How to project a new point to PCA new basis?

Question

For example, I have 9 variables and 362 cases. I've made PCA calculation, and found out that first 3 PCA coordinates are enough for me. Now, I have new point in my 9-dimensional structure, and I want to project it to principal component system coordinate. How to get its new coordinates? ``` %# 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'); ``` How to get coordinates of a new…

Accepted Answer

Main fallacy was in operation that converts points to new basis:

```
y = (W*x')';
```

Wikipedia says:

The projected vectors are the columns of the matrix

```
Y = W*·Z, 
```

where `Y is L×N, W is M×L, Z is M×N`,

but `pca()` returns `W` of size `L×M` and `Y` of size `NxL`

so, correct equation in Matlab is:

```
y = x*W
```

Below is the corrected code:

```
[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
```

How to project a new point to PCA new basis?

Solution

Problem