How can give weight to feature before PCA

Question

I wonder how can I give weight to my feature before employing PCA. I mean somehow weighted PCA. Because I know that one of the features is better than others and want to give importance to it in creating components (It is not possible to select only that feature. I should have others impact too)

Answer 1

After standardizing your data you can multiply the features with weights to assign weights before the principal component analysis. Giving higher weights means the variance within the feature goes up, which makes it more important.

Standardizing (mean 0 and variance 1) is important for PCA because it is looking for a new orthogonal basis where the origin stays the same, so having your data centered around the origin is good. The first principal component is the direction with the most variance, by scaling a certain feature with a weight of more than 1 will increase the variance on this axis and thus give more weight to this axis, pulling the first principal component in it's direction.

Answer 2

PCA is unsupervised method for finding the most important components. I don't see a reason why you should want add a weight. If you know what features are important, why use PCA at all? Or perform PCA on the features where you are unsure about the importance.

Further, components are created in directions with highest variance and the importance is measured by eigenvalues. So I can imagine you can somehow increase the variance (like Jan van der Vegt proposed). But that's a sorcery with very questionable output.