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=====Causal emergence in continuous variables=====

=====Causal emergence in continuous variables=====

Furthermore, in the paper <ref name="Chvykov_causal_geometry">{{cite journal|author1=Chvykov P|author2=Hoel E.|title=Causal Geometry|journal=Entropy|year=2021|volume=23|issue=1|page=24|url=https://doi.org/10.3390/e2}}</ref>, Hoel et al. proposed the theoretical framework of [[causal geometry]],  with the aim of generalizing causal emergence theory to function mappings  and dynamical systems with continuous states. This article defines <math>EI</math> for [[random function mapping]], and also introduces the concepts of intervention noise and [[causal geometry]], and compares this concept  with [[information geometry]]. [[Liu Kaiwei]] et al.<ref name="An_exact_theory_of_causal_emergence">{{cite journal|author1=Liu K|author2=Yuan B|author3=Zhang J|title=An Exact Theory of Causal Emergence for Linear Stochastic Iteration Systems|journal=Entropy|year=2024|volume=26|issue=8|page=618|url=https://arxiv.org/pdf/2405.09207}}</ref> developed an exact analytical framework for causal emergence in [[random iterative dynamical systems]].

Furthermore, in the paper <ref name="Chvykov_causal_geometry">{{cite journal|author1=Chvykov P|author2=Hoel E.|title=Causal Geometry|journal=Entropy|year=2021|volume=23|issue=1|page=24|url=https://doi.org/10.3390/e2}}</ref>, Hoel et al. proposed the theoretical framework of [[causal geometry]],  with the aim of generalizing causal emergence theory to function mappings  and dynamical systems with continuous states. This article defines <math>EI</math> for [[random function mapping]], and also introduces the concepts of intervention noise and [[causal geometry]], and compares this concept  with [[information geometry]]. Liu et al.<ref name="An_exact_theory_of_causal_emergence">{{cite journal|author1=Liu K|author2=Yuan B|author3=Zhang J|title=An Exact Theory of Causal Emergence for Linear Stochastic Iteration Systems|journal=Entropy|year=2024|volume=26|issue=8|page=618|url=https://arxiv.org/pdf/2405.09207}}</ref> developed an exact analytical framework for causal emergence in [[random iterative dynamical systems]].

====Rosas's causal emergence theory====

====Rosas's causal emergence theory====