切比雪夫不等式

Through above study, the theoretical analysis of the inherent fault tolerance of feed-forward neural networks has been extended from hard-limit activation functions to differentiable activation functions; 2. By using above method, fault tolerance of discrete Hopfield feedback neural networks have been also analyzed. It means that the above method can be applied not only for feed-forward neural networks but also for feedback neural networks; 3. For the feed-forward neural network with the Sigmoid activation function, Chebyshev Inequality method was presented.

这项研究将以前仅限于对硬限幅作用函数前向神经网络固有容错性能的理论分析推广到具有可微作用函数的前向神经网络； 2 针对离散型Hopfield反馈神经网络，利用上面提出的方法，对其同样进行了容错性分析，得出许多有用的结论和计算公式，说明上面提出的方法不仅适用于前向神经网络，同时也适用于反馈神经网络； 3 针对具有Sigmoid作用函数的前向神经网络提出了一种切比雪夫不等式法。

The famous Chebyshev's Inequality is usually used when estimating the boundary from above or below of probability .

在估计某些事件的概率的上下界时，常用到著名的切比雪夫不等式。

And then , the author analyzes its complete and necessary condition for foundation of Chebyshev's Ineuquality. Furthermore, the paper makes a demonstration again for Chebyshev's Inequality with the method of modern probability.

最后给出了切比雪夫不等式其等号成立的充要条件，并用现代概率方法重新证明了切比雪夫不等式。

Moreover, with its help people can also evaluate the fault-tolerant performance of ANN. In this paper, Chebyshev Inequality method, as a kind of evaluating thought, was firstly concerned in the faulttolerance analysis of ANN; 4. The variation of faulty behaviors of ANNs during the course of learning was studied by taking the discrete Hopfield neural network and Hebb learning algorithm as examples.

这种方法首次将切比雪夫不等式引入人工神经网络容错性分析当中，从而提出了对人工神经网络容错性分析采用估算的思想； 4 以离散型Hopfield神经网络和Hebb学习算法为例，着重研究了在学习过程中人工神经网络故障性能的变化。

Second, On the basis of analysis of density distribution of sequence points on the chaotic orbit of one-dimension chaotic mapping, a mutative interval chaotic optimization algorithm is presented in search for the solution optimums of the non-linear constrained problems with multi-variables and multi-peak values. With the objective function probability measurement approaching a limit 1, the convergence of the algorithm on the global optimal solution is proofed by aid of Chebyshev inequality.

二、基于一维混沌映射混沌点集的概率测度分析，提出了一种适于解决多变量多峰值非线性约束问题的新的混沌优化算法——变区间混沌优化算法，并应用切比雪夫不等式论证了算法依概率收敛于全局最优解。

In modeling heavy tails,we focus on multivariate t distributions and derive the moment generating function of the quadratic approximation.Then we use Fourier transform,Chebychev's Inequality and numerical technique inversion to approximate VaR value of portfolio,compared with Cornish-Fisher Expansions normal model and Delta-Normal model.

在推导出多个外汇期权的投资组合的二次近拟的矩母函数表达式基础上，本文使用傅里叶变换、切比雪夫不等式、数值转换计算求出投资组合的VaR的值，并和基于多元正态分布Cornish-Fisher模型以及基于Delta-正态模型计算所得的VaR值了进行比较。

The split between the two groups can hardly be papered over.

这两个团体间的分歧难以掩饰。

This approach not only encourages a greater number of responses, but minimizes the likelihood of stale groupthink.

这种做法不仅鼓励了更多的反应，而且减少跟风的可能性。