## 2 Principles and Methods
### Principles of Optimized Random Forest Algorithm  
#### 2.2.1 Principles of Random Forest Algorithm  
**Definition 1**.: _A Random Forest is an ensemble classifier consisting of a group of decision tree classifiers (\(\text{h}(X,\,\theta_{k})\), \(k\) = 1, 2, \(\ldots\), \(K\)). Each tree is formed by a random vector \(\{\theta_{k}\}\), which is independently and identically distributed. \(K\) denotes the number of decision trees in the random forest. For a given independent variable \(X\), each decision tree classifier votes to decide the optimal classification result._  
Essentially, a random forest is a classifier that aggregates multiple decision trees; if a decision tree is considered an expert in the classification task, then a random forest is a collection of such experts collaboratively classifying a given task.  
The steps to generate a random forest are as follows:  
Step 1. From the original training dataset, apply the bootstrap method to randomly draw K new bootstrap samples with replacement, constructing K classification regression trees. Each time, the samples not drawn form the K out-of-bag (OOB) datasets.  
Step 2. If there are n features, randomly select m\({}_{\text{try}}\) features (m\({}_{\text{try}}\leq\) n) at each node of each tree. Compute the information contained in each feature and select the one with the highest classification capability for node splitting.  
Step 3. Grow each tree to the maximum extent without pruning.  
Step 4. Combine the generated trees to form a random forest. Classify new data using this forest, with the classification result determined by the majority vote of the tree classifiers.  
**Definition 2**.: _Given a set of classifiers h1(X), h2(X), \(\ldots\), hk(X), where each classifier's training set is randomly sampled from the original dataset (Y, X) that follows a random distribution, the margin function is defined as:_  
\[mg(X,Y)=av_{k}I(h_{k}(X)=Y)-\underset{j\
eq Y}{\max}av_{k}I(h_{k}(X)=j) \tag{10}\]  
_where \(\leavevmode\
obreak\ 1(\cdot)\)is the indicator function._  
The margin function measures the extent to which the average number of correct classifications exceeds the average number of incorrect classifications. The larger the margin value, the more reliable the classification prediction.  
**Definition 3**.: _The generalization error is defined as:_  
\[PE^{*}=P_{X\leavevmode\
obreak\,Y}(mg(X\leavevmode\
obreak\,Y)<0) \tag{11}\]  
_where \(P\) covers the space of X and Y._  
In a random forest, when there are enough decision tree classifiers, \(h_{k}(X)=h(X\leavevmode\
obreak\,\theta_{k})\) follows the strong law of large numbers.  
**Theorem 1**.: _As the number of decision trees in the random forest increases, for all sequences \(\theta_{1}\leavevmode\
obreak\,\theta_{2}\leavevmode\
obreak\,\cdots\leavevmode \
obreak\,\theta_{k}\leavevmode\
obreak\,PE^{*}\)almost surely converges to:_  
\[P_{X,Y}\Bigg{\{}P_{\theta}(h(X,\theta)=Y)-\max P_{\theta}\Bigg{(}h(X \leavevmode\
obreak\,\theta)=j\Bigg{)}<0\Bigg{\}} \tag{12}\]  
Theorem 1 indicates that the random forest does not suffer from the overfitting problem as the number of decision trees increases, but it may have a certain degree of generalization error within limits.