# Is Principal Component Analysis a parametric method?

Principal component analysis assumes that the features are distributed by a Gaussian. Does this make Principal Component Analysis a parametric approach?

I can't seem to find a concrete answer saying that PCA is indeed a parametric approach. As we know, that parametric approaches assume some distribution of the input data.

• The first statement is contentious. PCA is, or can be thought of, as a deterministic eigenvalue-eigenvector calculation. It may well be that it works best with data near multivariate Gaussian in distribution, but that is a matter of ideal conditions for ease of interpretation, not a matter of anything assumed to be true. – Nick Cox Mar 7 '16 at 19:20
• But it is well known that it doesn't achieve useful results if the data isn't normally distributed – user46925 Mar 7 '16 at 19:27
• Who says? I often use PCA with non-normal distributions and often find it useful. It's nonlinearity which messes things up. – Nick Cox Mar 7 '16 at 19:31
• Zero, PCA does not require normality. It is, however, sensitive to ouliers. – ttnphns Mar 7 '16 at 20:04
• Is there an agreed upon definition of what makes a method parametric? I would have thought it was something about the parameters. – Jeremy Miles Mar 7 '16 at 22:38

We do not have to make any distributional assumptions in order to extract the Principal Component directions from a covariance or correlation matrix. To see this, recall the problem PCA solves. For a $p\times p$ covariance matrix $\mathbf{\Sigma}$ and all vectors $\mathbf{x} \in \mathbb{R}^p$ we would like to maximize
$$\mathbf{x}^{\prime} \mathbf{\Sigma} \mathbf{x} \tag{1}$$
subject to the condition that $\mathbf{x}$ is a unit vector, i.e. $\mathbf{x}^\prime \mathbf{x} = 1$. The solution is the eigenvector of the covariance (or correlation) matrix corresponding to the largest eigenvalue. The second principal component is the vector that maximizes $(1)$ subject to the additional condition that is perpendicular to the first direction and likewise for the third, forth and millionth principal component (always $\leq p$) . All we require is that the maximizers are unit vectors and perpendicular to the previous directions.
• asymptotic results what are those? – user46925 Mar 7 '16 at 19:35