## 3 The Procedures of Protein Feature Representation and Classification
### Feature Representation  
The GPCR protein sequences are initially represented using the pseudo amino acid composition (PseAAC) method. With this method, the protein sequences are transformed into numeric. The following steps are taken to convert protein sequences to numerical features using the PseAAC method. Protein P can be written as follows in the PseAAC method:  
\[\text{PseAAC}=\text{P}_{1},\,\text{P}_{2},...,\,\text{P}_{20},...,\,\text{P} _{\cdot}, \tag{1}\]  
where  
\[\wedge=20+n\lambda. \tag{2}\]  
The frequency of amino acids in the sequence is represented by the first twenty elements in Equation (1), from \(\text{P}_{1}\) to \(\text{P}_{20}\). The amino acid correlation rank is represented by the symbol \(\lambda\) where \(\lambda=1,\,\ldots,\,m\). This rank value is an integer that is non-negative and less than the length of the protein sequence. The number of physiochemical characteristics of the amino acid employed is represented by the n value. This study used the same six physiochemical properties of amino acids as [66], namely the mass properties, the pK group from \(\alpha-NH_{3}^{+}\), pK from the \(\alpha\)-COOH group, and the pI group at 25 \({}^{\circ}\)C. The weighting factor w was made to put more weight on PseAAC than amino acid composition. PseAAC features can be retrieved from the PseAAC web server [67]. There are two different kinds of PseAAC, type I and type II. Researchers [66] found that PseAAC type II works better as a protein feature. This is because, unlike type I PseAAC, type II PseAAC takes into account the impact of physiochemical characteristics in the calculation [68]. According to Equation (2), the total number of features created for each protein sequence based on the parameters determined above is =20 + 6 \(\times\) 25, or 170 features. Following that, the PseAAC features are normalised, with each feature scaled to the limit range [0, 1] so that the feature value is in a small range. Normalising features prevents large values from dominating small values. Additionally, this helps simplify the computation process. The formula for feature normalisation is in Equation (3):  
\[x_{N}=\frac{x_{0}-min_{0}}{max_{v}-min_{v}}\, \tag{3}\]  
where \(x_{N}\) is the value of the feature after the normalisation process, \(x_{0}\) is the original value of the feature, \(max_{v}\) and \(min_{v}\) are the maximum value and minimum value of the entire features. The wavelet transform is then calculated using the normalised PseAAC features. The scaling function and the wavelet function are the two most important functions in DWT. Using the scaling function, it is possible to obtain low-frequency features for the normalised PseAAC. Approximation coefficients are low-frequency features that contain global information. The definition of the scaling function \(\phi(x)\) is as in Equation (4) [69]:  
\[\phi_{a,b}(x)\equiv\frac{1}{\sqrt{|a|}}\phi\bigg{(}\frac{x-b}{a}\bigg{)}, \tag{4}\]where \(x\), \(a\), and \(b\) denote the data, scaling, and translation parameters, respectively. To obtain detail coefficients or local features, the wavelet function \(\psi(x)\) is used. The definition of the wavelet function is in Equation (5) [69]:  
\[\psi_{a,b}(x)\equiv\frac{1}{\sqrt{|a|}}\psi\bigg{(}\frac{x-b}{a}\bigg{)}, \tag{5}\]  
where \(x\), \(a\), and \(b\) denote the data, scaling, and translation parameters, respectively. To form the coefficient vector, both approximation and detail coefficients are combined depending on the wavelet family and decomposition level chosen. The detail coefficients are maintained in this study because they may provide useful information. Different wavelet family types can be used to implement DWT. Scaling and wavelet functions differ for each type of wavelet family [34]. It is essential to choose the right kind of family in order to make effective use of DWT [70]. In this particular research, there are 82 different wavelet families that can be used to describe GPCR proteins.