Learning Causal Structure from

Undersampled Time Series

David DanksPhilosophy (& Psychology)

Carnegie Mellon

Sergey PlisMind Research Network

The Situation

Causal timescale Measurement timescale≠

~100 ms ~2 sec????

Unknown extent!

What causal inferences can be made in this situation?

Two Challenges

1. Forwards inference: Given a causal structure at causal timescale, what is implied structure at (undersampled) measurement timescale?

2. Backwards inference: Given inferred causal structure at measurement timescale (with unknown undersampling), what structures at causal timescale are possible?

Representation

Causal timescale Measurement timescale

undersample by 2

(Alternate, Better) Representation

1

2 3

4

56

1 1

2 2

3 3

4 4

5 5

6 6

1

2 3

4

56

1 1

2 2

3 3

4 4

5 5

6 6

(Alternate, Better) Representation

Causal timescale Measurement timescale

Forwards Inference

• Restatement: Given G1 and undersample rate u, what is Gu?

• Note: X → Y in Gu iff X → ... → Y of length u in G1

• Forwards inference = finding paths of particular lengths

• ⇒ Easy “black box” for forwards inference

• Special case Q: What is true about Gu for many different u?

Forwards Inference

• Undersampling destroys information:

Forwards Inference

• Strongly Connected Component (SCC): Maximal set of nodes S s.t. ∀X,Y ∃ path from X to Y

• SCCs are obviously cyclic• SCC ! set LS of simple loops

(i.e., no repeat nodes)• gcd(LS) := greatest common

divisor of simple loop lengths

Key notion!

3 Forward Theorems(importance of gcd...)

• When does Gu stabilize as u → ∞?

• When is SCC structure stable across all u?

• To what do SCCs converge (when they do)?

Backwards Inference

• Restatement: Given Gu, what are the possible <G1, u> pairs?

• Massive underdetermination for large u

• SCC-graph GS over nodes for SCCs := Si → Sj iff ∃Xi∈Si,Xj∈Sj Xi → Xj

• Encodes high-level between-SCC structure• Ignores where & how the SCCs connect

• Provably, always a DAG

Backwards Inference

• Constancy of SCC-graph:

• ⇒ Given G, we can efficiently recover SCCs & between-SCC structure in G1!

• Polynomial SCC identification algorithms

Backwards Inference

• What about within-SCC structure?• Super-clique ⇒ No internal information• Not-yet-super-clique ⇒ ????

• Reason for hope:

uniquely discoverable!

Fully-general learning algorithm still in development...

Thanks!

Research partially supported by: National Science Foundation