# Research on Intrusion Detection Based on an Enhanced Random Forest Algorithm
## 3 Cost-Sensitive Random Forest Algorithm  
### Introduction to Intrusion Detection  
The redundancy and high-dimensionality of intrusion-detection datasets can significantly impact classification algorithms. On one hand, high-dimensional datasets increase the computational complexity of classifiers and consume more storage resources. On the other hand, redundant features reduce model usability and may introduce noise and unnecessary information, affecting the classification performance. To mitigate the adverse effects of high-dimensional data, we introduce the Kernel Principal Component Analysis (KPCA) algorithm [33]. This approach maps the original dataset's samples to a higher-dimensional space and then projects and reduces the high-dimensional sample data while maximizing the variance in the projected data. This results in a reduction in data dimensions and the removal of redundant information. The working principle is as follows:  
Let each column in the sample be \(x_{i}\), the sample set is \(X=[x_{1},x_{2},\ldots,x_{N}]\). Now use a nonlinear map \(\phi\) of vector \(x_{i}\) in \(X\) to a higher dimensional space \(\tau\) (recorded as the  
Figure 1: Key steps and improvement ideas. dimension), obtain the \(D\times N\) new matrix for A \(\phi(X)=[\varphi(x_{1}),\varphi(x_{2}),\ldots,\varphi(x_{n})]\) deduces that the covariance matrix in \(\tau\) against \(\phi(X)\) is as follows:  
\[C_{\tau}=\frac{1}{N}\phi(X)[\phi(X)]^{T}=\frac{1}{N}\sum_{i=1}^{N}\varphi(x_{i} )[\varphi(x_{i})]^{T}\]  
The eigenvalues for solving the covariance matrix are as follows:  
\[C_{\tau}p=\lambda p\]  
where the \(D\)-dimensional column vector \(p\) is the weight vector of the feature space, which can be expressed as a linear combination of \(\varphi(x_{i})\), namely  
\[p=\sum_{i=1}^{N}a_{i}\varphi(x_{i})=\ \phi(X)a\]  
By defining the matrix \(K=[\phi(X)]^{T}\phi(X)\), you can solve for non-zero eigenvalues:  
\[Ka=\lambda a\]  
While the KPCA algorithm demonstrates advantages in handling non-linear relationships and complex data structures, it also has certain limitations. The algorithm's kernel function parameters need to be adjusted according to the data's characteristics, and different parameter choices can result in varying dimensionality reduction outcomes. To address this, we integrate the Bald Eagle Search (BES) optimization algorithm [34], which effectively explores the solution space, reduces the reliance on initial conditions, and completes parameter optimization for KPCA. The utilization of the BES algorithm for optimizing the KPCA involves the following six steps:  
(1) Initialize the algorithm parameters, and initialize the number of condor population \(npop\), the number of algorithm iterations \(Maxlt\), and the fitness function \(fobj\). \[BestSol.\cos t=\inf\]  
(2) Objective function evaluation to calculate the fitness of each individual's current position. \[pop.\cos t(i)=fobj(pop.pos(i,:))\]  
(3) Select the search space, randomly select the search area, and determine the optimal search position as \(P_{i,new}\). \[P_{i,new}=P_{best}\cdot\alpha\cdot rand\cdot(P_{mean}-P_{i})\]  
(4) Search for space prey, find the best dive position, update the condor position. \[P_{i,new}=P_{i}+x(i)\cdot(P_{i}-P_{mean})+y(i)\cdot(P_{i}-P_{i+1})\]  
(5) Diving to capture prey, a rapid dive from the best position in the search space to the target prey, the rest of the population also moves to the best position and attacks the prey. rand is a random number between 0 and 1. \[P_{i,new}=rand\cdot P_{best}+P_{i}+x(i)\cdot(P_{i}-c_{1}\cdot P_{mean})+y(i) \cdot(P_{i}-c_{2}\cdot P_{best})\]  
(6) Determine whether the end condition is reached. If so, output the optimal result; otherwise, repeat step 2-step 6. When using the BES optimization algorithm to improve the KPCA algorithm, the fitness function setting is the key point.