# Steganalysis of Quantization Index Modulation Steganography in G.723.1 Codec
## 4 Quantitative Model
### PCA Dimensionality Reduction  
The steganographic detection feature has too many dimensionalities, which not only increases the classifier training time and prediction time, but also even causes \"dimensionality disaster\" [19]. At the same time, the training error will be too small which is not conducive to the classifier to estimate the statistical properties of the sample. In order to greatly reduce the amount of calculation, The A and M are subjected to dimensionality reduction by the PCA.  
The principal component analysis method first appeared in 1901 and was originally applied to non-random variables. With the advancement of science and technology and the development of computer software, the principal component analysis method has been extended to more fields, such as system evaluation, principal component regression analysis, selection of variable subsets, etc. PCA is based on the loss of information as little as possible, and uses the idea of dimensionality reduction to manage a large number of variables with correlation, which greatly reduces the amount of calculation, and the obtained variables are more effective for analyzing problems. These variables that reflect most of the characteristics of the original data are called principal components, and the information contained in each principal component does not overlap each other. The PCA calculation step [7] is as follows.  
Primitive matrix  
\[\mathbf{Y}=\left[\begin{array}{cccc}y_{11}&y_{12}&\cdots&y_{1p}\\ y_{21}&y_{22}&\cdots&y_{2p}\\ \vdots&\vdots&\ddots&\vdots\\ y_{N1}&y_{N2}&\cdots&y_{Np}\end{array}\right] \tag{11}\]Calculating the covariance matrix  
\[cov\Big{(}y_{i},y_{j}\Big{)}=\frac{1}{n-1}{\sum_{k=1}^{n}\Big{(}y_{k,i}-\overline {y_{i}}\Big{)}\Big{(}y_{k,j}-\overline{y_{j}}\Big{)}} \tag{12}\]  
where  
\[i,j\in[1,p],\overline{y_{i}}=\frac{1}{n}{\sum_{k=1}^{n}y_{ki}},\overline{y_{j} }=\frac{1}{n}{\sum_{k=1}^{n}y_{kj}} \tag{13}\]  
The covariance matrix \(C\) can be expressed as  
\[C=\left[\begin{array}{cccc}x_{11}&x_{12}&\cdots&x_{1p}\\ x_{21}&x_{22}&\cdots&x_{2p}\\ \vdots&\vdots&\ddots&\vdots\\ x_{p1}&x_{p2}&\cdots&x_{pp}\end{array}\right] \tag{14}\]  
where  
\[x_{ij}=cov\Big{(}y_{i},y_{j}\Big{)} \tag{15}\]  
The eigenvalues of \(C\) and the corresponding eigenvectors are calculated. The feature vector here is the principal component. The selection of principal components is based on the contribution rate introduced. The cumulative contribution rate is required to be more than 85%, which can reflect most of the characteristics of the original data. It is found through experiments that when the number of selected feature values is 120, the contribution rate can reach more than 85%.  
In order to test the impact of PCA dimensionality reduction, we carry out the following experiments. We randomly select 100 frames of speech and respectively calculate the change rate of \(A\) and \(M\). The results are expressed in \(\gamma 1\) and \(\gamma 2\) respectively. The change rates are shown in Figure 3 [18].  
As can be seen from Figure 3, the rate of change \(\gamma 1\) and \(\gamma 2\) are both above 65%. Obviously, the dimensionality matrix is very sensitive to QIM steganography. It is fully shown that the matrix not only makes the calculation work simple but also can well reflect the long-term characteristics and the correlation after the dimensionality reduction.