# Can I linearly combine the results of PCA with the variance of each feature?

I'm trying to reduce the dimension of a dataset from 8 features to 1 using the principal component analysis (PCA) algorithm. The reduced dataset needs to be in 1 dimension(D) so I can use it for matrix factorization and other algorithms.

However, after applying PCA, the reduced 1D dataset only keeps 45% of the variance so I'm loosing a lot of information. I tried to reduce the dimension keeping 95% of the variance but the resulting dataset has 3D.

I know the variance of each feature in the reduced dataset. Therefore, I was thinking of using the variances as weights and linearly combine them with the reduced values so I'd reduce the dataset further to 1D, for example:

value_1D = variance_f1*value_f1 + variance_f2*value_f2 + variance_f3*value_f3


Would it we correct to do it? Do you know any other alternative?

• It may be correct or reasonable in some specific investigatory context, but not generally. Components are uncorrelated, therefore constructing a linear combination of them is like adding apples to oranges. Sometimes we do add both together. Some psychometric scales, for example, are constructed disregarding any statistical correlation between the items being included, following purely nonstatistical reasons (such scale may be called battery of items). So yes, you may do it. The question is what for and how you are going to apply the result. – ttnphns Sep 13 '15 at 17:15
• Thank you, I need the 1D dataset for matrix factorization to build a collaborative filtering recommender system. In this matrix, rows are users and columns are items. The value in each cell would be the 1D value for user-item. The features are: impressions, shares, clicks and more. All these features represent positive feedback. I tried to linearly combine the reduced dataset with the variances but the results don't look good as there are negative values so when I combine them it looks as a "negative feedback". However, If I use absolute numbers then the results are much better. – joanfihu Sep 14 '15 at 17:39