Uncertainty measure is an important tool for analyzing imprecise and ambiguous data. Some information entropy models in rough set theory have been defined for various information systems. However, there are relatively few studies on evaluating uncertainty in fuzzy rough set. In this paper, we propose a new complement information entropy model in fuzzy rough set based on arbitrary fuzzy relation, which takes inner-class and outer-class information into consideration. The corresponding definitions of complement conditional entropy, complement joint entropy, complement mutual information and complement information granularity are also presented. The properties of these definitions are analyzed, which show complement information entropy shares some similar properties with Shannon's entropy. Moreover, a generalized information entropy model is proposed by introducing probability distribution into fuzzy approximate space. This model can be used to measure uncertainty of data with the different sample distributions. Applications of the proposed entropy measures in feature importance evaluation and feature selection are studied with data set experiments. Experimental results show that the proposed method is effective and adaptable to different classifiers.Communicated by V. Loia.

Presently, many researches have been carried out on rough set based data reduction. However, this method encounters a problem when dealing with real-valued data and fuzzy information. By lucubrating the theory of fuzzy rough set and utilizing the definition of information entropy presented in literature [5], the information entropy model of fuzzy rough set has been constructed. Then the conditional information entropy of attributes is adopted to measure the significance of attributes. On this condition, a heuristic fuzzy-rough data reduction method based on information entropy (E-FRDR) has been put forward. Finally, the method is validated by an example that indicates the method is feasible.

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With the development of high-precision inertial navigation systems, the deflection of vertical (DOV), gravity disturbance, is still one of the main error sources that restrict navigation accuracy. For the DOV compensation of the Strapdown Inertial Navigation System (SINS) problem, the influences of the calculation degree of the spherical harmonic coefficient and the calculation error of the DOV on the compensation effect were studied. Based on the SINS error model, the error propagation characteristics of the DOV in SINS were analyzed. In addition, the high-precision global gravity field spherical harmonic model EIGEN-6C4 was established and the influence comparative analysis of the calculation degree of the spherical harmonic coefficient on the DOV compensation of SINS in different regions was carried out. Besides, the influence of the calculation error of the DOV on the compensation was emphatically analyzed. Finally, the vehicle experiment verified the feasibility of compensation in SINS based on the gravity field spherical harmonic model. The simulation and experiment results show that it is necessary to consider the influence of the calculation degree and the calculation error of the DOV on the compensation for long-time high-precision SINS with the position accuracy of 0.3 nm/h, while the SINS with general requirements for position accuracy can ignore the impact.