Distributional data analysis: Difference between revisions

Revision as of 05:50, 31 December 2023

Revision as of 05:51, 31 December 2023

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Let <math>\mathcal{F}</math> be a space of distributions <math>\nu</math> and let <math>d</math> be a metric on <math>\mathcal{F}</math> so that <math>(\mathcal{F}, d)</math> forms a [[metric space]]. There are various metrics available for <math>d</math>.<ref>{{Cite book|last1=Deza|first1=M.M.|last2=Deza|first2=E.|title=Encyclopedia of distances|publisher=Springer|year=2013}}</ref>

For example, suppose <math>\nu_1, \; \nu_2 \in \mathcal{F}</math>, and let <math>f_1</math> and <math>f_2</math> be the density functions of <math>\nu_1</math> and <math>\nu_2</math>, respectively. The Fisher-Rao metric is defined as
<math display="block"> d_{FR}(f_1, f_2) = \arccos \left( \int_D \sqrt{f_1(x) f_2(x)} dx \right). </math>.

For univariate distributions, let <math>Q_1</math> and <math>Q_2</math> be the quantile functions of <math>\nu_1</math> and <math>\nu_2</math>. Denote the <math>L^p</math>-Wasserstein space as <math>\mathcal{W}_p</math>, which is the space of distributions with finite <math>p</math>-th moments. Then, for <math>\nu_1, \; \nu_2 \in \mathcal{W}_p</math>, the <math>L^p</math>-[[Wasserstein metric]] is defined as

<math display="block"> d_{W_p}(\nu_1, \nu_2) = \left( \int_0^1 [Q_1(s) - Q_2(s)]^p ds \right)^{1/p}. </math>.

== Mean and variance ==