This paper explores the homogeneity of coefficient functions in nonlinear models with functional coefficients and identifies the underlying semiparametric modelling structure. With initial kernel estimates, we combine the classic hierarchical clustering method with a generalised version of the information criterion to estimate the number of clusters, each of which has a common functional coefficient, and determine the membership of each cluster. To identify a possible semi-varying coefficient modelling framework, we further introduce a penalised local least squares method to determine zero coefficients, non-zero constant coefficients and functional coefficients which vary with an index variable. Through the nonparametric kernel-based cluster analysis and the penalised approach, we can substantially reduce the number of unknown parametric and nonparametric components in the models, thereby achieving the aim of dimension reduction. Under some regularity conditions, we establish the asymptotic properties for the proposed methods including the consistency of the homogeneity pursuit. Numerical studies, including Monte-Carlo experiments and two empirical applications, are given to demonstrate the finite-sample performance of our methods.
Maps obtained from various sensors or surveying instruments provide important geographic information for users. Buildings are critical components of various city-area maps. In this paper, we present a constrained building boundary simplification method based on the partial total least squares (PTLS) method. Simplification of building boundaries may cause some data quality problems, such as geometric displacements, right angles changed into non-right angles, and area inconsistency at different scales. Therefore, in this paper, a linear fitting model based on the PTLS method is constructed for building boundaries with the aim of reducing the total positional difference. Furthermore, end-point, right-angle, and area constraints are constructed at the same time to maintain the right angles and areas of the building. The simplification results are obtained by solving the proposed constrained model by using the PTLS method with constraints. The proposed method was applied in a building boundary simplification experiment. The results showed that the proposed method maintains the right angles and areas of buildings and reduces the geometric displacements of buildings after the simplification.
The classical composite midpoint rectangle rule for computing Cauchy principal value integrals on an interval is studied. By using a piecewise constant interpolant to approximate the density function, an extended error expansion and its corresponding superconvergence results are obtained. The superconvergence phenomenon shows that the convergence rate of the midpoint rectangle rule is higher than that of the general Riemann integral when the singular point coincides with some priori known points. Finally, several numerical examples are presented to demonstrate the accuracy and effectiveness of the theoretical analysis. This research is meaningful to improve the accuracy of the collocation method for singular integrals.
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