Bayesian hierarchical models for analysing spatial point-based data at a grid level: A comparison of approaches

Spatial data are now prevalent in a wide range of fields including environmental and health science. This has led to the development of a range of approaches for analysing patterns in these data. In this paper, we compare several Bayesian hierarchical models for analysing point-based data based on the discretization of the study region, resulting in grid-based spatial data. The approaches considered include two parametric models and a semiparametric model. We highlight the methodology and computation for each approach. Two simulation studies are undertaken to compare the performance of these models for various structures of simulated point-based data which resemble environmental data. A case study of a real dataset is also conducted to demonstrate a practical application of the modelling approaches. Goodness-of-fit statistics are computed to compare estimates of the intensity functions. The deviance information criterion is also considered as an alternative model evaluation criterion. The results suggest that the adaptive Gaussian Markov random field model performs well for highly sparse point-based data where there are large variations or clustering across the space; whereas the discretized log Gaussian Cox process produces good fit in dense and clustered point-based data. One should generally consider the nature and structure of the point-based data in order to choose the appropriate method in modelling a discretized spatial point-based data.

Assessing uncertainty in pollutant wash-off modelling via model validation

Stormwater pollution is linked to stream ecosystem degradation. In predicting stormwater pollution, various types of modelling techniques are adopted. The accuracy of predictions provided by these models depends on the data quality, appropriate estimation of model parameters, and the validation undertaken. It is well understood that available water quality datasets in urban areas span only relatively short time scales unlike water quantity data, which limits the applicability of the developed models in engineering and ecological assessment of urban waterways. This paper presents the application of leave-one-out (LOO) and Monte Carlo cross validation (MCCV) procedures in a Monte Carlo framework for the validation and estimation of uncertainty associated with pollutant wash-off when models are developed using a limited dataset. It was found that the application of MCCV is likely to result in a more realistic measure of model coefficients than LOO. Most importantly, MCCV and LOO were found to be effective in model validation when dealing with a small sample size which hinders detailed model validation and can undermine the effectiveness of stormwater quality management strategies.

Dosimetry of cone-defined stereotactic radiosurgery fields with a commercial synthetic diamond detector

Purpose Small field x-ray beam dosimetry is difficult due to a lack of lateral electronic equilibrium, source occlusion, high dose gradients and detector volume averaging. Currently there is no single definitive detector recommended for small field dosimetry. The objective of this work was to evaluate the performance of a new commercial synthetic diamond detector, namely the PTW 60019 microDiamond, for the dosimetry of small x-ray fields as used in stereotactic radiosurgery (SRS). Methods Small field sizes were defined by BrainLAB circular cones (4 – 30 mm diameter) on a Novalis Trilogy linear accelerator and using the 6 MV SRS x-ray beam mode for all measurements. Percentage depth doses were measured and compared to an IBA SFD and a PTW 60012 E diode. Cross profiles were measured and compared to an IBA SFD diode. Field factors, Ω_(Q_clin,Q_msr)^(f_clin,f_msr ), were calculated by Monte Carlo methods using BEAMnrc and correction factors, k_(Q_clin,Q_msr)^(f_clin,f_msr ), were derived for the PTW 60019 microDiamond detector. Results For the small fields of 4 to 30 mm diameter, there were dose differences in the PDDs of up to 1.5% when compared to an IBA SFD and PTW 60012 E diode detector. For the cross profile measurements the penumbra values varied, depending upon the orientation of the detector. The field factors, Ω_(Q_clin,Q_msr)^(f_clin,f_msr ), were calculated for these field diameters at a depth of 1.4 cm in water and they were within 2.7% of published values for a similar linear accelerator. The corrections factors, k_(Q_clin,Q_msr)^(f_clin,f_msr ), were derived for the PTW 60019 microDiamond detector. Conclusions We conclude that the new PTW 60019 microDiamond detector is generally suitable for relative dosimetry in small 6 MV SRS beams for a Novalis Trilogy linear equipped with circular cones.

An experimental approach to air gap optimisation for a correction-less small field diode

Aim A recent Monte Carlo based study has shown that it is possible to design a diode that measures small field output factors equivalent to that in water. This is accomplished by placing an appropriate sized air gap above the silicon chip (1) with experimental results subsequently confirming that a particular Monte Carlo design was accurate (2). The aim of this work was to test if a new correction-less diode could be designed using an entirely experimental methodology. Method: All measurements were performed on a Varian iX at a depth of 5 cm, SSD of 95 cm and field sizes of 5, 6, 8, 10, 20 and 30 mm. Firstly, the experimental transfer of kq,clin,kq,msr from a commonly used diode detector (IBA, stereotactic field diode (SFD)) to another diode detector (Sun Nuclear, unshielded diode, (EDGEe)) was tested. These results were compared to Monte Carlo calculated values of the EDGEe. Secondly, the air gap above the EDGEe silicon chip was optimised empirically. Nine different air gap “tops” were placed above the EDGEe (air depth = 0.3, 0.6, 0.9 mm; air width = 3.06, 4.59, 6.13 mm). The sensitivity of the EDGEe was plotted as a function of air gap thickness for the field sizes measured. Results: The transfer of kq,clin,kq,msr from the SFD to the EDGEe was correct to within the simulation and measurement uncertainties. The EDGEe detector can be made “correction-less” for field sizes of 5 and 6 mm, but was ∼2% from being “correction-less” at field sizes of 8 and 10 mm. Conclusion Different materials will perturb small fields in different ways. A detector is only “correction-less” if all these perturbations happen to cancel out. Designing a “correction-less” diode is a complicated process, thus it is reasonable to expect that Monte Carlo simulations should play an important role.

The use of penumbrae to define and analyse non-standard fields

Aim A new method of penumbral analysis is implemented which allows an unambiguous determination of field size and penumbra size and quality for small fields and other non-standard fields. Both source occlusion and lateral electronic disequilibrium will affect the size and shape of cross-axis profile penumbrae; each is examined in detail. Method A new method of penumbral analysis is implemented where the square of the derivative of the cross-axis profile is plotted. The resultant graph displays two peaks in the place of the two penumbrae. This allows a strong visualisation of the quality of a field penumbra, as well as a mathematically consistent method of determining field size (distance between the two peak’s maxima), and penumbra (full-widthtenth-maximum of peak). Cross-axis profiles were simulated in a water phantom at a depth of 5 cm using Monte Carlo modelling, for field sizes between 5 and 30 mm. The field size and penumbra size of each field was calculated using the method above, as well as traditional definitions set out in IEC976. The effect of source occlusion and lateral electronic disequilibrium on the penumbrae was isolated by repeating the simulations removing electron transport and using an electron spot size of 0 mm, respectively. Results All field sizes calculated using the traditional and proposed methods agreed within 0.2 mm. The penumbra size measured using the proposed method was systematically 1.8 mm larger than the traditional method at all field sizes. The size of the source had a larger effect on the size of the penumbra than did lateral electronic disequilibrium, particularly at very small field sizes. Conclusion Traditional methods of calculating field size and penumbra are proved to be mathematically adequate for small fields. However, the field size definition proposed in this study would be more robust amongst other nonstandard fields, such as flattening filter free. Source occlusion plays a bigger role than lateral electronic disequilibrium in small field penumbra size.

Approximate Bayesian Computation for astronomical model analysis : a case study in galaxy demographics and morphological transformation at high redshift

Approximate Bayesian Computation’ (ABC) represents a powerful methodology for the analysis of complex stochastic systems for which the likelihood of the observed data under an arbitrary set of input parameters may be entirely intractable – the latter condition rendering useless the standard machinery of tractable likelihood-based, Bayesian statistical inference [e.g. conventional Markov chain Monte Carlo (MCMC) simulation]. In this paper, we demonstrate the potential of ABC for astronomical model analysis by application to a case study in the morphological transformation of high-redshift galaxies. To this end, we develop, first, a stochastic model for the competing processes of merging and secular evolution in the early Universe, and secondly, through an ABC-based comparison against the observed demographics of massive (Mgal > 1011 M⊙) galaxies (at 1.5 < z < 3) in the Cosmic Assembly Near-IR Deep Extragalatic Legacy Survey (CANDELS)/Extended Groth Strip (EGS) data set we derive posterior probability densities for the key parameters of this model. The ‘Sequential Monte Carlo’ implementation of ABC exhibited herein, featuring both a self-generating target sequence and self-refining MCMC kernel, is amongst the most efficient of contemporary approaches to this important statistical algorithm. We highlight as well through our chosen case study the value of careful summary statistic selection, and demonstrate two modern strategies for assessment and optimization in this regard. Ultimately, our ABC analysis of the high-redshift morphological mix returns tight constraints on the evolving merger rate in the early Universe and favours major merging (with disc survival or rapid reformation) over secular evolution as the mechanism most responsible for building up the first generation of bulges in early-type discs.