FPCA – Functional Principal Component Analysis (FPCA): What is it All About

Question

Functional principal component analysis (FPCA) is something I have stumbled upon and never got to understand. What is it all about?

See "A survey of functional principal component
analysis" by Shang, 2011, and I'm citing:

PCA runs into serious difficulties in analyzing
functional data because of the “curse of dimensionality” (Bellman 1961). The “curse
of dimensionality” originates from data sparsity in high-dimensional space. Even
if the geometric properties of PCA remain valid, and even if numerical techniques
deliver stable results, the sample covariance matrix is sometimes a poor estimate
of the population covariance matrix. To overcome this difficulty, FPCA provides
a much more informative way of examining the sample covariance structure than
PCA […]

I just don't get it. What is the drawback this paper is describing? Isn't PCA supposed to be the ultimate method to handle situations like the “curse of dimensionality”?

Answer 1

Exactly, as you state in the question and as @tdc puts in his answer, in case of extremely high dimensions even if the geometric properties of PCA remain valid, the covariance matrix is no longer a good estimate of the real population covariance.

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There's a very interesting paper "Functional Principal Component Analysis of fMRI Data" (pdf) where they use functional PCA to visualize the variance:

...As in other explorative techniques, the objective is that of providing an initial assessment that will give the data a chance “to speak for themselves” before an appropriate model is chosen. [...]

In the paper they explain how exactly they've done it, and also provide theoretical reasoning:

The decisive advantage of this approach consists in the possibility of specifying a set of assumptions in the choice of the basis function set and in the error functional minimized by the fit. These assumptions will be weaker than the specification of a predefined hemodynamic function and a set of events or conditions as in F-masking, thus preserving the exploratory character of the procedure; however, the assumptions might remain stringent enough to overcome the difficulties of ordinary PCA.