The difference between the conditional distributions' average standard deviation in both directions: $$C_{CDS} = CDS(X|Y) - CDS(Y|X)$$

University of Michigan - Ann Arbor

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Introduced by Fonollosa, the standard deviation of the conditional distributions (CDS) is a regression-based feature that captures distribution asymmetry for causality detection. The CDS score measures the spread of the conditional distributions after normalization of the bins:

$$CDS(X,Y) = \sqrt{ \frac{1}{M} \sum_{y=0}^{M-1} var_x (p_n(Y=y|X=x)) }$$

Here, $$p_n(Y=y|X=x)$$ represents the normalized conditional probability and $$var_x$$ is the sample variance over $$x$$. This feature achieves good performance even when used alone, typically yielding a score of 0.69 on the Tübingen dataset.

Standard Deviation of the Conditional Distributions (CDS)

Several cause-effect pair filters (not involving any learning) have been proposed as causation coefficients, based on various data generative assumptions. Here are some examples: - R-squared $$C_{R2}$$ - Input/Residual Independence $$C_{IR}$$ - Entropy $$C_H$$ - Slope $$C_S$$ - Conditional Distribution Standard Deviation $$C_{CDS}$$

Examples of Cause-Effect Pair Filters as Causation Coefficients

 Isabelle Guyon, Alexander Statnikov, Berna Bakir Batu (2019)
Cause Effect Pairs in Machine Learning (The Springer Series on Challenges in Machine Learning)
https://www.amazon.com/Effect-Machine-Learning-Springer-Challenges/dp/3030218090

Cause Effect Pairs in Machine Learning

Conditional Distribution Standard Deviation (Cause-Effect Pair Filter)

The difference between statistics of independence between input and residual: $$C_{IR} = I(R_y, X) - I(R_x, Y)$$

Input/Residual Independence (Cause-Effect Pair Filter)

The difference between the R-squared statistics measuring goodness of fit in both directions: $$C_{R2} = R_y^2 - R_x^2$$

R-squared (Cause-Effect Pair Filter)

The difference in (discretized) entropy of the marginal distributions: $$C_H = H_x - H_y$$, with $$H_x = \sum_{k=1}^m p_k \log p_k$$ using $$m = 20$$ bins.

Entropy (Cause-Effect Pair Filter)

The difference in average slope in both directions: $$C_S = S_y - S_x$$, with $$S_y = \sum_{k=1}^{m-1} p_k \frac{\Delta y_k}{\Delta x_k}$$ using $$m = 20$$ intervals.

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