动力学特征，这时候CPS负载更像是幂函数而不是指数函数。第二项描述了幂指数是如何成为表示CPS组件之间相互作用的函数的。更准确地说，它叙述了随机过程在时间t内获得前一项实现（即标度项a/y）的加权和（即经由广义函数g(y,t)）的值a的概率。支持使用广义函数g(y,t)的原因是，在很多实际情况下，随机过程a(t)的多重分形性，如果它存在的话，会被一系列因素影响，例如，视频包的长度可变是因为输入流的变动性。
To better understand the advantages such a formalism brings from a modeling perspective, we can multiply with ak both terms in Equation 1 and integrate over the space of all magnitudes of a ,and obtain, under various constraints on the scaling distribution g(y,t), a dynamic equation for the higher-order moments Mk(t) of a(t):
为了更好地从建模的角度理解这种方法带来的好处，我们可以在方程1的每一项上都乘以ak并做积分，然后在标度广义函数g(y,t)的约束下获得一个a的高阶矩动态方程Mk(t)：
The nonlinear relationship of the exponent t(k)： 指数t(k)的非线性关系是：
in the expression of the higher-order moments Mk(t)in Equation (2) represents a multifractal signature (i.e., a bell-like distribution of fractal dimensions as shown in Figure 3b).Simply speaking, it implies that the distribution of a ( t ) consists of a superposition of some power law functions. From a practical standpoint, this behavior requires new control strategies based on nonlinear state equations.
在方程2中的高阶矩方程Mk(t)代表了一个多重分形信号（即一个钟形的如图3b中所示的分形维数的广义函数）。简单来说，它表示了关于a(t)的广义函数是由一些幂函数的叠加组成的。从实践的角度，这个特性要求新的控制策略要符合非线性状态方程。
A different modeling approach to Equation 1 is to capture the fractal characteristics of CPS workloads via fractional derivatives. Generally speaking, the fractional derivative5,10 of a probability
distribution P(a, t) consists of a convolution between the distribution P(a,t)of a certain metric(e.g., communication volume) and a memory kernel(e.g., power laws for capturing memory effects) characterizing the CPS workload.
方程1的另一种建模方法是通过分数阶微分获得CPS负载的多重分形特征。一般来说，P(a,t)的概率分布的分数阶微分是由介于具有某一度量标准的广义函数P(a,t)和具有CPS负载特性的内存内核之间卷积组成。
From a practical standpoint, we can capture the monofractal behavior exhibited in Figure 3a by using a single space/time fractional derivative which relies on a memory kernel with a single power law exponent. By way of contrast, a dynamical equation for P(a, t)capturing its multifractal behavior (see Figure 3b) requires a weighted sum of fractal (fractional) derivatives.
从实践的角度，我们可以使用单一的空间或时间的依赖于单个幂指数记忆核的分数阶导数来掌握图3a中显示的单分形特性。通过对照，P(a,t)的动态方程需要加权过的分数阶导数之和来获得它的多重分形特性（见图3b）。
Figure 5. From real-world data to models: By analyzing the higher-order moments of process a(t), both in time and frequency, we can decide whether a linear or nonlinear model is more appropriate. Moreover, if the distribution of interevent times is exponential, then classical linear systems theory is applicable (a). Instead, if the interevent times follow a power law, then a fractional differential equation may be used (b). For nonzero higher-order moments and multifractal behavior, Equation 1 may prove an adequate model (c).
To better emphasize the distinction between previous approaches and a statistical physics-inspired approach to CPS workload characterization, we present in Figure 5 a simple methodology for determining which model is suitable, given the statistical nature of network traffic. To simplify the analysis, the basic question to ask is whether the generic stochastic process a(t) in Figure 5 can be modeled accurately via a time-linear relationship.
为了更加强调对于CPS负载的特征描述的两种方法，以前的方法和现在的统计物理激发方法，之间的不同，我们在图5中展示了一种决定哪种建模更加合适的方法论，图中给出了网络通信量的统计属性。为了简化分析，需要讨论的基本问题是图5中的一般随机过程是否能够通过一个线性的关系来准确的建模。
Traditional tools for detecting the linearity of any process a(t) are based on investigating the higher-order statistics of the stochastic process a(t) both in time (e.g., third-order moment) and frequency (e.g., bispectrum or bicoherence) domains. For instance,we can consider the investigation of bicoherence b(f1,f2)：
探测任何过程的线性的传统工具都是基于对随机过程的高阶统计量的研究，无论是时域（例如三阶矩阵）还是频域（例如双频谱或双相干谱）。举个例子，我们可以认为对双相干谱的研究b(f1,f2)：
which is proportional with the bispectrum B(f1, f2)： 其中的比例项双频谱B(f1,f2)的表达式是:
That is, the bicoherence b (f1,f2) is proportional with the two-point Fourier
transform of
the
third-order
moment and inversely
proportional with the power spectrum S (f1). As Figure 5 shows, computing bicoherence (Equation 4) needs
换言之，双相干谱等于比例项三阶矩阵的二维傅里叶变换乘以比例项功率谱S(f1)的倒数。就像图5中显示的，计算双相干谱（方程4）需要知道：
（1）the third-order moment M 3( t1, t2) (i.e., the joint correlation of three shifted versions of the CPS workload a ( t ), a ( t+t1), and a ( t+t2) for two time lags, say t1=40 and t2=90),
（2）the 2D Fourier of M3(t1,t2) obtaining the bispectrum B(f1,f2) (Equation 5), and
（3）normalizing the bispectrum B ( f1, f2) with respect to the power spectrum—that is,
(1)三阶矩阵M3(t1,t2)（即，CPS负载a ( t ), a ( t+t1), a ( t+t2)的三个转变版本的联合相关，其中t1,t2代表时间间隔，可以假设为t1=40 and t2=90。
(2)M3(t1,t2)的二维傅里叶值包括双频谱B(f1,f2)（方程5），和 (3)使关于功率谱的双频谱正规化，功率谱的表达式是，
If the bicoherence remains constant for any two frequencies f1 and f2 , then a linear relationship can be assumed as a good approximation model for process a( t). Otherwise, the stochastic process a (t ) is said to exhibit a nonlinear behavior.
如果双相干谱在任何两个频率下都保持恒定，那么我们可以假设一个线性关系作为过程a(t)充分近似的模型。否则，随机过程a(t)就表现为非线性特征。
If, in addition to linearity, the distribution of inter-event times (i.e., the time between two consecutive changes in the magnitude of a(t)) is exponential, then the stochastic process can be described by a classical master equation and the linear time-invariant (LTI) system theory can be applied to study the system at hand. In contrast, if the distribution of inter-event times follows a power law, then the stochastic process a(t)is said to possess a fractal behavior that can be modeled via a fractional master equation with a single fractal exponent (i.e., the classical integer first-order time derivative becomes a fractional derivative of order 0 如果除了线性分布，时间间隔的分布（即介于两个连续变化的过程a(t)之间的时间大小）呈指数形式，这是随机过程可以用经典主方程来描述，并且线性时不变系统理论将可以应用到系统的研究中。相比之下，如果时间间隔的分布遵循幂次定律，那么随机过程a(t)可以说是拥有了分形特性，可以通过具有单个标度指数（即经典整数一阶时间导数变成了阶数介于0到1的分数阶导数）的分数阶主方程来建模。为了掌握随机过程的的非线性和多重分形特征，我们可以用分布函数来反映多重分形谱中发现的每一个分形维数的重要性。很明显地，线性时不变解析和控制方法并不适用于分析这种系统，因为它们在收敛到理想状态是非常缓慢，或者根本就没法收敛。因此，无论是单分形还是多重分形随机过程，都需要基于高阶矩阵解析的控制范式，这些我们将在接下来讨论。 Implication of the new formalism in CPS design 在CPS设计中新形式体系的含义 At this stage, a natural issue to address is: What are the main implications of mono- or multifractal behavior from a