used with reports of positive outcomes. The aim of the current paper, is to present the rationale and brief description of a Virtual Iraq/Afghanistan PTSD VR therapy application and present initial findings from its use with PTSD patients. Thus far, Virtual Iraq/Afghanistan consists of a series of customizable virtual scenarios designed to represent relevant Middle Eastern VR contexts for exposure therapy, including a city and desert road convoy environment. User-centered design feedback, needed to iteratively evolve the system, was gathered from returning Iraq War veterans in the USA and from a system deployed in Iraq and tested by an Army Combat Stress Control Team. Results from an open clinical trial at San Diego Naval Medical Center of the first 20 treatment completers indicate that 16 no longer met PTSD screening criteria at post-treatment, with only one not maintaining treatment gains at 3 month follow-up.

From short-range barotropic modelling to extended-range global weather prediction: a 40-year perspective

At the end of the 20th century, we can look back on a spectacular development of numerical weather prediction, which has, practically uninterrupted, been going on since the middle of the century. High-resolution predictions for more than a week ahead for any part of the globe are now routinely produced and anyone with an Internet connection can access many of these forecasts for anywhere in the world. Extended predictions for several seasons ahead are also being done — the latest El Niño event in 1997/1998 is an example of such a successful prediction. The great achievement is due to a number of factors including the progress in computational technology and the establishment of global observing systems, combined with a systematic research program with an overall strategy towards building comprehensive prediction systems for climate and weather. In this article, I will discuss the different evolutionary steps in this development and the way new scientific ideas have contributed to efficiently explore the computing power and in using observations from new types of observing systems. Weather prediction is not an exact science due to unavoidable errors in initial data and in the models. To quantify the reliability of a forecast is therefore essential and probably more so the longer the forecasts are. Ensemble prediction is thus a new and important concept in weather and climate prediction, which I believe will become a routine aspect of weather prediction in the future. The limit between weather and climate prediction is becoming more and more diffuse and in the final part of this article I will outline the way I think development may proceed in the future.

Instantaneous gelation in Smoluchowski’s coagulation equation revisited

We study the solutions of the Smoluchowski coagulation equation with a regularization term which removes clusters from the system when their mass exceeds a specified cutoff size, M. We focus primarily on collision kernels which would exhibit an instantaneous gelation transition in the absence of any regularization. Numerical simulations demonstrate that for such kernels with monodisperse initial data, the regularized gelation time decreasesas M increases, consistent with the expectation that the gelation time is zero in the unregularized system. This decrease appears to be a logarithmically slow function of M, indicating that instantaneously gelling kernels may still be justifiable as physical models despite the fact that they are highly singular in the absence of a cutoff. We also study the case when a source of monomers is introduced in the regularized system. In this case a stationary state is reached. We present a complete analytic description of this regularized stationary state for the model kernel, K(m1,m2)=max{m1,m2}ν, which gels instantaneously when M→∞ if ν>1. The stationary cluster size distribution decays as a stretched exponential for small cluster sizes and crosses over to a power law decay with exponent ν for large cluster sizes. The total particle density in the stationary state slowly vanishes as [(ν−1)logM]−1/2 when M→∞. The approach to the stationary state is nontrivial: Oscillations about the stationary state emerge from the interplay between the monomer injection and the cutoff, M, which decay very slowly when M is large. A quantitative analysis of these oscillations is provided for the addition model which describes the situation in which clusters can only grow by absorbing monomers.

On the structure of infinity-harmonic maps

Let H ∈ C 2(ℝ N×n ), H ≥ 0. The PDE system arises as the Euler-Lagrange PDE of vectorial variational problems for the functional E ∞(u, Ω) = ‖H(Du)‖ L ∞(Ω) defined on maps u: Ω ⊆ ℝ n → ℝ N . (1) first appeared in the author's recent work. The scalar case though has a long history initiated by Aronsson. Herein we study the solutions of (1) with emphasis on the case of n = 2 ≤ N with H the Euclidean norm on ℝ N×n , which we call the “∞-Laplacian”. By establishing a rigidity theorem for rank-one maps of independent interest, we analyse a phenomenon of separation of the solutions to phases with qualitatively different behaviour. As a corollary, we extend to N ≥ 2 the Aronsson-Evans-Yu theorem regarding non existence of zeros of |Du| and prove a maximum principle. We further characterise all H for which (1) is elliptic and also study the initial value problem for the ODE system arising for n = 1 but with H(·, u, u′) depending on all the arguments.