I have a large number of variables describing match performance.
I ran a PCA in order to reduce the number of variables and I obtained 3 principal components.
I'd like to obtain only one index describing match performance in football. Is it possible to combine the 3 principal components I obtained?
I read that there are 2 possible ways to do it:
Re-running a PCA on the new factors obtained
Multiplying the new factors with the variance explained. An example: PC1 explains 46% of the variance (0.46), PC2 30% (0.30) and PC3 12% (0.12). The total index will be calculated: Performance index= 0.46*PC1 + 0.30*PC2 + 0.12*PC3.
Are those two solutions appropriate? If not, is there a way to combine different components?
The first principal component is - by construction - the best single feature (or score, if you like) to explain your data. If you only want one element, you should simply use the First Principal Component.
The first approach you propose will return you exactly the same principal components you already have, since these are already built to maximise variance on their axes and to be orthogonal, running the decomposition again would issue the same results. The second approach would just give you some linear combination of your original features with no particular properties.
The beauty of (unrotated) principal components is that they are mutually orthogonal. There is no problem in just adding them up. There is also no surprise in how they will behave. The behavior of the sum will be exactly the same as if you included the three components as predictors but with equal regression weights, rather than allowing their regression weights to differ. Just add them together. You will them have a composite variable that accounts for a larger share of total variance and combines the functionality of the three components.