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This case report describes entropy based on statistics of the novel coronavirus infection in the Russian Federation. Entropy (in bits) determined on the basis of the ratio of patients with clinical manifestations of the disease and asymptomatic carrier state, increased from 0.7219 by April 1, 2020, when the ratio of such patients was 4/5 to 1/5, to 0.8813 by April 14, 2020, when the ratio of the two groups of patients was 7/10 to 3/10. On May 3, 2020, the ratio of the two groups leveled off, reaching % to %, so the entropy was 1.0000. An analysis of the data from April 19, 2020 to June 1, 2020 allowed us to clarify the average proportion between patients with and without symptoms, which is 14/25 to 11/25 with a certain bias towards patients with clinical manifestations of the novel coronavirus infection, and the entropy for this ratio is 0.9896. We found a transformation of entropy of the novel coronavirus infection which is explained by different proportions of patients with and without symptoms of the disease. [ABSTRACT FROM AUTHOR]

Shannon entropy is used to measure information uncertainty, while the information dimension is used to measure information complexity. Given two probability distributions, the difference can be measured by relative entropy. However, the existing relative entropy does not consider the effect of information dimension. To improve the existing entropy, a new relative entropy is presented in this paper. The information fractal dimension is considered in the proposed relative entropy. The new relative entropy is more generalized than the initial relative entropy. When dimension is not considered, it will degenerate to the initial relative entropy. Another interesting point is that the new relative entropy may have negative values when calculating. The physical meaning is still under exploration. Finally, some application examples are provided to exemplify the utilization of the proposed relative entropy. [ABSTRACT FROM AUTHOR]

In group decision making (GDM), due to complexity of various factors, decision makers (DMs) often provide incomplete preference relations (PRs) in preference matrix. Interval-valued probabilistic uncertain linguistic term set (IVPULTS) is a flexible and accurate tool to depict evaluation information of experts. In this paper, we mainly propose a new concept pertaining to interval-valued probabilistic uncertain linguistic preference relation (IVPULPR) that applies the IVPULTS to preference relations. Firstly, some new basic theoretical concepts of IVPULTS are developed including ordered IVPULTS, normalization method and new expectation function. Secondly, we establish several goal programming models to estimate the unknown elements in incomplete IVPULPR and propose the expected additive consistency of IVPULPR. To improve the consistency level, two optimization models are constructed based on the idea of minimum adjustment. Thirdly, we derive the experts' weights in terms of information uncertainty, where a new method to measure information uncertainty of IVPULTS is proposed. For the sake of improving group consensus, we construct a group consensus index (GCI) and two optimization models depending on the adjustment mechanism of expert weight. Finally, a complete GDM framework with incomplete IVPULPR is devised based on the analysis of IVPULPR consistency and group consensus. Through an experiment analysis by using an UCI dataset, we find that the proposed GDM model can not only precisely express fuzzy preference information of DMs, but also ensure achievement of acceptable consistency and group consensus under the condition of not changing the initial preference as much as possible. [ABSTRACT FROM AUTHOR]

• We propose a novel Wavelet-based downsampling module (HWD) for CNNs. To the best of our knowledge, our method is the first attempt to explore feasibility by prohibiting (impeding) information loss in the downsampling stage of DCNNs for the semantic segmentation task. • We explore the measurement of information uncertainty across layers in CNNs, and propose a novel metric, named Feature Entropy Index (FEI), to evaluate the information uncertainty or feature importance between the downsampled feature maps and the prediction results. • The proposed HWD can be directly replaced the strided convolution or pooling layer without significant increase of computation overhead and be easily integrated into the current segmentation architectures. Comprehensive experiments demonstrate the effectiveness of the HWD module when comparing with seven state-of-the-art segmentation methods. Downsampling operations such as max pooling or strided convolution are ubiquitously utilized in Convolutional Neural Networks (CNNs) to aggregate local features, enlarge receptive field, and minimize computational overhead. However, for a semantic segmentation task, pooling features over the local neighbourhood may result in the loss of important spatial information, which is conducive for pixel-wise predictions. To address this issue, we introduce a simple yet effective pooling operation called the Haar Wavelet-based Downsampling (HWD) module. This module can be easily integrated into CNNs to enhance the performance of semantic segmentation models. The core idea of HWD is to apply Haar wavelet transform for reducing the spatial resolution of feature maps while preserving as much information as possible. Furthermore, to investigate the benefits of HWD, we propose a novel metric, named as feature entropy index (FEI), which measures the degree of information uncertainty after downsampling in CNNs. Specifically, the FEI can be used to indicate the ability of downsampling methods to preserve essential information in semantic segmentation. Our comprehensive experiments demonstrate that the proposed HWD module could (1) effectively improve the segmentation performance across different modality image datasets with various CNN architectures, and (2) efficiently reduce information uncertainty compared to the conventional downsampling methods. Our implementation are available at https://github.com/apple1986/HWD. [ABSTRACT FROM AUTHOR]

Dempster–Shafer theory has been widely used in many applications, especially in the measurement of information uncertainty. However, under the D-S theory, how to use the belief entropy to measure the uncertainty is still an open issue. In this paper, we list some significant properties. The main contribution of this paper is to propose a new entropy, for which some properties are discussed. Our new model has two components. The first is Nguyen entropy. The second component is the product of the cardinality of the frame of discernment (FOD) and Dubois entropy. In addition, under certain conditions, the new belief entropy can be transformed into Shannon entropy. Compared with the others, the new entropy considers the impact of FOD. Through some numerical examples and simulation, the proposed belief entropy is proven to be able to measure uncertainty accurately. [ABSTRACT FROM AUTHOR]