【模式识别】CART和GML AdaBoost MATLAB TOOLBOX

AdaBoost的训练和分类的结构都是类似的，能够參考前一篇《Boosting》。仅仅简单介绍一下GML。GML内部弱分类器使用的CART决策树。决策树的叶子表示决策，内部每一个分支都是决策过程。

从根部開始，每一个决策结果指向下一层决策，最后到达叶子，得到终于的决策结果。一个比較简单的示意例如以下图所看到的：

对于特征{x1,x2,x3}值为{4,-1,2}的样本，上图决策树的终于决策为y=-1，而对于{3,-2,3}的样本预測值为+1。
工具箱中CART根的构造算法例如以下：

1.?对于n为特征的每一个维度，分别找到阈值使得分类错误率最小
2.?选择错误率最小的第i维构造根节点：
a)?预測xi>Θ
b)?分别构造true/false子树分支。各自的分类结果作为叶子

在树中移动并终于到达叶子，我们把得到的预測结果错误分类一个样本的概率作为“树叶的错误率”。

整颗决策树的构造使用例如以下规则：

1.?构造根节点
2.?选择错误率最小的叶子
3.?仅适用于和所选叶子相关的训练数据构造节点
4.?使用构造的节点取代所选的叶子
5.?反复2-4步直到全部的叶子错误率为0，或者到达循环结束的循环次数。

其实，工具箱使用中，CART树的构造过程仅仅有树深一个可调參数，使用：

function tree_node = tree_node_w(max_splits)

% Step1: reading Data from the file
clear
clc
file_data = load(‘Ionosphere.txt‘);
Data = file_data(:,1:10)‘;
Labels = file_data(:,end)‘;
Labels = Labels*2 - 1;
MaxIter = 200; % boosting iterations
% Step2: splitting data to training and control set
TrainData   = Data(:,1:2:end);
TrainLabels = Labels(1:2:end);
ControlData   = Data(:,2:2:end);
ControlLabels = Labels(2:2:end);
% Step3: constructing weak learner
weak_learner = tree_node_w(3); % pass the number of tree splits to the constructor
% Step4: training with Gentle AdaBoost
[GLearners GWeights] = GentleAdaBoost(weak_learner, TrainData, TrainLabels, MaxIter);
% Step5: training with Modest AdaBoost
[MLearners MWeights] = ModestAdaBoost(weak_learner, TrainData, TrainLabels, MaxIter);
% Step5: training with Modest AdaBoost
[RLearners RWeights] = RealAdaBoost(weak_learner, TrainData, TrainLabels, MaxIter);
% Step6: evaluating on control set
ResultG = sign(Classify(GLearners, GWeights, ControlData));
ResultM = sign(Classify(MLearners, MWeights, ControlData));
ResultR = sign(Classify(RLearners, RWeights, ControlData));
% Step7: calculating error
ErrorG  = sum(ControlLabels ~= ResultG) / length(ControlLabels)
ErrorM  = sum(ControlLabels ~= ResultM) / length(ControlLabels)
ErrorR  = sum(ControlLabels ~= ResultR) / length(ControlLabels)

以上代码构造深度为3的CART决策树作为弱分类器。分别使用GentleAdaBoost，ModestAdaBoost和RealAdaBoost迭代200次训练AdaBoost分类器。

以上代码直接使用MaxIter调用函数训练与以下方法是等价的：

% Step4: iterativly running the training
for lrn_num = 1 : MaxIter
    clc;
    disp(strcat(‘Boosting step: ‘, num2str(lrn_num),‘/‘, num2str(MaxIter)));
    %training gentle adaboost
    [GLearners GWeights] = GentleAdaBoost(weak_learner, TrainData, TrainLabels, 1, GWeights, GLearners);
    %evaluating control error
    GControl = sign(Classify(GLearners, GWeights, ControlData));
    GAB_control_error(lrn_num) = GAB_control_error(lrn_num) + sum(GControl ~= ControlLabels) / length(ControlLabels);
    %training real adaboost
    [RLearners RWeights] = RealAdaBoost(weak_learner, TrainData, TrainLabels, 1, RWeights, RLearners);
    %evaluating control error
    RControl = sign(Classify(RLearners, RWeights, ControlData));
    RAB_control_error(lrn_num) = RAB_control_error(lrn_num) + sum(RControl ~= ControlLabels) / length(ControlLabels);
    %training modest adaboost
    [NuLearners NuWeights] = ModestAdaBoost(weak_learner, TrainData, TrainLabels, 1, NuWeights, NuLearners);
    %evaluating control error
    NuControl = sign(Classify(NuLearners, NuWeights, ControlData));
    MAB_control_error(lrn_num) = MAB_control_error(lrn_num) + sum(NuControl ~= ControlLabels) / length(ControlLabels);
end

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