如何将新点投影到PCA新的基础上？

Question

例如，我有9个变量和362个案例。我做了PCA计算，发现前3个PCA坐标对我来说已经足够了。

现在，我在我的9维结构中有了新的观点，并且我想将它投影到主成分系统坐标上。如何获得新的坐标？

%# here is data (362x9) 
load SomeData 

[W, Y] = pca(data, 'VariableWeights', 'variance', 'Centered', true); 

%# orthonormal coefficient matrix 
W = diag(std(data))\W; 

% Getting mean and weights of data (for future data) 
[data, mu, sigma] = zscore(data); 
sigma(sigma==0) = 1; 

%# New point in original 9dim system 
%# For example, it is the first point of our input data 
x = data(1,:); 
x = bsxfun(@minus,x, mu); 
x = bsxfun(@rdivide, x, sigma); 

%# New coordinates as principal components 
y0 = Y(1,:); %# point we should get in result 
y = (W*x')'; %# our result 

%# error 
sum(abs(y0 - y)) %# 142 => they are not the same point 

%# plot 
figure() 
plot(y0,'g'); hold on; 
plot(y,'r'); 

如何获得新的点的坐标投影到新的主成分的基础？

Answer 1

主要谬论是在转换点，新的基础操作：

y = (W*x')'; 

维基说：

投影向量矩阵

Y = W*·Z, 

其中Y is L×N, W is M×L, Z is M×N的列，

但pca()回报大小L×M的W和大小的YNxL

所以，在Matlab正确的公式是：

y = x*W 

下面是更正后的代码：

[W, Y] = pca(data, 'VariableWeights', 'variance', 'Centered', true); 
W = diag(std(data))\W; 

%# Getting mean and weights of data (for future data) 
[~, mu, we] = zscore(data); 
we(we==0) = 1; 

%# New point in original 9dim system 
%# For example, it is the first point of our input data 
x = data(1,:); 
x = bsxfun(@minus,x, mu); 
x = bsxfun(@rdivide, x, we); 

%# New coordinates as principal components 
y = x*W; 
y0 = Y(1,:); 
sum(abs(y0 - y)) %# 4.1883e-14 ~= 0