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A new solution to overcome the constraints of multimodality medical intra-subject image registration is proposed, using the mutual information (MI) of image histogram-oriented gradients as a new matching criterion. We present a rigid, multi-modal image registration algorithm based on linear transformation and oriented gradients for the alignment of T2-weighted (T2w) images (as a fixed reference) and diffusion tensor imaging (DTI) (b-values of 500 and 1250 s/mm2) as floating images of three patients to compensate for the motion during the acquisition process. Diffusion MRI is very sensitive to motion, especially when the intensity and duration of the gradient pulses (characterized by the b-value) increases. The proposed method relies on the whole brain surface and addresses the variability of anatomical features into an image stack. The sparse features refer to corners detected using the Harris corner detector operator, while dense features use all image pixels through the image histogram of oriented gradients (HOG) as a measure of the degree of statistical dependence between a pair of registered images. HOG as a dense feature is focused on the structure and extracts the oriented gradient image in the x and y directions. MI is used as an objective function for the optimization process. The entropy functions and joint entropy function are determined using the HOGs data. To determine the best image transformation, the fiducial registration error (FRE) measure is used. We compare the results against the MI-based intensities results computed using a statistical intensity relationship between corresponding pixels in source and target images. Our approach, which is devoted to the whole brain, shows improved registration accuracy, robustness, and computational cost compared with the registration algorithms, which use anatomical features or regions of interest areas with specific neuroanatomy. Despite the supplementary HOG computation task, the computation time is comparable for MI-based intensities and MI-based HOG methods.

To reduce hydrological disasters, it is necessary to operate rain gauge stations at locations where the spatio-temporal characteristics of rainfall can be reflected. Entropy has been widely used to evaluate the designs and uncertainties associated with rain gauge networks. In this study, the optimal rain gauge network in the Daegu and Gyeongbuk area, which requires the efficient use of water resources due to low annual precipitation and severe drought damage, was determined using conditional and joint entropy, and the selected network was quantitatively evaluated using the root mean square error (RMSE). To consider spatial distribution, prediction errors were generated using kriging. Four estimators used in entropy calculations were compared, and weighted entropy was calculated by weighting the precipitation. The optimal number of rain gauge stations was determined by calculating the RMSE reduction and the reduction ratio according to the number of selected rain gauge stations. Our findings show that the results of conditional entropy were better than those of joint entropy. The optimal rain gauge stations showed a tendency wherein peripheral rain gauge stations were selected first, with central stations being added afterward.

Complex systems often exhibit multiple levels of organization covering a wide range of physical scales, so the study of the hierarchical decomposition of their structure and function is frequently convenient. To better understand this phenomenon, we introduce a generalization of information theory that works with hierarchical partitions. We begin revisiting the recently introduced Hierarchical Mutual Information (HMI), and show that it can be written as a level by level summation of classical conditional mutual information terms. Then, we prove that the HMI is bounded from above by the corresponding hierarchical joint entropy. In this way, in analogy to the classical case, we derive hierarchical generalizations of many other classical information-theoretic quantities. In particular, we prove that, as opposed to its classical counterpart, the hierarchical generalization of the Variation of Information is not a metric distance, but it admits a transformation into one. Moreover, focusing on potential applications of the existing developments of the theory, we show how to adjust by chance the HMI. We also corroborate and analyze all the presented theoretical results with exhaustive numerical computations, and include an illustrative application example of the introduced formalism. Finally, we mention some open problems that should be eventually addressed for the proposed generalization of information theory to reach maturity., Comment: 6 figures

This paper uses quantitative eye tracking indicators to analyze the relationship between images of paintings and human viewing. First, we build the eye tracking fixation sequences through areas of interest (AOIs) into an information channel, the gaze channel. Although this channel can be interpreted as a generalization of a first-order Markov chain, we show that the gaze channel is fully independent of this interpretation, and stands even when first-order Markov chain modeling would no longer fit. The entropy of the equilibrium distribution and the conditional entropy of a Markov chain are extended with additional information-theoretic measures, such as joint entropy, mutual information, and conditional entropy of each area of interest. Then, the gaze information channel is applied to analyze a subset of Van Gogh paintings. Van Gogh artworks, classified by art critics into several periods, have been studied under computational aesthetics measures, which include the use of Kolmogorov complexity and permutation entropy. The gaze information channel paradigm allows the information-theoretic measures to analyze both individual gaze behavior and clustered behavior from observers and paintings. Finally, we show that there is a clear correlation between the gaze information channel quantities that come from direct human observation, and the computational aesthetics measures that do not rely on any human observation at all This work is supported by the National Natural Science Foundation of China under grant No. 61702359, and by grant TIN2016-75866-C3-3-R from Spanish Government.

This paper concerns the problem of optimal monitoring network layout using information-theoretical methods. Numerous different objectives based on information measures have been proposed in recent literature, often focusing simultaneously on maximum information and minimum dependence between the chosen locations for data collection stations. We discuss these objective functions and conclude that a single-objective optimization of joint entropy suffices to maximize the collection of information for a given number of stations. We argue that the widespread notion of minimizing redundancy, or dependence between monitored signals, as a secondary objective is not desirable and has no intrinsic justification. The negative effect of redundancy on total collected information is already accounted for in joint entropy, which measures total information net of any redundancies. In fact, for two networks of equal joint entropy, the one with a higher amount of redundant information should be preferred for reasons of robustness against failure. In attaining the maximum joint entropy objective, we investigate exhaustive optimization, a more computationally tractable greedy approach that adds one station at a time, and we introduce the “greedy drop” approach, where the full set of stations is reduced one at a time. We show that no greedy approach can exist that is guaranteed to reach the global optimum.

Computer Vision (CV) domain is widely used in the current era of automation and visual surveillance for the detection and classification of different objects in a diverse environment. The automatic machine inspection of different objects in the scenes is based on internal and external parameters like features that provide a huge amount of information related to the nature of an object in the scene. In this work, we propose a new automated method based on classical and deep learning feature selection. The proposed object classification method follows three steps. The data augmentation is performed in the first step to make the balance database. Later, Pyramid HOG (PHOG) and Central Symmetric LBP (CS-LBP) features are serially fused along with deep learning-based extracted features. The deep learning features are extracted from the pre-trained CNN model name Inception V3. In the third step, a new technique name Joint Entropy along with KNN (JEKNN) is employed to select the best features. The best-selected features are finally classified by well-known supervised learning methods and choose the best one based on higher accuracy. The proposed method is evaluated on Caltech101 balanced dataset and achieved maximum accuracy of 90.4% on Ensemble classifier which outperforms as compare to existing techniques.

In this work, we propose a theoretical framework based on maximum profile likelihood for pairwise and groupwise registration. By an asymptotic analysis, we demonstrate that maximum profile likelihood registration minimizes an upper bound on the joint entropy of the distribution that generates the joint image data. Further, we derive the congealing method for groupwise registration by optimizing the profile likelihood in closed form, and using coordinate ascent, or iterative model refinement. We also describe a method for feature based registration in the same framework and demonstrate it on groupwise tractographic registration. In the second part of the article, we propose an approach to deep metric registration that implements maximum likelihood registration using deep discriminative classifiers. We show further that this approach can be used for maximum profile likelihood registration to discharge the need for well-registered training data, using iterative model refinement. We demonstrate that the method succeeds on a challenging registration problem where the standard mutual information approach does not perform well.

Dense and distributed tactile sensors are critical for robots to achieve human-like manipulation skills. Soft robotic sensors are a potential technological solution to obtain the required high dimensional sensory information unobtrusively. However, the design of this new class of sensors is still based on human intuition or derived from traditional flex sensors. This work is a first step towards automated design of soft sensor morphologies based on optimization of information theory metrics and machine learning. Elementary simulation models are used to develop the optimized sensor morphologies that are more accurate and robust with the same number of sensors. Same configurations are replicated experimentally to validate the feasibility of such an approach for practical applications. Furthermore, we present a novel technique for drift compensation in soft strain sensors that allows us to obtain accurate contact localization. This work is an effort towards transferring the paradigm of morphological computation from soft actuator designing to soft sensor designing for high performance, resilient tactile sensory networks. © 2020 IEEE.

First, information theory is reviewed for readers who are unfamiliar with basic concepts and ideas. Information of events and entropy of sample spaces and/or random variables are introduced and the properties are discussed. To measure the differences between distributions, the Kullback–Leibler (KL) information is explained through a theoretical consideration. In order to treat the association between random variables, the joint entropy and the conditional entropy are also introduced. Second, basic statistics are considered from a viewpoint of entropy. The log-likelihood ratio test statistic is interpreted in entropy. By introducing entropy of continuous variables, t and F statistics are also discussed through KL information.

This paper proposes an efficient method to compute entropy and joint entropy of images. Entropy of images is used to determine its quality. It is defined as the randomness or uncertainty present in the image. Similarly, joint entropy is a measure of the uncertainty present in the overlapped region of two images. Entropy and joint entropy computations are vital in several image processing applications. Intensity based image registration is done by maximizing the mutual information between two images. Mutual information is nothing but the difference between sum of individual entropies and joint entropy of two images. Image registration has applications, especially in the medical field, e.g. diagnosis and treatment of diseases. The entropy and joint entropy computation methods proposed in this paper are computationally less expensive than the standard methods. Entropy computation takes 78.60% less time than the standard method while computational time of joint entropy is reduced by 83.59%. This increase in efficiency comes at the cost of an error of 1.52% in entropy and 4.54% in joint entropy.

We present a new system to optimize feature extraction from 2D-topological data like images in the context of deep learning using correlation among training samples and curriculum learning optimization (CLO). The system treats every sample as 2D random variable, where a pixel contained in the sample is modelled as an independent and identically distributed random variable (i.i.d) realization. With this modelling we utilize information-theoretic and statistical measures of random variables to rank individual training samples and relationship between samples to construct syllabus. The rank of each sample is then used when the sample is fed to the network during training. Comparative evaluation of multiple state-of-the-art networks, including, ResNet, GoogleNet, and VGG, on benchmark datasets demonstrate a syllabus that ranks samples using measures such as Joint Entropy between adjacent samples, can improve learning and significantly reduce the amount of training steps required to achieve desirable training accuracy. We present results that indicate our approach can produce robust feature maps that in turn contribute to reduction of loss by as much as factors of 9 compared to conventional, no-curriculum, training.

Software-defined networking (SDN) is a communication paradigm that brings cost efficiency and flexibility through software-defined functions resident on centralized controllers. Although SDN applications are introduced in a limited scope with related technologies still under development, operational SDN networks already face major security threats. Therefore, comprehensive and efficient solutions are crucial. Especially, large-scale security threats such as distributed-denial-of-service (DDoS) attacks are jeopardizing safety and availability of data and services in these systems. A DDoS attack is aimed at making resources unavailable to legitimate users via overloading systems with excessive superfluous traffic from distributed sources. In this paper, we describe and evaluate a joint entropy-based security scheme (JESS) to enhance the SDN security with the aim of a reinforced SDN architecture against DDoS attacks. In particular, our proposed model devises a statistical solution to detect and mitigate these hazards. To the best of our knowledge, JESS is the first model that utilizes joint entropy for DDoS detection and mitigation in the SDN environment. Since it relies on a statistical model, it mitigates not only known attacks but also unfamiliar types in an efficient manner.

As one classic technique of computer vision, image retrieval could retrieve the target images from hundreds of thousands of images effectively. Furthermore, with the rapid development of deep learning, the quality of retrieval is increased obviously. However, under normal conditions, the high-quality retrieval is supported by a large number of learning instances. The large number of learning instances not only need much human source in the process of selection, but also need much computing source in the process of computation. More importantly, for some special categories, it’s difficult to obtain a large number of learning instances. Aiming at the problem above, we proposed one joint entropy based learning model which could reduce the number of learning instances through optimizing the distribution of learning instances. Firstly, the learning instances are pre-selected using improved watershed segmentation method. Then, joint entropy model is used for reducing the possibility of double, useless even mistaken instances existence. After that, a database using a large number of images is built up. Sufficient experiments based on the database show the model’s superiority that our model not only could reduce the number of learning instances but also could keep the accuracy of retrieval.

The signature matrix of an n-component three-state network (system), which depends only on the network structure, is a useful tool for comparing the reliability and stochastic properties of networks. In this paper, we consider a three-state network with states up, partial performance, and down. We assume that the network remains in state up, for a random time T1 and then moves to state partial performance until it fails at time T>T1. The signature-based expressions for the conditional entropy of T given T1, the joint entropy, Kullback-Leibler (K-L) information, and mutual information of the lifetimes T and T1 are presented. It is shown that the K-L information, and mutual information between T1 and T depend only on the network structure (i.e., depend only to the signature matrix of the network). Some signature-based stochastic comparisons are also made to compare the K-L of the state lifetimes in two different three-state networks. Upper and lower bounds for the K-L divergence and mutual information between T1 and T are investigated. Finally the results are extended to n-component multi-state networks. Several examples are examined graphically and numerically.

We propose a mathematical model of the entropy estimator in the Linux random number generator. First, we construct a probability model for random event times in entropy sources, and then precisely derive probability distributions for the first, second, and third time differences. Second, we obtain the probability distribution for the minimum of absolute values of these differences, which is used for the estimated entropy in the Linux system. Moreover, we provide several simulations that display the accuracy of our results for various parameters.

With the growth of deep learning, object recognition has received increasing interests and its accuracy has been improved significantly in the past few years, However, high-quality recognition largely depends on a large number of learning instances. If the number of learning instances is reduced, it’s difficult to maintain realistic recognition accuracy. Moreover, traditional methods usually don’t consider the semantic relationship between different regions. Actually, semantic constraint would contribute to improve the recognition accuracy effectively. Aiming at the problems above, we proposed one semantic constraint based object recognition method. On the one hand, instance-based transfer learning model could make use of learning instances of other categories to maintain realistic recognition accuracy. On the other hand, semantic constraint between different regions simulated as joint entropy is used to recognize target object more accurately. At last, adequate experiments using a large number of images show that our model not only could reduce the number of learning instances but also could achieve realistic recognition.

Recently, Sati and Gupta (2015) have introduced a generalized cumulative residual entropy based on the non-additive Tsallis entropy. The cumulative residual entropy, introduced by Rao et al. (2004) is a generalized measure of uncertainty which is applied in reliability and image alignment and non-additive measures of entropy. This entropy finds justifications in many physical, biological and chemical phenomena. In this paper, we derive the weighted form of this measure and call it Weighted Cumulative Residual Tsallis Entropy (WCRTE). Being a “length-biased” shift-dependent information measure, WCRTE is related to the differential information in which higher weight is assigned to large values of observed random variables. Based on the dynamic version of this new information measure, we propose ageing classes and it is shown that it can uniquely determine the survival function and Rayleigh distribution. Several properties, including linear transformations, bounds and related results to stochastic orders are obtained for these measures. Also, we identify classes of distributions in which some well-known distributions are maximum dynamic version of WCRTE. The empirical WCRTE is finally proposed to estimate the new information measure.

The convergence and divergence of oscillatory behavior of different brain regions are very important for the procedure of information processing. Measurements of coupling or correlation are very useful to study the difference of brain activities. In this study, EEG signals were collected from ten subjects under two conditions, i.e. eyes closed state and idle with eyes open. We propose a nonlinear algorithm, symbolic joint entropy, to compare the coupling strength among the frontal, temporal, parietal and occipital lobes and between two different states. Instead of decomposing the EEG into different frequency bands (theta, alpha, beta, gamma etc.), the novel algorithm is to investigate the coupling from the entire spectrum of brain wave activities above 4Hz. The coupling coefficients in two states with different time delay steps are compared and the group statistics are presented as well. We find that the coupling coefficient of eyes open state with delay consistently lower than that of eyes close state across the group except for one subject, whereas the results without delay are not consistent. The differences between two brain states with non-zero delay can reveal the intrinsic inter-region coupling better. We also use the well-known Henon map data to validate the algorithm proposed in this paper. The result shows that the method is robust and has a great potential for other physiologic time series.

We propose a new definition of entropy of basic probability assignments (BPAs) in the Dempster–Shafer (DS) theory of belief functions, which is interpreted as a measure of total uncertainty in the BPA. Our definition is different from those proposed by Hohle, Smets, Yager, Nguyen, Dubois–Prade, Lamata–Moral, Klir–Ramer, Klir–Parviz, Pal et al., Maeda–Ichihashi, Harmanec–Klir, Abellan–Moral, Jousselme et al., Pouly et al., and Deng. We state a list of six desired properties of entropy for DS belief functions theory, four of which are motivated by Shannon's definition of entropy of probability functions, and the remaining two are requirements that adapt this measure to the philosophy of the DS theory. Three of our six desired properties are different from the five properties proposed by Klir and Wierman. We demonstrate that our definition satisfies all six properties in our list, whereas none of the existing definitions do. Our new definition has two components. The first component is Shannon's entropy of an equivalent probability mass function obtained using the plausibility transform, which constitutes the conflict measure of entropy. The second component is Dubois–Prade's definition of entropy of basic probability assignments in the DS theory, which constitutes the non-specificity measure of entropy. Our new definition is the sum of these two components. Our definition does not satisfy the subadditivity property. Whether there exists a definition that satisfies our six properties plus subadditivity remains an open question.

Recycling of plastic bottles is a fundamental strategy to reduce the need of fossil resources and protect global environments. The collection of post-consumer plastic bottles (PCPBs) is a key node of the recycling of plastics due to its influences on the subsequent life cycle stages. However, there is still a lack of quantitative assessment on the quality of PCPBs, especially considering the contamination and the mixture of different plastics in the collection stage. This study provides an indicator based on the joint entropy method to evaluate the quality of PCPBs in collection stage, which includes components, polymer types and colors three levels. This indicator (Quality-value) is applied to evaluate the quality of PCPBs’ collection system in Beijing city, China, which is based on an on-site survey including 27 samples gathered from 4 distribution centers and 5 collection sites. The results showed that the quality of PCPBs decreased through the collection system of Beijing, which degraded from post-consumer bottles (0.320

Human activity monitoring in the video sequences is an intriguing computer vision domain which incorporates colossal applications, e.g., surveillance systems, human-computer interaction, and traffic control systems. In this research, our primary focus is in proposing a hybrid strategy for efficient classification of human activities from a given video sequence. The proposed method integrates four major steps: (a) segment the moving objects by fusing novel uniform segmentation and expectation maximization, (b) extract a new set of fused features using local binary patterns with histogram oriented gradient and Harlick features, (c) feature selection by novel Euclidean distance and joint entropy-PCA-based method, and (d) feature classification using multi-class support vector machine. The three benchmark datasets (MIT, CAVIAR, and BMW-10) are used for training the classifier for human classification; and for testing, we utilized multi-camera pedestrian videos along with MSR Action dataset, INRIA, and CASIA dataset. Additionally, the results are also validated using dataset recorded by our research group. For action recognition, four publicly available datasets are selected such as Weizmann, KTH, UIUC, and Muhavi to achieve recognition rates of 95.80, 99.30, 99, and 99.40%, respectively, which confirm the authenticity of our proposed work. Promising results are achieved in terms of greater precision compared to existing techniques.

Over the past two decades, the fragility of modern Logistics Networks (LNs) has been exposed by unexpected events with extreme impacts outside the realm of regular expectations, which are known as the black swan events. As a response to the growing dysfunctions of the global LNs due to the detrimental effects of the black swan events, this article develops a quantitative vulnerability assessment method by paying simultaneous attention to various structural properties of these networks. It adopts three prototypical examples of centrality indices known as betweenness, closeness, and eigenvector centrality. This work also develops a vulnerability index from the joint entropy of centrality values, which can be interpreted as a certain extent of risk to disruptions in supply provision in LNs. The proposed entropy-based vulnerability index enables a far more accurate analysis of vulnerability by measuring the degree of homogeneity and heterogeneity of centrality values. The suggested index is considered as an objective function that must be minimized to reduce the deleterious effects of the black swan events. To illustrate the proposed vulnerability assessment method, this paper analyzes a real-world global maritime network, concluding that the juxtaposition of centrality measures provides a richer insight into the vulnerability analysis of LNs. The results also favor the fact that increasing the homogeneity of a network through decentralization yields lower vulnerability and builds extra robustness into the networks.

Extreme precipitation events have been increasing in recent decades and have been causing impacts to human life and property, which requires expansion of existing networks to provide essential information for hydrometeorological analysis. This study developed an entropy and copula-based approach for precipitation monitoring network expansion, aiming at adding stations in ungauged areas with high value of monitoring (VOM), which was estimated through information content and redundancy. The approach was applied to a network in the Taihu Lake basin, China. We grouped the adjacent stations based on Thiessen polygons, computed VOM for each group and identified candidate locations for additional stations. The entropy measures, e.g., joint entropy and total correlation, required to compute VOM were estimated through the principle of maximum entropy (POME) and copula-based joint distribution fitting. We evaluated the performance of five copula families and adopted the Student t-copula as the optimal copula model. Results showed that orographic effect and network density were the main factors that influence VOM variation, i.e., high VOM appeared in the southwest mountain area and the east area with sparsely distributed stations. We also compared the results with those derived from three data quantization methods, among which the rounding technique produced consistent results. The final network schemes were generated by adding stations following a strategy that supports decision preference on the positioning scheme and station number, and were evaluated through the Kriging interpolation error. The proposed approach provides an efficient and a practical way for strengthening a precipitation monitoring system.

This paper considers a fuzzyc-means (FCM) clustering algorithm in combination with deterministic annealing and the Tsallis entropy maximization. The Tsallis entropy is aq-parameter extension of the Shannon entropy. By maximizing the Tsallis entropy within the framework of FCM, statistical mechanical membership functions can be derived. One of the major considerations when using this method is how to determine appropriate values forqand the highest annealing temperature,Thigh, for a given data set. Accordingly, in this paper, a method for determining these values simultaneously without introducing any additional parameters is presented, where the membership function is approximated using a series expansion method. The results of experiments indicate that the proposed method is effective, and bothqandThighcan be determined automatically and algebraically from a given data set.

Shannon's entropy and its variants have been applied to measure uncertainty in a variety of special binary relations. However, few studies have been conducted on uncertainty of general binary relations. In this study, we present a unified form of uncertainty measures for general binary relations. We redefine the concepts of entropy, joint entropy, conditional entropy, and mutual information. These uncertainty measures are generalizations of corresponding measures of special relations. We study the relationship between these measures and examine important properties. Finally, numerical experiments are performed to identify applications of the proposed uncertainty measures. Comparing with existing uncertainty measures, the proposed method not only addresses the uncertainty of heterogeneous data sets, but also exhibit better performance in attribute reduction. This study can provide a fundamental framework for uncertainty theories of special rough set models.

The cumulative entropy is an information measure which is alternative to the differential entropy. Indeed, the cumulative entropy of a random lifetime X can be expressed as the expectation of its mean inactivity time evaluated at X . In this paper we propose a new generalized cumulative entropy based on Tsallis entropy (CTE) and its dynamic version (DCTE). We study some properties and characterization results for this measure.

Cloud computing provides enormous storage capacity and huge computation power. Cloud enables users to deploy data-intensive applications without substructure investment. Intermediate data are generated under the cloud applications and stored again in cloud. An opponent may analyse multiple intermediate data sets to access the privacy sensitive information.The proposed technique to have good privacy and utility trade off, joint entropy with adaptive optimization process is used to maintain the privacy in cloud. Here optimal entropy value process using adaptive particle swarm optimization (APSO) process. After get the optimal entropy database difference model was processed, Entropy and database difference ratio is taken as the evaluation matrices and is tested using various datasets. The technique is also compared to the existing PSO optimization process on privacy preservation in cloud. An adaption feature of Particle swarm optimization show that APSO enhances privacy and preserving cost of intermediate data sets can be significantly saved.

In this article, we propose high order, semi-discrete, entropy stable, finite difference schemes for Ten-Moment Gaussian Closure equations. The crucial components of these schemes are an entropy conservative flux and suitable high order entropy dissipative operators to ensure entropy stability. We design two numerical fluxes, one is approximately entropy conservative, and another is entropy conservative flux. For the construction of appropriate entropy dissipative operators, we also derive entropy scaled right eigenvectors. This is used for sign preserving reconstruction of scaled entropy variables, which results in second and third order entropy stable schemes. We also extend these schemes to a plasma flow model with source term. Several numerical results are presented for homogeneous and non-homogeneous cases to demonstrate stability and performance of these schemes.

In this paper, we perform a multiscale entropy analysis on the Dow Jones Industrial Average Index using the Shannon entropy. The stock index shows the characteristic of multi-scale entropy that caused by noise in the market. The entropy is demonstrated to have significant predictive ability for the stock index in both long-term and short-term, and empirical results verify that noise does exist in the market and can affect stock price. It has important implications on market participants such as noise traders.

This paper proposes some new distance measures between interval-valued neutrosophic sets (IvNSs) and their similarity measures. Then, some entropy measures of IvNS based on the distances are proposed as the extension of the entropy measures of interval-valued intuitionistic fuzzy sets (IvIFSs). Also, we investigate the relationship between the presented entropy measures and the similarity measures for IvNSs. Finally, the comparison of the new entropy measures with existing entropy measures for IvNSs is given by the numerical and decision-making examples to demonstrate that the proposed new entropy measures for IvNSs are effective and reasonable and more intelligible in representing the degree of fuzziness of IvNSs than the existing ones.

Genome-wide association study is especially challenging in detecting high-order disease-causing models due to model diversity, possible low or even no marginal effect of the model, and extraordinary search and computations. In this paper, we propose a niche harmony search algorithm where joint entropy is utilized as a heuristic factor to guide the search for low or no marginal effect model, and two computationally lightweight scores are selected to evaluate and adapt to diverse of disease models. In order to obtain all possible suspected pathogenic models, niche technique merges with HS, which serves as a taboo region to avoid HS trapping into local search. From the resultant set of candidate SNP-combinations, we use G-test statistic for testing true positives. Experiments were performed on twenty typical simulation datasets in which 12 models are with marginal effect and eight ones are with no marginal effect. Our results indicate that the proposed algorithm has very high detection power for searching suspected disease models in the first stage and it is superior to some typical existing approaches in both detection power and CPU runtime for all these datasets. Application to age-related macular degeneration (AMD) demonstrates our method is promising in detecting high-order disease-causing models.