Post provided by Yair Daon

Who am I?

I’m Yair Daon, a mathematician-turned-epidemiologist at Bar-Ilan University’s Faculty of Medicine. Most days I stare at time-series curves that claim one thing “drives” another. When those claims are wrong, public-health decisions can drift off course. Our new Methods in Ecology & Evolution paper introduces a fast way to shout “no!” before that happens.

A two-minute primer for the non-specialist

Embedding & attractors: Imagine watching a mechanical clock through a key-hole: you record only the tip of the second hand. A theorem of Takens (Takens, 1981) implies that by stacking today’s position, yesterday’s, the day before, and so on, you can still reconstruct the hidden gears. The smallest stack that works is the embedding dimension. The curl of points you uncover is the system’s attractor; its “thickness” is the attractor dimension. This can help us understand causation in a key way: If Cause Y causes Outcome X, then things needed to explain Outcome X = things needed to explain Cause Y + maybe more things. “Things needed to explain” determine the attractor dimension, so Cause’s must be smaller than Outcome’s.  

Sugihara et al. (2012) taught ecologists a clever test: if past flu counts let you predict humidity, then flu encodes all information on humidity, hence humidity causes flu. Elegant? Yes. But in strongly synchronised systems the method sometimes shouts “cause!”  when it shouldn’t (Baskerville & Cobey, 2017) .

A puzzle in 2022

In 2022 I was playing with embedding dimensions — how many coordinates you need to rebuild a system’s dynamics from a single observable. Recalling a conversation from 2017 with Sugihara, I thought of using the embedding dimension as a criterion for causality. My former MSc advisor, Omri Sarig, suggested I look at the dimension of the attractor instead. Around the same time, I was reading some of Judea Pearl’s books on causal inference and noted that “causal assumptions are encoded (…) in the missing links” (Pearl, 2016). While Sugihara’s test is excellent at detecting links, if you want to refute a link you need a complementary test. The need felt even more urgent after chatting with my postdoc advisors Uri Obolski and Amit Huppert about the “Does influenza drive absolute humidity?” paper. Refuting causal relations suddenly felt both urgent and feasible.

When the phone cracked

The engineering hurdle was smoothing and the breakthrough came in a phone-repair shop. While I waited for a shattered screen to be replaced, I scribbled on a napkin and realised that the optimal hard threshold (Gavish & Donoho, 2014) — originally a denoising trick for matrices — could solve the filtering headache in causal tests. Too much freedom in smoothing lets anyone manufacture a convenient answer; Gavish’s threshold, folded into Singular Spectrum Analysis, gave me a robust, almost parameter-free way to let the data speak. Only one number is needed (the window length) and as long as that window is a small multiple of the system’s period, results barely change.

Meet BCAD

The resulting workflow is surprisingly simple:

• Smooth noisy series via Singular Spectrum Analysis and hard threshold.

• Estimate each series’ attractor dimension.

• Bootstrap attractors to find confidence intervals.

• Refute: if dim X < dim Y for many bootstrap samples — then Y does not cause X.

Why bother? Because rejecting false causal relations is cool, interesting and saves huge modelling effort. In “Does influenza drive absolute humidity?” (Baskerville & Cobey, 2017), CCM often claimed the virus affected the weather. BCAD, run on US state data, rejected that backwards link in 46 of 48 states while preserving the correct forward link (humidity causes flu).

Caveats in bold

Strong synchrony hurts. When driver and response series are nearly identical, their attractors merge and BCAD loses power. Medium and weak synchronization pose no problem, though.

Data length matters. Reliable dimension estimates need reasonably long, contiguous records. We do not know exactly what the limit is, though, but very short or gap-ridden series can definitely deceive us.

We spell out these limits—and a few algorithmic safeguards—in the paper.

People who lit the path

A special thanks to Irena Vankova for dragging me to George Sugihara’s seminar at the Courant Institute ten years ago, to Omri Sarig for the original attractor-dimension spark, and to Uri and Amit, who provided endless optimism and reality-checks throughout the project.

Grab the code, kick the tyres

Paper: https://doi.org/10.1111/2041-210X.70066

Code & tutorial: https://github.com/yairdaon/BCAD.

Bonus: numba-accelerated cross-immunization model (Gog and Swinton, 2002) at https://github.com/yairdaon/fts.

I’m on X/Twitter as @YairDaon. Tag / DM me if you are want to better understand causal structures in your system of interest, spot a bug, or hit a weird corner-case. When it comes to causal relations, refuting matters as much as linking.

References:

Takens, Floris. “Detecting strange attractors in uid turbulence.” Dynamical Systems and Turbulence 898 (1981): 366.

Sugihara, George, et al. “Detecting causality in complex ecosystems.” science 338.6106 (2012): 496-500.

Baskerville, Edward B., and Cobey, Sarah. “Does influenza drive absolute humidity?.” Proceedings of the National Academy of Sciences 114.12 (2017): E2270-E2271.

Pearl, Judea. “Causal inference in statistics: An overview.” (2009): 96-146.

Gavish, Matan, and Donoho, David L. “The optimal hard threshold for singular values is 4/√3.” IEEE Transactions on Information Theory 60.8 (2014): 5040-5053.