数据科学和机器学习（第 16 部分）：全新面貌的决策树·进阶篇
（2/3）·上篇代码被吐槽不够简洁，这回把节点、拆分、递归一次讲透，为下篇森林算法铺路
决策树里的子节点到底装了什么
内部节点在决策树里不是终点，它只负责抛出一个测试条件，真正分流靠的是挂在其下的子节点。每个子节点对应测试条件的一种结果，并承载满足条件后的那部分数据子集，递归下去直到叶子节点给出类标签。 拿一个极简的水果分类树举例：根节点测试「颜色是红色吗」，为真走左子节点、为假走右子节点，左右两边都是叶子——左边标苹果，右边标橙子。这种结构说明子节点本质是「条件分支后的数据切片容器」。 MQL5 里通常用指针把子节点链到父节点上，下面两行就是最基础的左右子节点声明，注释标明了指向关系。 外汇与贵金属行情用这类树做状态分流时，注意过拟合高风险：样本外品种可能让分支失效。
Node *left_child; class=class="str">"cmt">//left child Node Node *right_child; class=class="str">"cmt">//right child Node
「三类决策树算法怎么选」
做价格行为分类，先得认清楚手里的树长什么样。CART 既能分类也能回归，分类时按基尼杂质切分节点，回归时换均方误差最小化，一条算法两条路。 ID3 只干分类的活，靠熵和信息增益决定每次拿哪个特征劈数据，逻辑直接但容易偏向取值多的字段。C4.5 是 ID3 的修正版，用增益比率压住了多级别属性的偏好，分类更稳。 我们后面要落地的分类模型，选的是 ID3 路线：以信息增益、杂质计算加分类特征为核心，不碰回归分支。外汇与贵金属波动受事件驱动明显，用历史 K 线形态训练这类树，信号失效概率不低，仅作辅助过滤。
◍ ID3 怎么挑特征：信息增益与两种杂质度量
ID3 在每个内部节点靠信息增益决定拆哪个特征。增益量化的是一次拆分后数据集熵（或无序度）的下降量：下降越多，子集类标签越同质，模型在该节点的判别越干净。 工程实现里常把熵和基尼指数都留作选项。两者都是杂质函数，基尼指数为 1 减各类概率平方和，熵为 -Σp·log2(p)，数值越低代表纯度越高；在外汇或贵金属行情分类里，切换这两种度量可能改变树的分叉形态，但都不会凭空提高预测命中率，杠杆品种高风险依旧。 下面这段 MQL5 把两种模式写进同一个 information_gain 接口：先按子节点样本占比算权重，再用父节点杂质减加权子节点杂质得到增益。 double CDecisionTree::information_gain(vector &parent, vector &left_child, vector &right_child) 函数签名：输入父集与左右子集向量，返回本次拆分的增益值。 double weight_left = left_child.Size() / (double)parent.Size(), 算左子集占父集的比例权重。 weight_right = right_child.Size() / (double)parent.Size(); 算右子集占父集的比例权重。 double gain =0; 初始化增益为 0。 switch(m_mode) 按成员变量 m_mode 选择杂质度量模式。 case MODE_GINI: 若走基尼模式。 gain = gini_index(parent) - ( (weight_left*gini_index(left_child)) + (weight_right*gini_index(right_child)) ); 父基尼减左右加权基尼得增益。 break; case MODE_ENTROPY: 若走熵模式。 gain = entropy(parent) - ( (weight_left*entropy(left_child)) + (weight_right*entropy(right_child)) ); 父熵减左右加权熵得增益。 break; return gain; 返回计算出的增益。 double CDecisionTree::entropy(vector &y) 熵计算函数，输入标签向量 y。 vector class_labels = matrix_utils.Unique_count(y); 统计各分类出现频次。 vector p_cls = class_labels / double(y.Size()); 转成每类概率。 vector entropy = (-1 * p_cls) * log2(p_cls); 逐类算 -p·log2(p)。 return entropy.Sum(); 求和得总熵。 double CDecisionTree::gini_index(vector &y) 基尼函数，输入标签向量 y。 vector unique = matrix_utils.Unique_count(y); 各类频次。 vector probabilities = unique / (double)y.Size(); 各类概率。 return 1.0 - MathPow(probabilities, 2).Sum();
- 减概率平方和后得基尼。
开 MT5 把这段塞进自己的树类，改 m_mode 跑同一批 K 线标签，能直接比对两种度量的分叉差异。
class="type">class="kw">double CDecisionTree::information_gain(vector &parent, vector &left_child, vector &right_child) { class="type">class="kw">double weight_left = left_child.Size() / (class="type">class="kw">double)parent.Size(), weight_right = right_child.Size() / (class="type">class="kw">double)parent.Size(); class="type">class="kw">double gain =class="num">0; class="kw">switch(m_mode) { case MODE_GINI: gain = gini_index(parent) - ( (weight_left*gini_index(left_child)) + (weight_right*gini_index(right_child)) ); break; case MODE_ENTROPY: gain = entropy(parent) - ( (weight_left*entropy(left_child)) + (weight_right*entropy(right_child)) ); break; } class="kw">return gain; } class="type">class="kw">double CDecisionTree::entropy(vector &y) { vector class_labels = matrix_utils.Unique_count(y); vector p_cls = class_labels / class="type">class="kw">double(y.Size()); vector entropy = (-class="num">1 * p_cls) * log2(p_cls); class="kw">return entropy.Sum(); } class="type">class="kw">double CDecisionTree::gini_index(vector &y) { vector unique = matrix_utils.Unique_count(y); vector probabilities = unique / (class="type">class="kw">double)y.Size(); class="kw">return class="num">1.0 - MathPow(probabilities, class="num">2).Sum(); }
树怎么长出来：拆分与递归建树
决策树分类常用基尼杂质或熵做拆分准则，回归则多用均方误差。核心动作是用一个阈值把样本劈成左右两堆：特征值小于等于阈值的进 dataset_left，其余进 dataset_right，再回传含特征索引、阈值和信息增益的 split_info 结构。
算法并不会随便切一刀，而是遍历所有特征、取该特征下的唯一值作为候选阈值，逐个算信息增益，保留增益最大的那次拆分。下面这段 MQL5 结构和方法就是干这事的基础件。
[CODE]
//A struct containing splitted data information
struct split_info
{
uint feature_index;
double threshold;
matrix dataset_left,
dataset_right;
double info_gain;
};
split_info CDecisionTree::split_data(const matrix &data, uint feature_index, double threshold=0.5)
{
int left_size=0, right_size =0;
vector row = {};
split_info split;
ulong cols = data.Cols();
split.dataset_left.Resize(0, cols);
split.dataset_right.Resize(0, cols);
for (ulong i=0; i<data.Rows(); i)
{
row = data.Row(i);
if (row[feature_index] <= threshold)
{
left_size++;
split.dataset_left.Resize(left_size, cols);
split.dataset_left.Row(row, left_size-1);
}
else
{
right_size++;
split.dataset_right.Resize(right_size, cols);
split.dataset_right.Row(row, right_size-1);
}
}
return split;
}
split_info CDecisionTree::get_best_split(matrix &data, uint num_features)
{
double max_info_gain = -DBL_MAX;
vector feature_values = {};
vector left_v={}, right_v={}, y_v={};
//---
split_info best_split;
split_info split;
for (uint i=0; i<num_features; i)
{
feature_values = data.Col(i);
vector possible_thresholds = matrix_utils.Unique(feature_values); //Find unique values in the feature, representing possible thresholds for splitting.
for (uint j=0; j<possible_thresholds.Size(); j)
{
split = this.split_data(data, i, possible_thresholds[j]);
if (split.dataset_left.Rows()>0 && split.dataset_right.Rows() > 0)
{
y_v = data.Col(data.Cols()-1);
right_v = split.dataset_right.Col(split.dataset_right.Cols()-1);
代码逐行拆：split_info 结构存了特征列号、切分阈值、左右子矩阵和增益值；split_data 默认阈值 0.5，按行扫数据，满足 <=阈值 的塞进 left 矩阵，否则进 right 矩阵，两个矩阵都先 Resize 到 0 列宽再动态扩行。get_best_split 里 max_info_gain 初始化为 -DBL_MAX，保证任意真实增益都能覆盖它；用 matrix_utils.Unique 提唯一值当候选阈值，仅当左右子集都非空才计算增益，避免单边空树。
建树是递归的：如果 best_split.info_gain > 0 就继续，对左右子集分别 build_tree(curr_depth+1)，否则生成叶节点并调 calculate_leaf_value 填值。把 build_tree 包在 fit 里更贴近 Python 习惯，预测时 make_predictions 递归比阈值，一路走到叶子返回结果。外汇和贵金属行情噪声大，直接拿原始价格训树容易过拟合，建议先算波动率特征再喂数据，回测胜率可能更稳。
class=class="str">"cmt">//A class="kw">struct containing splitted data information class="kw">struct split_info { class="type">uint feature_index; class="type">class="kw">double threshold; matrix dataset_left, dataset_right; class="type">class="kw">double info_gain; }; split_info CDecisionTree::split_data(const matrix &data, class="type">uint feature_index, class="type">class="kw">double threshold=class="num">0.5) { class="type">int left_size=class="num">0, right_size =class="num">0; vector row = {}; split_info split; class="type">ulong cols = data.Cols(); split.dataset_left.Resize(class="num">0, cols); split.dataset_right.Resize(class="num">0, cols); for (class="type">ulong i=class="num">0; i<data.Rows(); i) { row = data.Row(i); if (row[feature_index] <= threshold) { left_size++; split.dataset_left.Resize(left_size, cols); split.dataset_left.Row(row, left_size-class="num">1); } else { right_size++; split.dataset_right.Resize(right_size, cols); split.dataset_right.Row(row, right_size-class="num">1); } } class="kw">return split; } split_info CDecisionTree::get_best_split(matrix &data, class="type">uint num_features) { class="type">class="kw">double max_info_gain = -DBL_MAX; vector feature_values = {}; vector left_v={}, right_v={}, y_v={}; class=class="str">"cmt">//--- split_info best_split; split_info split; for (class="type">uint i=class="num">0; i<num_features; i) { feature_values = data.Col(i); vector possible_thresholds = matrix_utils.Unique(feature_values); class=class="str">"cmt">//Find unique values in the feature, representing possible thresholds for splitting. for (class="type">uint j=class="num">0; j<possible_thresholds.Size(); j) { split = this.split_data(data, i, possible_thresholds[j]); if (split.dataset_left.Rows()>class="num">0 && split.dataset_right.Rows() > class="num">0) { y_v = data.Col(data.Cols()-class="num">1); right_v = split.dataset_right.Col(split.dataset_right.Cols()-class="num">1);
「遍历切分点挑信息增益最大的节点」
决策树在生长时，核心动作是穷举每个特征的候选阈值，算出左右子集的信息增益，保留截至当前最大的那一次切分。下面这段逻辑里，left_v 取左子集的最后一列作为切分后的标签观测，curr_info_gain 由 information_gain 函数对 y_v、left_v、right_v 三者算出。 当 curr_info_gain 大于 max_info_gain 时，代码把本次的 feature_index、threshold、左右数据集和增益值全部写进 best_split，并更新 max_info_gain。DEBUG_MODE 下的 printf 会打出类似「split left: [120x5] split right: [80x5] curr_info_gain: 0.231044 max_info_gain: 0.198332」的日志，你可以直接开 MT5 定义 DEBUG_MODE 宏来核对每次切分的样本数与增益变化。 build_tree 负责把输入矩阵拆成特征矩阵 X 和目标向量 Y，用 X.Rows() 与 X.Cols() 拿到样本数 samples 和特征数 features。只有 samples >= m_min_samples_split 且 curr_depth <= m_max_depth 才允许继续调用 get_best_split 往下分；这两个条件直接决定树会不会过深，外汇与贵金属行情噪声大，过深容易过拟合，调参时建议先盯这两个值。
left_v = split.dataset_left.Col(split.dataset_left.Cols()-class="num">1); class="type">class="kw">double curr_info_gain = this.information_gain(y_v, left_v, right_v); if (curr_info_gain > max_info_gain) class=class="str">"cmt">// Check if the current information gain is greater than the maximum observed so far. { class="macro">#ifdef DEBUG_MODE printf("split left: [%dx%d] split right: [%dx%d] curr_info_gain: %f max_info_gain: %f",split.dataset_left.Rows(),split.dataset_left.Cols(),split.dataset_right.Rows(),split.dataset_right.Cols(),curr_info_gain,max_info_gain); class="macro">#endif best_split.feature_index = i; best_split.threshold = possible_thresholds[j]; best_split.dataset_left = split.dataset_left; best_split.dataset_right = split.dataset_right; best_split.info_gain = curr_info_gain; max_info_gain = curr_info_gain; } } } class="kw">return best_split; } Node *CDecisionTree::build_tree(matrix &data, class="type">uint curr_depth=class="num">0) { matrix X; vector Y; matrix_utils.XandYSplitMatrices(data,X,Y); class=class="str">"cmt">//Split the class="kw">input matrix into feature matrix X and target vector Y. class="type">ulong samples = X.Rows(), features = X.Cols(); class=class="str">"cmt">//Get the number of samples and features in the dataset. Node *node= NULL; class=class="str">"cmt">// Initialize node pointer if (samples >= m_min_samples_split && curr_depth<=m_max_depth) { split_info best_split = this.get_best_split(data, (class="type">uint)features); class="macro">#ifdef DEBUG_MODE Print("best_split left: [",best_split.dataset_left.Rows(),"x",best_split.dataset_left.Cols(),"]\nbest_split right: [",best_split.dataset_right.Rows(),"x",best_split.dataset_right.Cols(),"]\nfeature_index: ",best_split.feature_index,"\nInfo gain: ",best_split.info_gain,"\nThreshold: ",best_split.threshold); class="macro">#endif
◍ 递归建树与预测落点的代码骨架
决策树在信息增益大于 0 时才继续分裂，否则直接落成叶子节点并写入 calculate_leaf_value 算出的数值。build_tree 里左右子集各带 curr_depth+1 递归往下走，节点保存 feature_index、threshold 与 info_gain，这条链路决定了后续预测时走左还是走右。 fit 函数把特征矩阵 x 与标签 y 按列拼接成 data，再交給 build_tree 生成 root；predict 对矩阵逐行调单样本预测，返回长度等于 x.Rows() 的向量，你在 MT5 里跑完可直接比对信号序列。 make_predictions 是真正的落点逻辑：遇到 leaf_value 非 NULL 就返回该值，否则按 x[tree.feature_index] 与 threshold 的大小关系递归进左或右子树。DEBUG_MODE 下会 printf 出 threshold、feature_index 与 leaf_value，方便你抓某次 Split 异常。 外汇与贵金属行情受杠杆与跳空影响，树模型信号仅代表历史样本下的概率倾向，实盘前务必用策略测试器跑多品种验证。
if (best_split.info_gain > class="num">0) { Node *left_child = this.build_tree(best_split.dataset_left, curr_depth+class="num">1); Node *right_child = this.build_tree(best_split.dataset_right, curr_depth+class="num">1); node = new Node(best_split.feature_index,best_split.threshold,left_child,right_child,best_split.info_gain); class="kw">return node; } } node = new Node(); node.leaf_value = this.calculate_leaf_value(Y); class="kw">return node; } node = new Node(best_split.feature_index, best_split.threshold, left_child, right_child, best_split.info_gain); node = new Node(); node.value = this.calculate_leaf_value(Y); class="kw">return node; class="type">void CDecisionTree::fit(matrix &x, vector &y) { matrix data = matrix_utils.concatenate(x, y, class="num">1); this.root = this.build_tree(data); } vector CDecisionTree::predict(matrix &x) { vector ret(x.Rows()); for (class="type">ulong i=class="num">0; i<x.Rows(); i++) ret[i] = this.predict(x.Row(i)); class="kw">return ret; } class="type">class="kw">double CDecisionTree::predict(vector &x) { class="kw">return this.make_predictions(x, this.root); } class="type">class="kw">double CDecisionTree::make_predictions(vector &x, const Node &tree) { if (tree.leaf_value != NULL) class=class="str">"cmt">// This is a leaf leaf_value class="kw">return tree.leaf_value; class="type">class="kw">double feature_value = x[tree.feature_index]; class="type">class="kw">double pred = class="num">0; class="macro">#ifdef DEBUG_MODE printf("Tree.threshold %f tree.feature_index %d leaf_value %f",tree.threshold,tree.feature_index,tree.leaf_value); class="macro">#endif if (feature_value <= tree.threshold) { pred = this.make_predictions(x, tree.left_child); } else { pred = this.make_predictions(x, tree.right_child); } class="kw">return pred; } if (feature_value <= tree.threshold): pred = this.make_predictions(x, *tree.left_child); pred = this.make_predictions(x, *tree.right_child); class="kw">return pred;