In sklearn.decomposition.PCA, why are components_ negative?

Question

I\'m trying to follow along with Abdi & Williams - Principal Component Analysis (2010) and build principal components through SVD, using numpy.linalg.svd.

When I

Answer 1

After some digging, I've cleared up some, but not all, of my confusion on this. This issue has been covered on stats.stackexchange here. The mathematical answer is that "PCA is a simple mathematical transformation. If you change the signs of the component(s), you do not change the variance that is contained in the first component." However, in this case (with sklearn.PCA), the source of ambiguity is much more specific: in the source (line 391) for PCA you have:

U, S, V = linalg.svd(X, full_matrices=False)
# flip eigenvectors' sign to enforce deterministic output
U, V = svd_flip(U, V)

components_ = V

svd_flip, in turn, is defined here. But why the signs are being flipped to "ensure a deterministic output," I'm not sure. (U, S, V have already been found at this point...). So while sklearn's implementation is not incorrect, I don't think it's all that intuitive. Anyone in finance who is familiar with the concept of a beta (coefficient) will know that the first principal component is most likely something similar to a broad market index. Problem is, the sklearn implementation will get you strong negative loadings to that first principal component.

My solution is a dumbed-down version that does not implement svd_flip. It's pretty barebones in that it doesn't have sklearn parameters such as svd_solver, but does have a number of methods specifically geared towards this purpose.