Pak Hang Lo

• A popular model for the locations of fibres or grains in composite materials is the inhomogeneous Poisson process in dimension 3. Its local intensity function may be estimated non-parametrically by local smoothing, e.g. by kernel estimates. They crucially depend on the choice of bandwidths as tuning parameters controlling the smoothness of the resulting function estimate. In this thesis, we propose a fast algorithm for learning suitable global and local bandwidths from the data. It is well-known, that intensity estimation is closely related to probability density estimation. As a by-product of our study, we show that the difference is asymptotically negligible regarding the choice of good bandwidths, and, hence, we focus on density estimation. There are quite a number of data-driven bandwidth selection methods for kernel density estimates. cross-validation is a popular one and frequently proposed to estimate the optimal bandwidth. However, if the sample size is very large, it becomes computational expensive. In material science, in particular, it is very common to have several thousand up to several million points. Another type of bandwidth selection is a solve-the-equation plug-in approach which involves replacing the unknown quantities in the asymptotically optimal bandwidth formula by their estimates. In this thesis, we develop such an iterative fast plug-in algorithm for estimating the optimal global and local bandwidth for density and intensity estimation with a focus on 2- and 3-dimensional data. It is based on a detailed asymptotics of the estimators of the intensity function and of its second derivatives and integrals of second derivatives which appear in the formulae for asymptotically optimal bandwidths. These asymptotics are utilised to determine the exact number of iteration steps and some tuning parameters. For both global and local case, fewer than 10 iterations suffice. Simulation studies show that the estimated intensity by local bandwidth can better indicate the variation of local intensity than that by global bandwidth. Finally, the algorithm is applied to two real data sets from test bodies of fibre-reinforced high-performance concrete, clearly showing some inhomogeneity of the fibre intensity.