如何向PCA新基础投射新点？

Question

例如，我有9个变量和362个案例。我已经进行了PCA计算，发现前3个PCA坐标足够我。

现在，我在9维结构中有了新的观点，我想把它投射到主成分系统坐标。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');

如何获得投射到新主成分基础的新点的坐标？

Answer 1

正在运作的主要谬误将积分转换为新的基础：

y = (W*x')';

维基百科说：

  

投影向量是矩阵的列

Y = W*·Z, 

     

其中Y is L×N, W is M×L, Z is M×N，

但pca()返回W L×M，尺寸Y和NxL <{1}}

所以，Matlab中的正确方程是：

y = x*W

以下是更正后的代码：

[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