Distributional data analysis: Difference between revisions

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Let the reference measure <math>\nu_0</math> be the Wasserstein mean <math>\mu_\oplus</math>.

Then, a ''principal geodesic subspace (PGS)'' of dimension <math>k</math> with respect to <math>\mu_\oplus</math> is a set <math>G_k = \operatorname{argmin}_{G \in \text{CG}_{\nu_\oplus, k}(\mathcal{W}_2)} K_{W_2}(G)</math>.<ref name="gpca1">{{Cite journal|last1=Bigot|first1=J.|last2=Gouet|first2=R.|last3=Klein|first3=T.|last4=López|first4=A.|date=2017|title=Geodesic PCA in the Wasserstein space by convex PCA|journal=Annales de l'institut Henri Poincare (B) Probability and Statistics|volume=53|issue=1|pages=1–26|doi=10.1214/15-AIHP706|bibcode=2017AnIHP..53....1B |s2cid=49256652 |url=https://hal.archives-ouvertes.fr/hal-01978864/file/AIHP706.pdf }}</ref><ref name="gpca2">{{Cite journal|last1=Cazelles|first1=E.|last2=Seguy|first2=V.|last3=Bigot|first3=J.|last4=Cuturi|first4=M.|last5=Papadakis|first5=N.|date=2018|title=Geodesic PCA versus Log-PCA of histograms in the Wasserstein space|journal=SIAM Journal on Scientific Computing|volume=40|issue=2|pages=B429–B456|doi=10.1137/17M1143459 |bibcode=2018SJSC...40B.429C }}</ref>

Note that the tangent space <math>T_{\mu_\oplus}</math> is a subspace of <math>L^2_{\mu_\oplus}</math>, the Hilbert space of <math>{\mu_\oplus}</math>-square-integrable functions. Obtaining the PGS is equivalent to performing PCA in <math>L^2_{\mu_\oplus}</math> under constraints to lie in the convex and closed subset.<ref name="gpca2"/> Therefore, a simple approximation of the Wasserstein Geodesic PCA is the Log FPCA by relaxing the geodesicity constraint, while alternative techniques are suggested.<ref name="gpca1"/><ref name="gpca2"/>