MATLAB实例：PCA降维

5.1,3.5,1.4,0.2,1
4.9,3.0,1.4,0.2,1
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4.6,3.1,1.5,0.2,1
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5.4,3.9,1.7,0.4,1
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5.0,3.4,1.5,0.2,1
4.4,2.9,1.4,0.2,1
4.9,3.1,1.5,0.1,1
5.4,3.7,1.5,0.2,1
4.8,3.4,1.6,0.2,1
4.8,3.0,1.4,0.1,1
4.3,3.0,1.1,0.1,1
5.8,4.0,1.2,0.2,1
5.7,4.4,1.5,0.4,1
5.4,3.9,1.3,0.4,1
5.1,3.5,1.4,0.3,1
5.7,3.8,1.7,0.3,1
5.1,3.8,1.5,0.3,1
5.4,3.4,1.7,0.2,1
5.1,3.7,1.5,0.4,1
4.6,3.6,1.0,0.2,1
5.1,3.3,1.7,0.5,1
4.8,3.4,1.9,0.2,1
5.0,3.0,1.6,0.2,1
5.0,3.4,1.6,0.4,1
5.2,3.5,1.5,0.2,1
5.2,3.4,1.4,0.2,1
4.7,3.2,1.6,0.2,1
4.8,3.1,1.6,0.2,1
5.4,3.4,1.5,0.4,1
5.2,4.1,1.5,0.1,1
5.5,4.2,1.4,0.2,1
4.9,3.1,1.5,0.1,1
5.0,3.2,1.2,0.2,1
5.5,3.5,1.3,0.2,1
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4.4,3.0,1.3,0.2,1
5.1,3.4,1.5,0.2,1
5.0,3.5,1.3,0.3,1
4.5,2.3,1.3,0.3,1
4.4,3.2,1.3,0.2,1
5.0,3.5,1.6,0.6,1
5.1,3.8,1.9,0.4,1
4.8,3.0,1.4,0.3,1
5.1,3.8,1.6,0.2,1
4.6,3.2,1.4,0.2,1
5.3,3.7,1.5,0.2,1
5.0,3.3,1.4,0.2,1
7.0,3.2,4.7,1.4,2
6.4,3.2,4.5,1.5,2
6.9,3.1,4.9,1.5,2
5.5,2.3,4.0,1.3,2
6.5,2.8,4.6,1.5,2
5.7,2.8,4.5,1.3,2
6.3,3.3,4.7,1.6,2
4.9,2.4,3.3,1.0,2
6.6,2.9,4.6,1.3,2
5.2,2.7,3.9,1.4,2
5.0,2.0,3.5,1.0,2
5.9,3.0,4.2,1.5,2
6.0,2.2,4.0,1.0,2
6.1,2.9,4.7,1.4,2
5.6,2.9,3.6,1.3,2
6.7,3.1,4.4,1.4,2
5.6,3.0,4.5,1.5,2
5.8,2.7,4.1,1.0,2
6.2,2.2,4.5,1.5,2
5.6,2.5,3.9,1.1,2
5.9,3.2,4.8,1.8,2
6.1,2.8,4.0,1.3,2
6.3,2.5,4.9,1.5,2
6.1,2.8,4.7,1.2,2
6.4,2.9,4.3,1.3,2
6.6,3.0,4.4,1.4,2
6.8,2.8,4.8,1.4,2
6.7,3.0,5.0,1.7,2
6.0,2.9,4.5,1.5,2
5.7,2.6,3.5,1.0,2
5.5,2.4,3.8,1.1,2
5.5,2.4,3.7,1.0,2
5.8,2.7,3.9,1.2,2
6.0,2.7,5.1,1.6,2
5.4,3.0,4.5,1.5,2
6.0,3.4,4.5,1.6,2
6.7,3.1,4.7,1.5,2
6.3,2.3,4.4,1.3,2
5.6,3.0,4.1,1.3,2
5.5,2.5,4.0,1.3,2
5.5,2.6,4.4,1.2,2
6.1,3.0,4.6,1.4,2
5.8,2.6,4.0,1.2,2
5.0,2.3,3.3,1.0,2
5.6,2.7,4.2,1.3,2
5.7,3.0,4.2,1.2,2
5.7,2.9,4.2,1.3,2
6.2,2.9,4.3,1.3,2
5.1,2.5,3.0,1.1,2
5.7,2.8,4.1,1.3,2
6.3,3.3,6.0,2.5,3
5.8,2.7,5.1,1.9,3
7.1,3.0,5.9,2.1,3
6.3,2.9,5.6,1.8,3
6.5,3.0,5.8,2.2,3
7.6,3.0,6.6,2.1,3
4.9,2.5,4.5,1.7,3
7.3,2.9,6.3,1.8,3
6.7,2.5,5.8,1.8,3
7.2,3.6,6.1,2.5,3
6.5,3.2,5.1,2.0,3
6.4,2.7,5.3,1.9,3
6.8,3.0,5.5,2.1,3
5.7,2.5,5.0,2.0,3
5.8,2.8,5.1,2.4,3
6.4,3.2,5.3,2.3,3
6.5,3.0,5.5,1.8,3
7.7,3.8,6.7,2.2,3
7.7,2.6,6.9,2.3,3
6.0,2.2,5.0,1.5,3
6.9,3.2,5.7,2.3,3
5.6,2.8,4.9,2.0,3
7.7,2.8,6.7,2.0,3
6.3,2.7,4.9,1.8,3
6.7,3.3,5.7,2.1,3
7.2,3.2,6.0,1.8,3
6.2,2.8,4.8,1.8,3
6.1,3.0,4.9,1.8,3
6.4,2.8,5.6,2.1,3
7.2,3.0,5.8,1.6,3
7.4,2.8,6.1,1.9,3
7.9,3.8,6.4,2.0,3
6.4,2.8,5.6,2.2,3
6.3,2.8,5.1,1.5,3
6.1,2.6,5.6,1.4,3
7.7,3.0,6.1,2.3,3
6.3,3.4,5.6,2.4,3
6.4,3.1,5.5,1.8,3
6.0,3.0,4.8,1.8,3
6.9,3.1,5.4,2.1,3
6.7,3.1,5.6,2.4,3
6.9,3.1,5.1,2.3,3
5.8,2.7,5.1,1.9,3
6.8,3.2,5.9,2.3,3
6.7,3.3,5.7,2.5,3
6.7,3.0,5.2,2.3,3
6.3,2.5,5.0,1.9,3
6.5,3.0,5.2,2.0,3
6.2,3.4,5.4,2.3,3
5.9,3.0,5.1,1.8,3

function [COEFF,SCORE,latent,tsquared,explained,mu,data_PCA]=pca_demo()
x=load(\'iris.data\');
[~,d]=size(x);
k=d-1; %前k个主成分
x=zscore(x(:,1:d-1));  %归一化数据
[COEFF,SCORE,latent,tsquared,explained,mu]=pca(x);
% 1）获取样本数据 X ，样本为行，特征为列。
% 2）对样本数据中心化，得S（S = X的各列减去各列的均值）。
% 3）求 S 的协方差矩阵 C = cov(S)
% 4) 对协方差矩阵 C 进行特征分解 [P,Lambda] = eig(C);
% 5）结束。
% 1、输入参数 X 是一个 n 行 p 列的矩阵。每行代表一个样本观察数据，每列则代表一个属性，或特征。
% 2、COEFF 就是所需要的特征向量组成的矩阵，是一个 p 行 p 列的矩阵，没列表示一个出成分向量，经常也称为（协方差矩阵的）特征向量。并且是按照对应特征值降序排列的。所以，如果只需要前 k 个主成分向量，可通过：COEFF(:,1:k) 来获得。
% 3、SCORE 表示原数据在各主成分向量上的投影。但注意：是原数据经过中心化后在主成分向量上的投影。即通过：SCORE = x0*COEFF 求得。其中 x0 是中心平移后的 X（注意：是对维度进行中心平移，而非样本。），因此在重建时，就需要加上这个平均值了。
% 4、latent 是一个列向量，表示特征值，并且按降序排列。
% 5、tsquared Hotelling的每个观测值X的T平方统计量
% 6、explained 由每个主成分解释的总方差的百分比
% 7、mu 每个变量X的估计平均值
% x= bsxfun(@minus,x,mean(x,1));
data_PCA=x*COEFF(:,1:k);
latent1=100*latent/sum(latent);%将latent总和统一为100，便于观察贡献率
pareto(latent1);%调用matla画图 pareto仅绘制累积分布的前95%，因此y中的部分元素并未显示
xlabel(\'Principal Component\');
ylabel(\'Variance Explained (%)\');
% 图中的线表示的累积变量解释程度
print(gcf,\'-dpng\',\'Iris PCA.png\');
iris_pac=data_PCA(:,1:2) ;
save iris_pca iris_pac

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function data=pca_data(data, choose)
% PCA降维，保留90%的特征信息 
data = normlization(data, choose); %归一化
score = 0.90; %保留90%的特征信息
[num,dim] = size(data);
xbar = mean(data,1);
means = bsxfun(@minus, data, xbar);
cov = means\'*means/num;
[V,D] = eig(cov);
eigval = diag(D);
[~,idx] = sort(eigval,\'descend\');
eigval = eigval(idx);
V = V(idx,:);
p = 0;
for i=1:dim
   perc = sum(eigval(1:i))/sum(eigval);
   if perc > score
       p = i;
       break;       
   end 
end
E = V(1:p,:);
data= means*E\';