## 2 Materials and Methods
### Improved Fisher Kernel Function  
The research on the kernel function of support vector machine is ultimately served for the discriminant function of support vector machine. Therefore, before improving Fisher kernel function, this section first addresses the discriminant function of support vector machine under normal circumstances, followed by the improvement of the Fisher kernel function, and combines it with the discriminant function of support vector machine. Fisher kernel similarity function, based on the PLSA model of Section 2.2, is derived in this section. Based on the generated model, a general method, called Fisher kernel algorithm, is proposed to extract the features of the promoter sequence. First, suppose there is an instance set \(X_{i}\), \(X_{i}\) and \(X\) whose similarity is represented by the kernel function \(K(X_{i},X)\), it corresponds to the label \(S_{i}\) (which can be taken as \(\pm 1\)). Corresponding to example \(X\), Label \(\hat{S}\) is obtained by summing all \(S_{i}\) weights, and \(\hat{S}=\text{sign}(\sum_{i}S_{i}\lambda_{i}K(X_{i},X))\) can be obtained. In the above formula, the coefficient \(\lambda_{i}\) serves as a free parameter, and the kernel function \(K\), to some extent, also serves as a free parameter. An optimal coefficient \(\lambda_{i}\) must be found so that the label \(\hat{S}\) has the largest probability value. Small changes which occurred in solving this optimal problem may exert a great impact, such as the change from support vector machine to generalized linear model. Another important issue is related to the selection of kernel functions. Logical regression model is considered in the generalized linear model, because the probability of label S and a vector parameter \(\theta\) in the logistic regression model is shown by \(P(S|X,\theta)=\delta\big{(}S\theta^{T}X\big{)}\), where \(\delta(\cdot)\) denotes a logical function, \(\delta(z)=\left(1+e^{-z}\right)^{-1}\); and the parameter \(\theta\) can be obtained by maximizing the following formula:  
\[\log P(S_{i}|X_{i},\theta)+\log P(\theta)=\sum_{i}\log\delta\big{(}S\theta^{T} X\big{)}-\frac{1}{2}\theta^{T}\Sigma^{-1}\theta+c \tag{8}\]  
where \(c\) denotes independent of \(\theta\); in terms of the kernel function \(k(\cdot)\), the inner product form \(K(X_{i},X)=X_{i}^{T}\Sigma X\) is adopted. In fact, the form of the kernel function will be universal as long as the feature vectors from the samples can replace these features. The following discussion further discusses how to make a general kernel function more effective. Under the guidance of the \"Mercer\" theory, an effective kernel function can simply describe the inner product between the defined feature vectors, that is,  
\[K\Big{(}X_{i},X_{j}\Big{)}=\phi(X_{i})^{T}\phi(X_{j}) \tag{9}\]  
The feature vectors in the formula are derived from some specific mappings \(X\rightarrow\phi(X)\), with the definite requirement for kernel function in the application. The key now lies in how to derive it from the generated probability model generally. It is known from the previous description that automatically defining a kernel function means assuming a metric relationship in the example. Thus, it is necessary to define these metric relationships directly in the generated model \(P(X|\theta)\). Since the goal is classification, then a latent variable should be included into the generated model, which is the categorical variable \(S\). In order to make this idea more general, a parametric model class \(P(X|\theta)\) is taken into consideration, where \(\theta\subset\Theta\). This probability model is defined as a Riemannian manifold \(M_{\Theta}\) by the local metric given by [PERSON]'s information matrix I. Here,  
\[ \begin{split}& I=E_{x}\big{\{}U_{x}U_{x}{}^{T}\big{\}}\\ & U_{X}=\log P(X|\theta)\end{split}  \tag{10}\]  
This expectation exceeds \(P(X|\theta)\). To simplify the model, this limits the independence of the matrices \(I\) and \(U_{X}\) in the parameter set \(\theta\).