In data processing, measurement of uncertainty is one of the significant evaluation tools, which can describe the uncertainty essence of data. So far, there are few measurable tools to study the uncertainty of type-2 fuzzy information systems (TFISs-2) (the expanded models of fuzzy information systems). This paper is devoted to looking for effective indicators to describe the uncertainty of TFISs-2. The fuzzy \(T_{\text {cos}}\)-similarity relations are first introduced, which are generated by TFISs-2 based on Gaussian kernel. Then, the fuzzy information structures are defined on account of this fuzzy \(T_{\text {cos}}\)-similarity relation. Next, two measures constructed from the upper and lower approximations are given for TFISs-2, that is, \(\delta\)-accuracy and \(\delta\)-roughness, which are used to reflect the degree of accuracy and inaccuracy of information by numerical form. Furthermore, by combining roughness and entropy, the \(\delta\)-rough entropy is investigated. Finally, the practicability of proposed measures is tested by a numerical experiment. The experimental results show that the \(\delta\)-rough entropy is efficacious and applicable for TFISs-2.

在数据处理中，不确定度的测量是重要的评估工具之一，可以描述数据的不确定性本质。到目前为止，几乎没有可测量的工具来研究2型模糊信息系统（TFISs-2）（模糊信息系统的扩展模型）的不确定性。本文致力于寻找有效的指标来描述TFISs-2的不确定性。首先介绍了模糊\（T _ {\ text {cos}} \）-相似性关系，它是由TFISs-2基于高斯核生成的。然后，基于该模糊\（T _ {\ text {cos}} \）-相似性关系定义模糊信息结构。接下来，针对TFISs-2，给出了根据上下近似构造的两个度量，即\（\ delta \）-accuracy和\（\ delta \）- roughness，它们通过数字形式反映信息的准确性和不准确性程度。此外，通过结合粗糙度和熵，研究了\（\ delta \） -粗糙熵。最后，通过数值实验验证了所提出措施的实用性。实验结果表明，\（\ delta \）粗糙熵是有效的，适用于TFISs-2。