## 2 Fundamental Theories
### Adaptive Parameter Optimization  
#### 3.2.1 Sine Cosine Algorithm  
Exploration and exploitation [35] are two phases in which the sine cosine algorithm (SCA) processes the optimization problem. In the phase of exploration, a set of random solutions are initiated as the outset of optimization process. With a strong randomness, SCA is able to search for feasible solutions quickly in the searching space. In the phase of exploitation, the random solutions change gradually, and the randomness is distinctly weaker than that of exploration phase, which is conducive to a better local searching.  
Let \(Z_{i}=(Z_{i1},Z_{i2},\ldots,Z_{il})^{T}\) be the position of \(i\)th (\(i=1\), \(2\),..., \(M\)) individual, whereupon each solution of the optimization problem corresponds to the position of corresponding individual in the searching space, where \(l\) is the dimension of individuals. The best position of individual \(i\) is \(P_{i}=(P_{i1},P_{i2},\ldots,P_{il})^{T}\). During iterations, the position of individual \(i\) will be updated by the following Equations [35]:  
\[\begin{array}{l}Z_{i}^{k+1}=Z_{i}^{k}+r_{1}\times\sin(r_{2})\times\left|r_{3 }P_{i}^{k}-Z_{i}^{k}\right|\\ Z_{i}^{k+1}=Z_{i}^{k}+r_{1}\times\cos(r_{2})\times\left|r_{3}P_{i}^{k}-Z_{i}^{ k}\right|\end{array} \tag{24}\]  
where \(Z_{i}^{k}\) is the position of individual \(i\) in \(k\)th iteration, \(r_{1}\), \(r_{2}\), and \(r_{3}\) are all random values.  
The above equations are synthesized to:  
\[Z_{i}^{k+1}=\left\{\begin{array}{ll}Z_{i}^{k}+r_{1}\times\sin(r_{2})\times \left|r_{3}P_{i}^{k}-Z_{i}^{k}\right|,&r_{4}<0.5\\ Z_{i}^{k}+r_{1}\times\cos(r_{2})\times\left|r_{3}P_{i}^{k}-Z_{i}^{k}\right|,&r_ {4}\geq 0.5\end{array}\right. \tag{25}\]  
where \(r_{4}\) is a random value within [0, 1].  
As the above equations show, the updating equation has four main parameters, i.e., \(r_{1}\), \(r_{2}\), \(r_{3}\) and \(r_{4}\). The parameter \(r_{1}\) contributes to determine the position's region of individual \(i\) at next iteration. The parameter \(r_{2}\) is a random number in the scope of [0, 2\(\pi\)], which defines distance that next movement should be towards. To stochastically emphasize (\(r_{3}>1\)) or deemphasize (\(r_{3}<1\)) the  
Figure 2: The flowchart of support vector data description with improved decision strategy (ID-SVDD).  
effect of the best position of individual, a random weight \(r_{3}\) within [0, 2] is brought in the equations. Finally, the parameter \(r_{4}\) is a random number in the range of [0, 1] to switch fairly between sine and cosine components, specifically when \(r_{4}<0.5\), sine component dominates iterations of the position of individual \(i\), otherwise cosine component dominates iterations.  
To seek feasible solutions in the searching space and eventually find the global optimum, exploration and exploitation phases should be in a balanced state. For this reason, the amplitudes of the sine and cosine functions are adaptively adjusted by changing \(r_{1}\) as follows [35]:  
\[r_{1}=a-t\frac{a}{T} \tag{26}\]  
where \(T\), \(t\) and \(a\) are the maximum number of iterations, the current number of iterations and a constant respectively.