How to project a new point to PCA new basis?

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 point projected to new principal component basis?

Answer 1

[Wei Wang]

When you specify a variable weight, the coefficient (W in your code) is not orthonormal, but the reconstruction rule is still Xcentered= score*coeff'. To get the score, you would have to do Xcentered/coeff' instead of Xcentered*coeff;

See the example below:

load hald;
data = ingredients;

% Weight and Mean:
wt = 1./var(data);
mu = mean(data);

% PCA, W is coefficient and Y is the score
[W,Y]=pca(data,'VariableWeights',wt,'centered',true);

% First observation of the centered data and its score
x1 = data(1,:)-mu;
y1 = Y(1,:)

% According to the reconstruction rule, we should have x1=y1*W'
% therefore, y1 = x1/W'
y = x1/W'

Answer 2

[Jun Ming Soh]

Given X is an n observations x p variables/parameters table, after calculating the PCA with 'Economy' set as false, size(coeff) = p x p, size(score) = n x p, size(percentage_variance) = p x 1.

[coeff, score, ~,~,percentage_variance] = pca(X, 'Economy', false);

If there is a new datapoint with size of 1 x p, the corresponding principle component will be

score_new = new_datapoint * coeff;

Answer 3

[Nikos Mp]

To project the old data to PC3 we should project them to PC1 then to PC2 then to PC3 ? Or we can do it using the data(:,1:3)*coeffs(:,1:3)