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Under the uncertain statistical framework by Liu [19], there is still a lack of an effective fitting method for uncertain linear models with Box-Cox transformation of response variables. For example, for the transformation parameter Λ , the uncertain least squares estimation will produce a severely low estimation result. In this paper, we propose uncertain Box-Cox regression analysis by utilizing the uncertainty theory to model the imprecise data and applying a generalized Box-Cox transformation indexed by its parameter to validate classic regression assumptions. We use rescaled least squares to estimate unknown parameters and provide an estimate for noises followed by residual analysis for these uncertain Box-Cox regression models. We also give the forecast values and confidence intervals and use a numerical example to demonstrate our methodology. Our work sets a uniformed framework for Box-Cox transformation on the uncertain regression, and extends such regression from linear to nonlinear cases, taking the Johnson-Schumacher growth model as an example.

We propose and numerically demonstrate a secure optical communication scheme in a small-world semiconductor lasers network based on cluster chaos synchronization and false-message scrambling. According to the symmetries of network, the semiconductor lasers are divided into sets of heterogeneous coupled clusters. The properties of cluster chaos synchronization, the performance and security of optical chaotic communication, as well as the influence of false message on the synchronization of clusters are systematically investigated. It is demonstrated that, by properly setting the parameters of the lasers network, high-quality chaos synchronization can be achieved between the intra-cluster lasers, whereas incoherent states are observed between inter-cluster lasers. Moreover, by inserting false messages into the inter-cluster couplings, the privacy of chaotic carriers transmitted over the inter-cluster public links can be significantly enhanced, as such secure bidirectional communication among intra-cluster lasers is achievable with a message bit rate of several Gbit/s. Compared with the conventional point-to-point communication systems, the proposed scheme supports multipoint-to-multipoint secure communications.

Regression analysis is to establish the regression relationship between dependent variables and independent variables. The data of traditional regression model are often assumed to be observed precisely. However this assumption holds only sometimes. Due to the influence of various uncertain factors, the data in reality is often inaccurate. Therefore, We treat real data as uncertain variables. Uncertain regression analysis is likely to provide an effective analysis method. Based on the uncertainty theory, the residual analysis of verhulst-pearl model is discussed in this paper. We use the least square method to estimate the parameters. And we also obtained the confidence interval of the response variables for the new explanatory variables. In the uncertain regression analysis, we propose a leave-p-out cross-validation method for model selection under imprecise observation. We end up the paper with a numerical example of uncertain Verhulst-Pearl regression model and show that the model has a better prediction accuracy.