Age and period are the most widely used parameters for forecasting mortality rates. Empirical mortality rate differences in multiple populations often show strong geometric patterns on the two-dimensional age–period plane. The idea of this paper is to take these geometric patterns as the starting point for the development of forecasts. A parametric approach is presented and discussed which uses simple techniques from spatial statistics. The proposed model is statistically parsimonious and yields forecasts that are consistent with the historical data and coherent for multiple populations.

A stochastic approach to conditional simulation of flow in randomly heterogeneous media is proposed with the combination of the Karhunen-Loeve expansion and the probabilistic collocation method (PCM). The conditional log hydraulic conductivity field is represented with the Karhunen-Loeve expansion, in terms of some deterministic functions and a set of independent Gaussian random variables. The propagation of uncertainty in the flow simulations is carried out through the PCM, which relies on the efficient polynomial chaos expansion used to represent the flow responses such as the hydraulic head. With the PCM, existing flow simulators can be employed for uncertainty quantification of flow in heterogeneous porous media when direct measurements of hydraulic conductivity are taken into consideration. With illustration of several numerical examples of groundwater flow, this study reveals that the proposed approach is able to accurately quantify uncertainty of the flow responses conditioning on hydraulic conductivity data, while the computational efforts are significantly reduced in comparison to the Monte Carlo simulations.

We introduce a class of Cox cluster processes called generalised shot noise Cox processes (GSNCPs), which extends the definition of shot noise Cox processes (SNCPs) in two directions: the point process that drives the shot noise is not necessarily Poisson, and the kernel of the shot noise can be random. Thereby, a very large class of models for aggregated or clustered point patterns is obtained. Due to the structure of GSNCPs, a number of useful results can be established. We focus first on deriving summary statistics for GSNCPs and, second, on how to simulate such processes. In particular, results on first- and second-order moment measures, reduced Palm distributions, the J-function, simulation with or without edge effects, and conditional simulation of the intensity function driving a GSNCP are given. Our results are exemplified in important special cases of GSNCPs, and we discuss their relation to the corresponding results for SNCPs.

Shot noise Cox processes constitute a large class of Cox and Poisson cluster processes in ℝd, including Neyman-Scott, Poisson-gamma and shot noise G Cox processes. It is demonstrated that, due to the structure of such models, a number of useful and general results can easily be established. The focus is on the probabilistic aspects with a view to statistical applications, particularly results for summary statistics, reduced Palm distributions, simulation with or without edge effects, conditional simulation of the intensity function and local and spatial Markov properties.

Geostatistical conditional simulations are used to get a family of non-smoothed three-dimensional (3D) images of the St. Lawrence krill aggregation from echointegration data at 38 and 120 kHz on a systematic grid of transects. These maps respect the inherent patchiness of krill. Their ensemble gives an histogram of the krill density estimates at any point of the 3D grid, not only the mean density as in kriging. The spatially consistent simulations are conditional to match the observations and their histogram and variogram. The 3D problem is by-passed by transposing to a short series of 2D ones, which simplifies the modelling of the autocorrelation function and has the additional advantage of reducing by a factor of 5 the number of points to simulate. The spatially structured krill density profile is summarised by the first few factors of a principal component analysis, where the variables are the volume backscattering strength at 120 kHz for the integrated vertical bins along the profiles and the observations are the different profiles. The principal component scores are simulated over a 2D-grid and then used to reconstruct the full 3D-image. The method adequately reproduces the histogram, variogram and mean profile of krill density, and cross-validations replicate the observations. It is as precise as an alternative kriging approach to estimate the mean density, but has the additional advantages of requiring less computing time for our particular application. These conditional simulations enable estimating the probability density function of the krill density at any point, an essential information for predator-prey interactions and other ecosystem studies.