Principal component analysis: Difference between revisions

Revision as of 19:52, 13 February 2025

Revision as of 19:54, 13 February 2025

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The applicability of PCA as described above is limited by certain (tacit) assumptions<ref>Jonathon Shlens, [https://arxiv.org/abs/1404.1100 A Tutorial on Principal Component Analysis.]</ref> made in its derivation. In particular, PCA can capture linear correlations between the features but fails when this assumption is violated (see Figure 6a in the reference). In some cases, coordinate transformations can restore the linearity assumption and PCA can then be applied (see [[Kernel principal component analysis|kernel PCA]]).

Another limitation is the mean-removal process before constructing the covariance matrix for PCA. In fields such as astronomy, all the signals are non-negative, and the mean-removal process will force the mean of some astrophysical exposures to be zero, which consequently creates unphysical negative fluxes,<ref name="soummer12"/> and forward modeling has to be performed to recover the true magnitude of the signals.<ref name="pueyo16">{{Cite journal|arxiv= 1604.06097 |last1= Pueyo|first1= Laurent |title= Detection and Characterization of Exoplanets using Projections on Karhunen Loeve Eigenimages: Forward Modeling |journal= The Astrophysical Journal |volume= 824|issue= 2|pages= 117|year= 2016|doi= 10.3847/0004-637X/824/2/117|bibcode = 2016ApJ...824..117P|s2cid= 118349503|doi-access= free}}</ref> As an alternative method, [[non-negative matrix factorization]] focusing only on the non-negative elements in the matrices is well-suited for astrophysical observations.<ref name="blantonRoweis07"/><ref name="zhu16"/><ref name="ren18"/> See more at the [[#Non-negative matrix factorization|the relation between PCA and non-negative matrix factorization]].

PCA is at a disadvantage if the data has not been standardized before applying the algorithm to it. PCA transforms the original data into data that is relevant to the principal components of that data, which means that the new data variables cannot be interpreted in the same ways that the originals were. They are linear interpretations of the original variables. Also, if PCA is not performed properly, there is a high likelihood of information loss.<ref>{{cite web | title=What are the Pros and cons of the PCA? | website=i2tutorials | date=September 1, 2019 | url=https://www.i2tutorials.com/what-are-the-pros-and-cons-of-the-pca/ | access-date=June 4, 2021}}</ref>