The Limits of Mathematics

This condensed version of chao-dyn/9509010 will be the main hand-out for a course on algorithmic information theory to be given 22-29 May 1996 at the Rovaniemi Institute of Technology, Rovaniemi, Finland (see announcement at http://www.rotol.fi/ ).Comment: LaTeX, 45 page

Analytic modeling of aerosol size distributions

Mathematical functions commonly used for representing aerosol size distributions are studied parametrically. Methods for obtaining best fit estimates of the parameters are described. A catalog of graphical plots depicting the parametric behavior of the functions is presented along with procedures for obtaining analytical representations of size distribution data by visual matching of the data with one of the plots. Examples of fitting the same data with equal accuracy by more than one analytic model are also given

Signature of a universal statistical description for drift-wave plasma turbulence

This Letter provides a theoretical interpretation of numerically generated probability density functions (PDFs) of intermittent plasma transport events. Specifically, nonlinear gyrokinetic simulations of ion-temperature-gradient turbulence produce time series of heat flux which exhibit manifestly non-Gaussian PDFs with enhanced tails. It is demonstrated that, after the removal of autocorrelations, the numerical PDFs can be matched with predictions from a fluid theoretical setup, based on the instanton method. This result points to a universality in the modeling of intermittent stochastic process, offering predictive capability.Comment: 4 pages, 5 figure

Determination of the Joint Confidence Region of Optimal Operating Conditions in Robust Design by Bootstrap Technique

Robust design has been widely recognized as a leading method in reducing variability and improving quality. Most of the engineering statistics literature mainly focuses on finding "point estimates" of the optimum operating conditions for robust design. Various procedures for calculating point estimates of the optimum operating conditions are considered. Although this point estimation procedure is important for continuous quality improvement, the immediate question is "how accurate are these optimum operating conditions?" The answer for this is to consider interval estimation for a single variable or joint confidence regions for multiple variables. In this paper, with the help of the bootstrap technique, we develop procedures for obtaining joint "confidence regions" for the optimum operating conditions. Two different procedures using Bonferroni and multivariate normal approximation are introduced. The proposed methods are illustrated and substantiated using a numerical example.Comment: Two tables, Three figure

A Bayesian spatio-temporal model of panel design data: airborne particle number concentration in Brisbane, Australia

This paper outlines a methodology for semi-parametric spatio-temporal modelling of data which is dense in time but sparse in space, obtained from a split panel design, the most feasible approach to covering space and time with limited equipment. The data are hourly averaged particle number concentration (PNC) and were collected, as part of the Ultrafine Particles from Transport Emissions and Child Health (UPTECH) project. Two weeks of continuous measurements were taken at each of a number of government primary schools in the Brisbane Metropolitan Area. The monitoring equipment was taken to each school sequentially. The school data are augmented by data from long term monitoring stations at three locations in Brisbane, Australia. Fitting the model helps describe the spatial and temporal variability at a subset of the UPTECH schools and the long-term monitoring sites. The temporal variation is modelled hierarchically with penalised random walk terms, one common to all sites and a term accounting for the remaining temporal trend at each site. Parameter estimates and their uncertainty are computed in a computationally efficient approximate Bayesian inference environment, R-INLA. The temporal part of the model explains daily and weekly cycles in PNC at the schools, which can be used to estimate the exposure of school children to ultrafine particles (UFPs) emitted by vehicles. At each school and long-term monitoring site, peaks in PNC can be attributed to the morning and afternoon rush hour traffic and new particle formation events. The spatial component of the model describes the school to school variation in mean PNC at each school and within each school ground. It is shown how the spatial model can be expanded to identify spatial patterns at the city scale with the inclusion of more spatial locations.Comment: Draft of this paper presented at ISBA 2012 as poster, part of UPTECH projec

Synthetic LISA: Simulating Time Delay Interferometry in a Model LISA

We report on three numerical experiments on the implementation of Time-Delay Interferometry (TDI) for LISA, performed with Synthetic LISA, a C++/Python package that we developed to simulate the LISA science process at the level of scientific and technical requirements. Specifically, we study the laser-noise residuals left by first-generation TDI when the LISA armlengths have a realistic time dependence; we characterize the armlength-measurements accuracies that are needed to have effective laser-noise cancellation in both first- and second-generation TDI; and we estimate the quantization and telemetry bitdepth needed for the phase measurements. Synthetic LISA generates synthetic time series of the LISA fundamental noises, as filtered through all the TDI observables; it also provides a streamlined module to compute the TDI responses to gravitational waves according to a full model of TDI, including the motion of the LISA array and the temporal and directional dependence of the armlengths. We discuss the theoretical model that underlies the simulation, its implementation, and its use in future investigations on system characterization and data-analysis prototyping for LISA.Comment: 18 pages, 14 EPS figures, REVTeX 4. Accepted PRD version. See http://www.vallis.org/syntheticlisa for information on the Synthetic LISA software packag

Don't know, can't know: Embracing deeper uncertainties when analysing risks

This article is available open access through the publisher’s website at the link below. Copyright @ 2011 The Royal Society.Numerous types of uncertainty arise when using formal models in the analysis of risks. Uncertainty is best seen as a relation, allowing a clear separation of the object, source and ‘owner’ of the uncertainty, and we argue that all expressions of uncertainty are constructed from judgements based on possibly inadequate assumptions, and are therefore contingent. We consider a five-level structure for assessing and communicating uncertainties, distinguishing three within-model levels—event, parameter and model uncertainty—and two extra-model levels concerning acknowledged and unknown inadequacies in the modelling process, including possible disagreements about the framing of the problem. We consider the forms of expression of uncertainty within the five levels, providing numerous examples of the way in which inadequacies in understanding are handled, and examining criticisms of the attempts taken by the Intergovernmental Panel on Climate Change to separate the likelihood of events from the confidence in the science. Expressing our confidence in the adequacy of the modelling process requires an assessment of the quality of the underlying evidence, and we draw on a scale that is widely used within evidence-based medicine. We conclude that the contingent nature of risk-modelling needs to be explicitly acknowledged in advice given to policy-makers, and that unconditional expressions of uncertainty remain an aspiration

Six Peaks Visible in the Redshift Distribution of 46,400 SDSS Quasars Agree with the Preferred Redshifts Predicted by the Decreasing Intrinsic Redshift Model

The redshift distribution of all 46,400 quasars in the Sloan Digital Sky Survey (SDSS) Quasar Catalog III, Third Data Release, is examined. Six Peaks that fall within the redshift window below z = 4, are visible. Their positions agree with the preferred redshift values predicted by the decreasing intrinsic redshift (DIR) model, even though this model was derived using completely independent evidence. A power spectrum analysis of the full dataset confirms the presence of a single, significant power peak at the expected redshift period. Power peaks with the predicted period are also obtained when the upper and lower halves of the redshift distribution are examined separately. The periodicity detected is in linear z, as opposed to log(1+z). Because the peaks in the SDSS quasar redshift distribution agree well with the preferred redshifts predicted by the intrinsic redshift relation, we conclude that this relation, and the peaks in the redshift distribution, likely both have the same origin, and this may be intrinsic redshifts, or a common selection effect. However, because of the way the intrinsic redshift relation was determined it seems unlikely that one selection effect could have been responsible for both.Comment: 12 pages, 12 figure, accepted for publication in the Astrophysical Journa

Application of Bayesian model averaging to measurements of the primordial power spectrum

Cosmological parameter uncertainties are often stated assuming a particular model, neglecting the model uncertainty, even when Bayesian model selection is unable to identify a conclusive best model. Bayesian model averaging is a method for assessing parameter uncertainties in situations where there is also uncertainty in the underlying model. We apply model averaging to the estimation of the parameters associated with the primordial power spectra of curvature and tensor perturbations. We use CosmoNest and MultiNest to compute the model Evidences and posteriors, using cosmic microwave data from WMAP, ACBAR, BOOMERanG and CBI, plus large-scale structure data from the SDSS DR7. We find that the model-averaged 95% credible interval for the spectral index using all of the data is 0.940 < n_s < 1.000, where n_s is specified at a pivot scale 0.015 Mpc^{-1}. For the tensors model averaging can tighten the credible upper limit, depending on prior assumptions.Comment: 7 pages with 7 figures include