In sklearn.decomposition.PCA, why are components_ negative?
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 display the components_ attribute from a fitted PCA with sklearn, they're of the exact same magnitude as the ones that I've manually computed, but some (not all) are of opposite sign. What's causing this?
Update: my (partial) answer below contains some additional info.
Take the following example data:
from pandas_datareader.data import DataReader as dr
import numpy as np
from sklearn.decomposition import PCA
from sklearn.preprocessing import scale
# sample data - shape (20, 3), each column standardized to N~(0,1)
rates = scale(dr(['DGS5', 'DGS10', 'DGS30'], 'fred',
start='2017-01-01', end='2017-02-01').pct_change().dropna())
# with sklearn PCA:
pca = PCA().fit(rates)
print(pca.components_)
[[-0.58365629 -0.58614003 -0.56194768]
[-0.43328092 -0.36048659 0.82602486]
[-0.68674084 0.72559581 -0.04356302]]
# compare to the manual method via SVD:
u, s, Vh = np.linalg.svd(np.asmatrix(rates), full_matrices=False)
print(Vh)
[[ 0.58365629 0.58614003 0.56194768]
[ 0.43328092 0.36048659 -0.82602486]
[-0.68674084 0.72559581 -0.04356302]]
# odd: some, but not all signs reversed
print(np.isclose(Vh, -1 * pca.components_))
[[ True True True]
[ True True True]
[False False False]]
Answer
As you figured out in your answer, the results of a singular value decomposition (SVD) are not unique in terms of singular vectors. Indeed, if the SVD of X is \sum_1^r \s_i u_i v_i^\top :
with the s_i ordered in decreasing fashion, then you can see that you can change the sign (i.e., "flip") of say u_1 and v_1, the minus signs will cancel so the formula will still hold.
This shows that the SVD is unique up to a change in sign in pairs of left and right singular vectors.
Since the PCA is just a SVD of X (or an eigenvalue decomposition of X^\top X), there is no guarantee that it does not return different results on the same X every time it is performed. Understandably, scikit learn implementation wants to avoid this: they guarantee that the left and right singular vectors returned (stored in U and V) are always the same, by imposing (which is arbitrary) that the largest coefficient of u_i in absolute value is positive.
As you can see reading the source: first they compute U and V with linalg.svd(). Then, for each vector u_i (i.e, row of U), if its largest element in absolute value is positive, they don't do anything. Otherwise, they change u_i to - u_i and the corresponding left singular vector, v_i, to - v_i. As told earlier, this does not change the SVD formula since the minus sign cancel out. However, now it is guaranteed that the U and V returned after this processing are always the same, since the indetermination on the sign has been removed.