## 3 System Optimization Model
### One-Dimension Normal Cloud Model
Besides, with another parameter \(He\), they can represent the extent of random dispersion of cloud drops. * \(He\) is the uncertainty degree of entropy \(En\), which reflects the randomness and fuzziness of entropy and is often used for measuring the uncertainty of entropy and determined by randomness and fuzziness of entropy. The overall features of the qualitative concepts can be denoted as cloud feature vector \(C(Ex,En,He)\). The normal cloud is a kind of uncertain transformation model from a qualitative concept to a quantitative representation. Correspondingly, the positive normal cloud generator establishes the mapping relation between the qualitative concept and the quantitative data. From Figure 1, we can see that the normal cloud has obvious geometric features, which can be used to discuss the properties of the normal cloud. The overall characteristics of the cloud can be depicted by regression curves and principal curves, which is from the expectation along the vertical direction and the orthogonal direction, respectively. Since obtaining the analytical formula is complicated it can only be obtained by a linear approximation method. The target area is divided into a number of square grids, and then the historical data monitored by nodes in each grid can be collected to construct the cloud model. In order to measure the similarity between the nodes dispersed in adjacent fields, the similarity degree of the expected curve is introduced. Assuming the sensor network is located in the square area with the length \(M\) of the sides, the region  
Figure 1: A typical normal cloud model. is divided into \(m\times m\) grids where the nodes are analyzed individually. Suppose \(P=\{p_{1},p_{2},\ldots,p_{n}\}\) is the data set collected at \(n\) time intervals and \(T\) is a qualitative concept of \(P\), and there exists a random value of \(\eta\in\mu_{T}(p_{i})\), \(\eta\in[0,1]\). In other words, \(\eta\) is the distribution of data interval of \(p_{i}\) to membership grade \(T\). To analyze the random cloud drop \(x\), normal random number \(En^{\prime}\) with expected value \(En\) and variance of \(He^{2}\) is generated by the digital features of the cloud. The probability density function \(f_{En^{\prime}}(x)\) can be expressed as:  
\[f_{En^{\prime}}(x)=\frac{1}{\sqrt{2\pi}He}\exp\{-\frac{(x-En)^{2}}{2 He^{2}}\} \tag{1}\]  
Normal cloud \(X\) is subject to normal distribution with the expected value \(Ex\) and variance of \(En^{\prime 2}\), then the probability density of \(X\) can be given:  
\[f_{X}(x)=\frac{1}{\sqrt{2\pi}\left|En^{\prime}\right|}\exp\{-\frac{(x-En)^{2}} {2 He^{\prime 2}}\} \tag{2}\]  
Next, we can get:  
\[f_{X}(x)=\int_{-\infty}^{+\infty}\frac{1}{\sqrt{2\pi}He\left|y\right|}\exp\{- \frac{(x-Ex)^{2}}{2y^{2}}-\frac{(y-En)^{2}}{2 He^{2}}\}dy \tag{3}\]  
Consequently, the function is a density function of normal distribution \(N(Ex^{2},En^{2})\), \(He=0\). Next, the degree of certainty \(Y_{i}\) can be regarded as a sample of random variable \(Y_{i}=e^{-\frac{(x_{i}-Ex)^{2}}{2(En^{\prime}_{i})^{2}}}\), and the probability density of \(Y_{i}\) can be given as:  
\[f_{Y_{i}}(y)=\frac{d(g_{Y_{i}}(y))}{dy}=\frac{1}{\sqrt{-\pi\ln y}} \tag{4}\]  
where \(g_{Y_{i}}=P\{Y\leq y\}=P\left\{e^{-\frac{(x_{i}-Ex)^{2}}{2(En^{\prime}_{i})^ {2}}}\leq y\right\}\). If \(x_{i}\) corresponds to the \(N\) dimension domain space, \((x_{i}y_{i})\) is a point of \(N+1\) dimension domain space and the set of all points that form cloud \((X,Y)\). While \(x_{i}\) is one-dimensional, the normal cloud \((X,Y)\) can be a two-dimensional random variable with the joint probability density:  
\[f_{X,Y}(x,y) =f_{Y}(y)f_{X}(x|Y=y)\] \[=\left\{ \begin{array}{ll}\frac{1}{2\pi He}\times\frac{1}{\ln y }\times\exp\{\frac{(x-Ex-\sqrt{-\pi\ln y}En)^{2}}{4 He^{2}\ln y}\},&(Ex\leq x< +\infty,0<y\leq 1)\\ \frac{1}{2\pi He}\times\frac{1}{\ln y}\times\exp\{\frac{(x-Ex+\sqrt{-\pi\ln y }En)^{2}}{4 He^{2}\ln y}\},&(-\infty<x<Ex,0<y\leq 1)\end{array} \right.