The FASK algorithm FASK (Fast Adjacency Skewness) appeals to - - PowerPoint PPT Presentation

The FASK algorithm

• FASK (Fast Adjacency Skewness) appeals to

Skewness.

• It runs the Fast Adjacency Search (FAS) to find

edges that can be found from linearity.

• It uses a heuristic skewness rule to add additional

edges to the graph.

• It uses two other skewness rules to orient all of the

edges in the graph.

• If they can be oriented as 2-cycles, orient them as such.
• Otherwise, if they can oriented one direction or the
• ther, do so.

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Sanchez-Romero, R., Ramsey, J. D., Zhang, K., Glymour, M. R., Huang, B., & Glymour, C. (2018). Causal Discovery of Feedback Networks with Functional Magnetic Resonance

• Imaging. bioRxiv, 245936.

A Kind of Skewness

• Let X be smoothly positively skewed about 0 (for

centered X) if for every b, the area under the p.d.f. from –b to 0 is greater than the area under the p.d.f. from 0 to b.

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• b

b

Assumptions

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Basic idea of the left-right rule

Lemma 1. Let X _||_ eY, X, eY smoothly positively skewed, X, eY centered, Y = aX + eY, a > 0. Then E(XeY) in regions A + B + C + D must necessarily be negative. The picture shows a = 1.

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And it follows after a few steps of algebra:

This is the Left-Right rule. It’s a pairwise orientation rule. This is almost saying corr(X, Y | X > 0) > corr(X, Y | Y > 0) but without the centering of the variables. Theorem 1 is true if X à Y, a > 0, X, Y, eX, and eY smoothly positively skewed Any problem you can transform into a problem like this you can give an answer to.

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Additional flips

• Need to “flip” the direction for each of the

following that holds:

• The skewness of X is negative.
• The skewness of Y is negative.
• Additionally, if corr(X, Y) < delta (default value -0.2) after

the previous flips

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Heuristic Skewness Adjacency Rule

• If X and Y are independent, then abs(corr(X, Y | X > 0) -

corr(X, Y | Y > 0)) = abs(0 – 0) = 0. So if this absolute difference is different from zero, then there must be a trek between X and Y.

• If it’s very different from zero, we add an adjacency

(heuristic).

• We use a cutoff of 0.3, which we got from experience with

fMRI data.

• It might be good to go back and condition on intermediaries

in the graph to see if the edge can be explained away. We didn’t do this.

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2-Cycles

• The Left-Right rule assumes X—Y is not a 2-cycle, so we have to
• rient those first.
• If X and Y have no adjacents, we simply check to see if corr(X, Y)

!= corr(X, Y | X > 0) or corr(X, Y | Y > 0), using a T-test.

• For control 2-cycles (where coefficient in opposite directions have
• pposite signs), we check signs of the differences. For X à Y

these will be the same, but for control 2-cycles they will be different.

• If X and Y have adjacents, we condition on subsets of the
• adjacents. (See theory for the Cyclic Causal Discovery algorithm

(Richardson and Spirtes).

• We don’t try to detect confounders, so confounders will look like

2-cycles in FASK output if they are detectable.

• We used a single-subject sample size of 500 since MRI scanning

protocols are better now.

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