Add Multivariate Functional PCA (not FPCA)

[dani2442]

Motivation

I am studying the multivariate FPCA framework presented by Berrendero et al. (2011) [1]. Unlike conventional FPCA or mFPCA approaches, it does not rely on the Karhunen–Loève expansion. Instead, this method compresses functions $L^2(\mathbb{R}, \mathbb{R}^p)$ into $L^2(\mathbb{R}, \mathbb{R}^k)$ where $k \ll p$, whereas the other approaches [2,3,4] compress the signal into a finite-dimensional space.

Also, I would like to volunteer to implement this method.

[1] Berrendero, J. R. et al. (Sept. 2011). “Principal components for multivariate functional data”. In:
Computational Statistics & Data Analysis 55.9, pp. 2619–2634. issn: 0167-9473. doi: 10.1016/
j . csda . 2011 . 03 . 011. url: https : / / www . sciencedirect . com / science / article / pii /
S0167947311001022 (visited on 04/10/2025).
[2] Chiou, Jeng-Min et al. (2014). “Multivariate Functional Principal Component Analysis: A Nor-
malization Approach”. In: Statistica Sinica 24.4. Publisher: Institute of Statistical Science,
Academia Sinica, pp. 1571–1596. issn: 1017-0405. url: https://www.jstor.org/stable/
24310959 (visited on 04/10/2025).
[3] Jacques, Julien et al. (2014). “Model-based clustering for multivariate functional data”. In: Com-
putational Statistics & Data Analysis 71.C. Publisher: Elsevier, pp. 92–106. issn: 0167-9473.
doi: 10.1016/j.csda.2012.12.004. url: https://EconPapers.repec.org/RePEc:eee:
csdana:v:71:y:2014:i:c:p:92-106 (visited on 04/19/2025).
[4] Ramsay, J. O. et al. (2005). Functional Data Analysis. en. Springer Series in Statistics. New York,
NY: Springer. isbn: 978-0-387-40080-8 978-0-387-22751-1. doi: 10.1007/b98888. url: http:
//link.springer.com/10.1007/b98888 (visited on 04/19/2025).

Desired functionality

Implement the F2FPCA (Functional-to-Functional PCA) class—note: while there is no established naming convention, I suggest this name—within the dim_reduction folder. The implementation should follow the structure of the original FPCA class and include the following methods:

• fit
• transform
• inverse_transform

Alternatives

No response

Additional context

No response

[vnmabus]

Hi, and sorry for the delay in answering. In principle any FDA method that is widely used is a possible candidate for addition. I am not really familiar with the MFPCA from Berrendero. We were thinking to implement the MFPCA explained in https://www.tandfonline.com/doi/abs/10.1080/01621459.2016.1273115 in the future. Do you know what are the differences between these two versions?

In any case, we can always implement both. However I would prefer if they both extend the functionality of the FPCA class (maybe with a new parameter to select the method) instead of creating a new one, unless the parameters or attributes are wildly different.

[dani2442]

Hi, no problem

We were thinking to implement the MFPCA explained in https://www.tandfonline.com/doi/abs/10.1080/01621459.2016.1273115 in the future. Do you know what are the differences between these two versions?
Yes, the paper that you mention is a generalization of the Multivariate FPCA. It admits a more general domain for the functions.

The approach proposed by Berrendero et. al is different. Instead of compressing the functions to a finite-dimensional space, it compresses to a multivariate function with fewer dimensions, i.e., $L^2(\mathbb{R}, \mathbb{R}^p \mapsto L^2(\mathbb{R}, \mathbb{R}^k)$ where $k\ll p$.

However I would prefer if they both extend the functionality of the FPCA class (maybe with a new parameter to select the method) instead of creating a new one, unless the parameters or attributes are wildly different.

Okay, makes sense. Yes, I can use the already built class.

Here is a diagram that I've made to explain the difference better: