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Abstract Causal inference in complex systems has been largely promoted by the proposal of some advanced temporal causation models. However, temporal models have serious limitations when time series data are not available or present insignificant variations, which causes a common challenge for earth system science. Meanwhile, there are few spatial causation models for fully exploring the rich spatial cross-sectional data in Earth systems. The generalized embedding theorem proves that observations can be combined together to construct the state space of the dynamic system, and if two variables are from the same dynamic system, they are causally linked. Inspired by this, here we show a Geographical Convergent Cross Mapping (GCCM) model for spatial causal inference with spatial cross-sectional data-based cross-mapping prediction in reconstructed state space. Three typical cases, where clearly existing causations cannot be measured through temporal models, demonstrate that GCCM could detect weak-moderate causations when the correlation is not significant. When the coupling between two variables is significant and strong, GCCM is advantageous in identifying the primary causation direction and better revealing the bidirectional asymmetric causation, overcoming the mirroring effect.
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This content will become publicly available on January 15, 2026
Extending empirical dynamic modeling to cross-sectional data beyond traditional time series
Abstract The foundation of Empirical dynamic modeling (EDM) is in representing time-series data as the trajectory of a dynamic system in a multidimensional state space rather than as a collection of traces of individual variables changing through time. Takens’s theorem provides a rigorous basis for adopting this state-space view of time-series data even from just a single time series, but there is considerable additional value to building out a state space with explicit covariates. Multivariate EDM case studies to-date, however, generally rely on building up understanding first from univariate to multivariate and use lag-coordinate embeddings for critical steps along the path of analysis. Here, we propose an alternative set of steps for multivariate EDM analysis when the traditional roadmap is not practicable. The general approach borrows ideas of random data projection from compressed sensing, but additional justification is described within the framework of Takens’s theorem. We then detail algorithms that implement this alternative method and validate through application to simulated model data. The model demonstrations are constructed to explicitly demonstrate the possibility for this approach to extend EDM application from time-series trajectories to effectively realizations of the underlying vector field, i.e. data sets that measure change over time with very short formal time series but are otherwise “big” in terms of number of variables and samples.
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- Award ID(s):
- 1660584
- PAR ID:
- 10570330
- Publisher / Repository:
- bioRxiv
- Date Published:
- Format(s):
- Medium: X
- Institution:
- bioRxiv
- Sponsoring Org:
- National Science Foundation
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