Queue length distribution of network packet traffic: Tsallis entropy maximization with fractional moments

A framework based on maximization of Tsallis entropy constrained by fractional moments is proposed to model queue length distribution of number of packets in network traffic exhibiting long-range behavior. For appropriate range of the Tsallis entropy parameter q, it is found that the first moment of number of packets may not exist Based on Euler summation formula, explicit expressions for mean queue length and buffer overflow probability exhibiting power law behavior are obtained. It is shown that in the limiting case as q tends to 1, one recovers the asymptotic results for buffer overflow probability depicting Weibull-like tail.     

q-Exponential product-form solution of packet distribution in queueing networks: maximization of Tsallis entropy

A maximum Tsallis entropy solution is presented to examine the effect of long-range dependence (LRD) of packet traffic on network of queues. An important finding is that usual product form solution of queueing networks does not hold. However, it is possible to preserve the product like structure in terms of q-product of q-exponential functions. A special case is considered when normalized q-expectation values of first moment and queue utilization at each node are available as the constraint. The joint state probability distribution is shown to depict asymptotically power law behavior.     

Tsallis Entropy Based q-Gaussian Density Model and Its Application in Measurement Accuracy Improvement

The central limit theorem guarantees the distribution of the measurand is Gaussian when the number of repeated measurement is infinity, but in many practical cases, the number of measurement times is limited to a given number. To overcome this contradiction, this paper firstly carries out the maximum likelihood estimation for parameter q in q-Gaussian density model developed under the maximum Tsallis entropy principle. Then the q-Gaussian probability density function is used in the particle filter to estimate and measure the nonlinear system. The estimated parameter q is related to the ratio between the measurement variance and the given variance, which indicates that the measurement accuracy cannot be improved if we only increase the repeated measurement times. Via using the proposed q-Gaussian density model, the measurement error (the average mean square error) of the estimation results can be reduced to a considerable level where the number of repeated measurement is limited. The experimental example is given to verify the proposed model and the measurement results prove the correctness and effectiveness of it.     

基于多尺度多方向Gabor变换的Tsallis熵阈值分割方法

为了能在统一框架内处理无模态、单模态、双模态或者多模态直方图情形下的自动阈值选取问题,该文提出一种基于多尺度多方向Gabor变换的Tsallis熵阈值分割方法(MGTE)。该方法先通过Gabor变换得到多尺度乘积图像,然后利用内外轮廓图像从多尺度乘积图像中重构1维直方图,并在重构1维直方图上采用Tsallis熵计算模型来选取4个方向Tsallis熵取最大值时对应的阈值,最后对4个方向的阈值进行加权求和作为最终分割阈值。将提出的方法和5个分割方法在4幅合成图像和40幅真实世界图像上进行了实验。结果表明提出的方法虽然计算效率不占优势,但它的分割适应性和分割精度有明显的提高。     

On the Usefulness of the Maximum Entropy Principle in the Bayesian Estimation of Reliability

The use of the maximum entropy principle for determining prior distribution is compared with other techniques in statistical-decision theory for estimating reliability. The comparison is made in the context of estimating the parameter representing the probability of success in a binomial model and the parameter representing the mean time to failure in a simple exponential model. The existence of partial prior information of varying degrees is also assumed. When a quadratic loss function is used, the techniques based on the maximum entropy principle lose much of their appeal.     

On uniqueness Theorems for Tsallis entropy and Tsallis relative entropy

The uniqueness theorem for Tsallis entropy was presented in H. Suyari, IEEE Trans. Inf. Theory, vol. 50, pp. 1783-1787, Aug. 2004 by introducing the generalized Shannon-Khinchin axiom. In the present correspondence, this result is generalized and simplified as follows: Generalization : The uniqueness theorem for Tsallis relative entropy is shown by means of the generalized Hobson's axiom. Simplification: The uniqueness theorem for Tsallis entropy is shown by means of the generalized Faddeev's axiom     

Application of entropy formulas in detection of denial‐of‐service attacks

The paper compares five entropy formulas (Shannon, Tsallis, Rényi, Bhatia‐Singh, and Ubriaco) and their application in the detection of distributed denial‐of‐service (DDoS) attacks. The Shannon formula has been used extensively for this purpose for more than a decade. The use of the Tsallis and Rényi formulas in this context has also been proposed. Bhatia‐Singh entropy is a novel information metric with promising results in initial applications in this area. Ubriaco proposed an entropy function based on the fractional calculus. In this paper, flow size distribution was used as the input for detection. The type of DDoS attack is SYN flood, and simulation was used to obtain the input dataset. The results show that the Rényi and Bhatia‐Singh detectors perform better than the rest. Rényi and Tsallis performed similarly with respect to the true positive rate, but Rényi had a much lower false positive rate. The Bhatia‐Singh detector had the best true positive rate but a higher false positive rate than Rényi. The Ubriaco detector performed similar to the Shannon detector. With respect to detection delay, Tsallis, Ubriaco, and Shannon produced similar results, with a slight advantage associated with the Ubriaco detector, while Rényi and Bhatia‐Singh had a larger detection delay than the former three.     

基于Tsallis交叉熵快速迭代的河流遥感图像分割

为了使河流遥感图像分割的精度和速度进一步提高,本文提出了一种基于二维Tsallis交叉熵快速迭代的河流遥感图像分割方法。鉴于现有的Tsallis交叉熵阈值法运算效率不够高,首先提出了一维Tsallis交叉熵阈值选取的快速迭代算法;然后导出了基于灰度级—邻域平均灰度级直方图的Tsallis交叉熵阈值选取公式,以进一步提高分割精度,并采用递推方式计算阈值选取准则函数中的中间变量,避免其重复运算,加快运算速度;最后,提出了二维Tsallis交叉熵阈值选取的快速迭代算法,推导出相应的公式,大大减少了运算量。大量实验结果表明,与近年来提出的4种阈值分割方法相比,本文方法在对河流遥感图像的分割效果及运行时间上均有明显优势,是河流检测与类型识别系统中可选择的一种快速有效的分割方法。      

Generalization of Shannon-Khinchin axioms to nonextensive systems and the uniqueness theorem for the nonextensive entropy

Tsallis entropy, one-parameter generalization of Shannon entropy, has been often discussed in statistical physics as a new information measure. This new information measure has provided many satisfactory physical interpretations in nonextensive systems exhibiting chaos or fractal. We present the generalized Shannon-Khinchin axioms to nonextensive systems and prove the uniqueness theorem rigorously. Our results show that Tsallis entropy is the simplest among all nonextensive entropies. By the detailed comparisons of our axioms with the previously presented two sets of axioms, we reveal the peculiarity of pseudoadditivity as an axiom. In this correspondence, the most fundamental basis for Tsallis entropy as information measure is established in the information-theoretic framework.     

基于混沌PSO或分解的二维Tsallis交叉熵阈值分割(英文)

The segmentation effect of Tsallis entropy method is superior to that of Shannon entropy method, and the computation speed of two-dimensional Shannon cross entropy method can be further improved by optimization. The existing two-dimensional Tsallis cross entropy method is not the strict two-dimensional extension. Thus two new methods of image thresholding using two-dimensional Tsallis cross entropy based on either Chaotic Particle Swarm Optimization (CPSO) or decomposition are proposed. The former uses CPSO...     

基于Tsallis熵的自适应红外图像边缘检测方法

针对红外图像边缘检测这一难题,结合红外图像的特点,提出了基于Tsallis熵的自适应红外图像边缘检测方法.该方法分别计算图像子空间的边缘与非边缘的Tsallis熵,根据子空间最优Tsallis熵,构造出子空间最佳阈值的评价函数,根据评价函数,选择不同方向的边缘检测模板,增强了图像的边缘信息,从而避免了单一模板造成的边缘丢失现象.实验结果表明,与传统的边缘检测方法相比,该方法对于红外图像可以最大程度上抑制噪声,有效地提高图像的边缘检测效果.     

On q-logistic and related models

Numerous types of asymmetrical distributions such as the logistic, Weibull, gamma, and beta distributions have been used for modeling various random phenomena such as those encountered in data engineering, pattern recognition, and reliability assessment studies. Several generalizations of the logistic distribution, and certain related models, are proposed in this paper. The corresponding density functions involve an additional parameter, denoted by q, which allows for increased flexibility for modeling purposes; in fact, the larger this parameter is, the lower the mode of the resulting distribution will be. Generalizations of the type-1 and type-2 beta distributions are introduced, along with their logistic-type counterparts; the moments and cumulants of the latter are also derived. Other extensions are discussed including a q-analog of the generalized type-2 beta model, a q-extended generalized logistic distribution, and q-analogs of generalizations of the Dirichlet distribution. As is shown graphically, the proposed univariate distributions can generate a wide array of unimodal or symmetric bimodal curves.     

边缘引导和轮廓约束下的跨域香农熵最大化导向的自动阈值选取方法

为了处理诸如高斯、伽马、极值、瑞利、均匀或贝塔等基本灰度分布情形下的阈值选取难题,本文提出了一种跨域香农熵最大化导向的自动阈值选取方法.该方法利用不变的引导边缘图像和变化的约束轮廓图像共同构造出一系列持续变化的一维灰度直方图,并采用香农熵作为熵计算模型,从而得以跨越图像中若干局部区域去计算跨域香农熵,并以最大跨域香农熵对应的阈值作为最终阈值.在40幅合成图像和50幅真实世界图像上的实验结果表明,该方法虽然在计算效率方面不优于Masi熵阈值方法、Tsallis熵阈值方法、局部香农熵阈值方法和迭代三类阈值方法,但在分割适应性方面有显著增强,且在误分割率方面有显著下降.     

Fast and accurate image registration using Tsallis entropy and simultaneous perturbation stochastic approximation

The Tsallis measure of mutual information is combined with the simultaneous perturbation stochastic approximation algorithm to register images. It is shown that Tsallis entropy can improve registration accuracy and speed of convergence, compared with Shannon entropy, in the calculation of mutual information. Simulation results show that the new algorithm achieves up to seven times faster convergence and four times more precise registration than using a classic form of entropy.     

海森堡模型中概率及相应熵的计算分析

介绍海森堡模型的不同位型[N,n] （N为海森堡链总格点数, n为格点中自旋向下的电子数）中的体现本征值获取难易程度的本征值获取概率及其相应信息熵（香农所定义的）和体现模型体系关联程度的自旋向下电子发现概率、每一粒子的von Neumann及体系的平均von Neumann熵,可为量子计算与信息传递提供启示性信息。研究结果：(1)事件发生概率大于（小于）50%时,信息熵随概率增加而减小（增加）。(2)不同位型[N,n],当n（N）同, N（n）增加时：本征值获取概率减小,其相应的信息熵正确反映本征值获取的难易程度;模型参数一定时,格点中自旋向下电子发现概率与每一粒子的von Neumann及体系的平均von Neumann熵都分别减小（增加）。(3)位型[N,n]相同时, 每一粒子的von Neumann及体系的平均von Neumann熵随参数变化时出现拐点,显示体系发生量子相变的信息。(4)同位型[N,n]且同参数时处于海森堡链对称位置粒子的von Neumann熵相同。     

蝙蝠优化的二维Tsallis熵多阈值SAR图像分割

针对智能优化SAR图像分割算法存在计算量大、易陷入局部最优、分割精度不够等问题,融合蝙蝠算法和二维Tsallis熵多阈值,提出了一种蝙蝠优化的二维Tsallis熵多阈值SAR图像分割算法。算法利用立方映射均匀化初始蝙蝠种群,引入Levy飞行特征加强算法跳出局部最优能力,使用Powell局部搜索加快算法收敛等3方面改进蝙蝠算法;同时将二维Tsallis熵单阈值分割方法扩展到多阈值分割,建立基于多阈值的选取方法,并结合改进的蝙蝠算法,将二维Tsallis熵多阈值应用于SAR图像分割中。仿真结果表明,与其他智能优化分割算法相比,本分割算法在边缘处理和分割精度上都有明显优势。     

A long-range dependent model for Internet traffic with power transformation

Internet traffic has been shown to have long-range dependence, and is often modeled by using the fractional Gaussian noise model. The fractional Gaussian noise model can capture the autocorrelation of a real trace, but cannot fit the marginal distribution when the trace has a non-Gaussian marginal distribution. In this letter, we use the inverted Box-Cox transformation to establish a long-range dependent Internet traffic model that can simultaneously capture both the long-range dependence parameter and the marginal distribution of a real trace.     

On minimizing distortion and relative entropy

A common approach for estimating a probability mass function w when given a prior q and moment constraints given by Aw/spl les/b is to minimize the relative entropy between w and q subject to the set of linear constraints. In such cases, the solution w is known to have exponential form. We consider the case in which the linear constraints are noisy, uncertain, infeasible, or otherwise "soft." A solution can then be obtained by minimizing both the relative entropy and violation of the constraints Aw/spl les/b. A penalty parameter /spl sigma/ weights the relative importance of these two objectives. We show that this penalty formulation also yields a solution w with exponential form. If the distortion is based on an /spl lscr//sub p/ norm, then the exponential form of w is shown to have exponential decay parameters that are bounded as a function of /spl sigma/. We also state conditions under which the solution w to the penalty formulation will result in zero distortion, so that the moment constraints hold exactly. These properties are useful in choosing penalty parameters, evaluating the impact of chosen penalty parameters, and proving properties about methods that use such penalty formulations.     

Two units connected in series with general bulk service and random breakdown in unit 2

We study a queueing model consisting of two units 1 and 2 connected in series with a finite intermediate waiting room. The customers in the buffer are served according to a general bulk service rule with exponential times. Unit 2 is in the up state for a random interval of time following an exponential distribution with parameter α and the distribution of time spent in the down state is exponential with parameter β. Here we obtain the steady state probability vector for the number of customers in the queue.     

Context-Sensitive Hidden Markov Models for Modeling Long-Range Dependencies in Symbol Sequences

The hidden Markov model (HMM) has been widely used in signal processing and digital communication applications. It is well known for its efficiency in modeling short-term dependencies between adjacent symbols. However, it cannot be used for modeling long-range interactions between symbols that are distant from each other. In this paper, we introduce the concept of context-sensitive HMM. The proposed model is capable of modeling strong pairwise correlations between distant symbols. Based on this model, we propose dynamic programming algorithms that can be used for finding the optimal state sequence and for computing the probability of an observed symbol string. Furthermore, we also introduce a parameter re-estimation algorithm, which can be used for optimizing the model parameters based on the given training sequences.