1. Identify causality through geometric methods. 2. Derivative of the function of the time series is related to transfer entropy. 3. Provide tool $$\textit{geoC}$$ to identify causality geometrically. 4. Correlation dimension can be used to determine causality.

University of Michigan - Ann Arbor

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Slightly out of scope, but a very interesting read : 
[arxiv.org/abs/2002.02078](https://arxiv.org/abs/2002.02078) 

On Geometry of Information Flow for Causal Inference

If there exists a system X, we ask if the data contained in X is sufficient to predict future states of X, or if there is a system Y that can help improve our predictions of X. If there is, then we say X is not closed.

Granger Causality

Movement of information between systems. 

Information flow

Summary : Image

Given two dynamical systems x, y such that :
$$x' = f_1(x, y)$$
$$y' = f_2(x, y)$$
Where $a'$ denotes the next iteration/future version of $a$, transfer entropy is defined as : 
$$T_{y \rightarrow x} =  D_{KL}((p(x'|x))||p(x'|x,y) )$$


Transfer Entropy

Asymmetric Space Transfer Operator Theorem

Theorem 2

Conditional Correlation Dimensional Geometric Information flow

Correlation Dimensional Geometric information Flow

Results

1. Developed geometric interpretation of information flow as causal inference measured by positive transfer entropy
2. Geometric description allows for newer and efficient computational methods for causal inference.
3. Also reveals geometry of underlying data

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