As a powerful tool for the representation of classifying knowledge, fuzzy classification rules can not only effectively deal with imprecise and uncertain data, but also possess readability and interpretability. Fuzzy granular rules, also to be a kind of fuzzy classification rules, can be induced by fuzzy information granules. It has been acknowledged that one of the important criteria for evaluating the quality of information granules comes from the principle of justifiable granularity. Unfortunately, the existing methods for extracting fuzzy granular rules fail to take into account the principle of justifiable granularity. In view of the advantages of the justifiable granularity principle in classifying knowledge, we propose in this paper a new method of extracting fuzzy granular rules using the justifiable granularity principle and a fusion strategy and establish a fuzzy granular rule-based classifier (FGRBC). Specifically, the justifiability of fuzzy granules is first presented according to both coverage and specificity of fuzzy granules, on which a rule extraction method is formulated to obtain a set of fuzzy granular rules. Furthermore, a fusion strategy is put forward to generate a set of fused rules. Then, the two sets of rules are combined and attribute reduction is performed on the combined rule set. Finally, the reduced combined rule set is employed to construct FGRBC. Moreover, performance of FGRBC is evaluated by numerical experiments and the results show that FGRBC is of satisfactory classification ability.

Multiscale estimation for geographically weighted regression (GWR) and the related models has attracted much attention due to their superiority. This kind of estimation method will not only improve the accuracy of the coefficient estimators but also reveal the underlying spatial scale of each explanatory variable. However, most of the existing multiscale estimation approaches are backfitting-based iterative procedures that are very time-consuming. To alleviate the computation complexity, we propose in this paper a non-iterative multiscale estimation method and its simplified scenario for spatial autoregressive geographically weighted regression (SARGWR) models, a kind of important GWR-related model that simultaneously takes into account spatial autocorrelation in the response variable and spatial heterogeneity in the regression relationship. In the proposed multiscale estimation methods, the two-stage least-squares (2SLS) based GWR and the local-linear GWR estimators of the regression coefficients with a shrunk bandwidth size are respectively taken to be the initial estimators to obtain the final multiscale estimators of the coefficients without iteration. A simulation study is conducted to assess the performance of the proposed multiscale estimation methods, and the results show that the proposed methods are much more efficient than the backfitting-based estimation procedure. In addition, the proposed methods can also yield accurate coefficient estimators and such variable-specific optimal bandwidth sizes that correctly reflect the underlying spatial scales of the explanatory variables. A real-life example is further provided to demonstrate the applicability of the proposed multiscale estimation methods.

Using local spatial statistics to explore local spatial association of geo-referenced data has attracted much attention. As is known, a local statistic is formulated at a particular sampling unit based on a prespecific proximity relationship and the observations in the neighborhood of this sampling unit. However, geostatistical data such as meteorological data and air pollution data are generally collected from meteorological or monitoring stations which are usually sparsely located or highly clustered over space. For such data, a local spatial statistic formulated at an isolate sampling point may be ineffective because of its distant neighbors, or the statistic is undefinable in the sub-regions where no observations are available, which limits the comprehensive exploration of local spatial association over the whole studied region. In order to overcome the predicament, a local-linear geographically weighted interpolation method is proposed in this paper to obtain the predictors of the underlying spatial process on a lattice spatial tessellation, on which a local spatial statistic can be well formulated at each interpolation point. Furthermore, the bootstrap test is suggested to identify the locations where local spatial association is significant using the interpolated-value-based local spatial statistics. Simulation with comparison to some existing interpolation and test methods is conducted to assess the performance of the proposed interpolation and the suggested test methods and a case study based on PM2.5 concentration data in Guangdong province, China, is used to demonstrate their applicability. The results show that the proposed interpolation method performs accurately in retrieving an underlying spatial process and the bootstrap test with the interpolated-value-based local statistics is powerful in identifying local patterns of spatial association.

Geographically and temporally weighted regression (GTWR) models have been widely used to explore spatiotemporal nonstationarity where all the regression coefficients are assumed to be varying over both space and time. In reality, however, constant, only temporally varying, and only spatially varying coefficients might also be possible depending on the underlying effects of the explanatory variables on the response variable. Therefore, the development of inference and estimation methods for such special types of the coefficients is essential to the deep understanding of spatiotemporal characteristics of the regression relationship. In this article, an average-based approach, relying on a modified estimation of the conventional GTWR models, is proposed to calibrate the GTWR models with the special types of the coefficients, on which a statistical test is formulated to simultaneously infer constant, temporally varying, and spatially varying coefficients. The simulation study shows that the test method is of valid Type I error and satisfactory power and the average-based estimation method yields more accurate estimators for the special types of the coefficients. A real-life example based on Beijing house prices is given to demonstrate the applicability of the test and estimation methods as well as the extensibility of the test in model selection.