Changlin Mei

Statistical Tests for Spatial Nonstationarity Based on the Geographically Weighted Regression Model

Geographically weighted regression (GWR) is a way of exploring spatial nonstationarity by calibrating a multiple regression model which allows different relationships to exist at different points in space. Nevertheless, formal testing procedures for spatial nonstationarity have not been developed since the inception of the model. In this paper the authors focus mainly on the development of statistical testing methods relating to this model. Some appropriate statistics for testing the goodness of fit of the GWR model and for testing variation of the parameters in the model are proposed and their approximated distributions are investigated. The work makes it possible to test spatial nonstationarity in a conventional statistical manner. To substantiate the theoretical arguments, some simulations are run to examine the power of the statistics for exploring spatial nonstationarity and the results are encouraging. To streamline the model, a stepwise procedure for choosing important independent variables is also formulated. In the last section, a prediction problem based on the GWR model is studied, and a confidence interval for the true value of the dependent variable at a new location is also established. The study paves the path for formal analysis of spatial nonstationarity on the basis of the GWR model.

Incomplete decision contexts: Approximate concept construction, rule acquisition and knowledge reduction

Incomplete decision contexts are a kind of decision formal contexts in which information about the relationship between some objects and attributes is not available or is lost. Knowledge discovery in incomplete decision contexts is of interest because such databases are frequently encountered in the real world. This paper mainly focuses on the issues of approximate concept construction, rule acquisition and knowledge reduction in incomplete decision contexts. We propose a novel method for building the approximate concept lattice of an incomplete context. Then, we present the notion of an approximate decision rule and an approach for extracting non-redundant approximate decision rules from an incomplete decision context. Furthermore, in order to make the rule acquisition easier and the extracted approximate decision rules more compact, a knowledge reduction framework with a reduction procedure for incomplete decision contexts is formulated by constructing a discernibility matrix and its associated Boolean function. Finally, some numerical experiments are conducted to assess the efficiency of the proposed method.

Concept learning via granular computing

n Abstractn n Concepts are the most fundamental units of cognition in philosophy and how to learn concepts from various aspects in the real world is the main concern within the domain of conceptual knowledge presentation and processing. In order to improve efficiency and flexibility of concept learning, in this paper we discuss concept learning via granular computing from the point of view of cognitive computing. More precisely, cognitive mechanism of forming concepts is analyzed based on the principles from philosophy and cognitive psychology, including how to model concept-forming cognitive operators, define cognitive concepts and establish cognitive concept structure. Granular computing is then combined with the cognitive concept structure to improve efficiency of concept learning. Furthermore, we put forward a cognitive computing system which is the initial environment to learn composite concepts and can integrate past experiences into itself for enhancing flexibility of concept learning. Also, we investigate cognitive processes whose aims are to deal with the problem of learning one exact or two approximate cognitive concepts from a given object set, attribute set or pair of object and attribute sets.n n

Knowledge reduction in real decision formal contexts

This study deals with the problem of knowledge reduction in decision formal contexts. From the perspective of rule acquisition, a new framework of knowledge reduction for decision formal contexts is formulated and a corresponding reduction method is also developed by using the discernibility matrix and Boolean function. The presented framework of knowledge reduction is for general decision formal contexts, and based on the proposed reduction method, knowledge hidden in a decision formal context can compactly be unravelled in the form of implication rules.

Testing for spatial autocorrelation among the residuals of the geographically weighted regression

Geographically weighted regression (GWR) is a useful technique for exploring spatial nonstationarity by calibrating, for example, a regression model which allows different relationships to exist at different points in space. In this line of research, many spatial data sets have been successfully analyzed and some statistical tests for spatial variation have been developed. However, an important assumption in these studies is that the disturbance terms of the GWR model are uncorrelated and of common variance. Similar to the case in the ordinary linear regression, spatial autocorrelation can invalidate the standard assumption of homoscedasticity of the disturbances and mislead the results of statistical inference. Therefore, developing some statistical methods to test for spatial autocorrelation is a very important issue. In this paper, two kinds of the statistical tests for spatial autocorrelation among the residuals of the GWR model are suggested. Also, an efficient approximation method for calculating the p-values of the test statistics is proposed. Some simulations are run to examine the performances of the proposed methods and the results are encouraging. The study not only makes it possible to test for spatial autocorrelation among the GWR residuals in a conventional statistical manner, but also provides a useful means for model validation.

A comparative study of multigranulation rough sets and concept lattices via rule acquisition

Transforming decision systems into formal decision contexts is studied.Relationship between AND decision rules and granular rules is discussed.Relationship between OR decision rules and disjunctive rules is investigated.Support and certainty factors of different rules are compared.Algorithm complexity of rule acquisition is analyzed. Recently, by combining rough set theory with granular computing, pessimistic and optimistic multigranulation rough sets have been proposed to derive AND and OR decision rules from decision systems. At the same time, by integrating granular computing and formal concept analysis, Willes concept lattice and object-oriented concept lattice were used to obtain granular rules and disjunctive rules from formal decision contexts. So, the problem of rule acquisition can bring rough set theory, granular computing and formal concept analysis together. In this study, to shed some light on the comparison and combination of rough set theory, granular computing and formal concept analysis, we investigate the relationship between multigranulation rough sets and concept lattices via rule acquisition. Some interesting results are obtained in this paper: (1) AND decision rules in pessimistic multigranulation rough sets are proved to be granular rules in concept lattices, but the inverse may not be true; (2) the combination of the truth parts of an OR decision rule in optimistic multigranulation rough sets is an item of the decomposition of a disjunctive rule in concept lattices; (3) a non-redundant disjunctive rule in concept lattices is shown to be the multi-combination of the truth parts of OR decision rules in optimistic multigranulation rough sets; and (4) the same rule is defined with a same certainty factor but a different support factor in multigranulation rough sets and concept lattices. Moreover, algorithm complexity analysis is made for the acquisition of AND decision rules, OR decision rules, granular rules and disjunctive rules.

Feature selection in mixed data

Feature selection in the data with different types of feature values, i.e., the heterogeneous or mixed data, is especially of practical importance because such types of data sets widely exist in real world. The key issue for feature selection in mixed data is how to properly deal with different types of the features or attributes in the data set. Motivated by the fuzzy rough set theory which allows different fuzzy relations to be defined for different types of attributes to measure the similarity between objects and in view of the effectiveness of entropy to measure information uncertainty, we propose in this paper a fuzzy rough set-based information entropy for feature selection in a mixed data set. It is proved that the newly-defined entropy meets the common requirement of monotonicity and can equivalently characterize the existing attribute reductions in the fuzzy rough set theory. Then, a feature selection algorithm is formulated based on the proposed entropy and a filter-wrapper method is suggested to select the best feature subset in terms of classification accuracy. An extensive numerical experiment is further conducted to assess the performance of the feature selection method and the results are satisfactory. HighlightsA novel fuzzy rough set-based information entropy is constructed for mixed data.The proposed entropy can equivalently characterize the existing attribute reductions in the fuzzy rough set theory.A feature selection algorithm is formulated based on the proposed entropy.A filter-wrapper method is suggested to select a best feature subset.

Testing the importance of the explanatory variables in a mixed geographically weighted regression model

A mixed geographically weighted regression (MGWR) model is a kind of regression model in which some coefficients of the explanatory variables are constant, but others vary spatially. It is a useful statistical modelling tool in a number of areas of spatial data analysis. After an MGWR model is identified and calibrated, which has been well studied recently, one of the important inference problems is to evaluate the influence of the explanatory variables in the constant-coefficient part on the response of the model. This is useful in the selection of the variables and for the purpose of explanation. In this paper, a statistical inference framework for this issue is suggested and, besides the F-approximation, which has been frequently used in the literature of the geographically weighted regression technique, a bootstrap procedure for deriving the p-value of the test is also suggested. The performance of the test is investigated by extensive simulations. It is demonstrated that both the F-approximation and the bootstrap procedure work satisfactorily.

Local linear estimation of spatially varying coefficient models: an improvement on the geographically weighted regression technique

Geographically weighted regression (GWR), as a useful method for exploring spatial non-stationarity of a regression relationship, has been applied to a variety of areas. In this method a spatially varying coefficient model is locally calibrated and the spatial-variation patterns of the locally estimated regression coefficients are taken as the main evidence of spatial nonstationarity for the underlying data-generating processes. Therefore, the validity of the analysis results drawn by GWR is closely dependent on the accuracy between the underlying coefficients and their estimates. Motivated by the local polynomial-modelling technique in statistics, we propose a local linear-based GWR for the spatially varying coefficient models, in which the coefficients are locally expanded as linear functions of the spatial coordinates and then estimated by the weighted least-squares procedure. Some theoretical and numerical comparisons with GWR are conducted and the results demonstrate that the proposed method can significantly improve GWR, not only in goodness-of-fit of the whole regression function but also in reducing bias of the coefficient estimates.

Fuzzy nonparametric regression based on local linear smoothing technique

In a great deal of literature on fuzzy regression analysis, most of research has focused on some predefined parametric forms of fuzzy regression relationships, especially on the fuzzy linear regression models. In many practical situations, it may be unrealistic to predetermine a fuzzy parametric regression relationship. In this paper, a fuzzy nonparametric model with crisp input and LR fuzzy output is considered and, based on the distance measure for fuzzy numbers suggested by Diamond [P. Diamond, Fuzzy least squares, Information Sciences 46 (1988) 141-157], the local linear smoothing technique in statistics with the cross-validation procedure for selecting the optimal value of the smoothing parameter is fuzzified to fit this model. Some simulation experiments are conducted to examine the performance of the proposed method and three real-world datasets are analyzed to illustrate the application of the proposed method. The results demonstrate that the proposed method works quite well not only in producing satisfactory estimate of the fuzzy regression function, but also in reducing the boundary effect significantly.