Proximal Graphical Event Model IBM Research Debarun Bhattacharjya, - - PowerPoint PPT Presentation

IBM Research

Proximal Graphical Event Model

Debarun Bhattacharjya, Dharmashankar Subramanian, Tian Gao

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Event datasets: Occurrences of various event types over time

• Examples: web logs; customer transactions; network notifications; political events; financial

events; insurance claims; health episodes; other medical events

• Notation: 𝑬 = 𝑚𝑗, 𝑢𝑗 , 𝑗 = 1, … , 𝑂; 𝑚𝑗 ∈ 𝑀, 𝑀 = 𝑁

– Assume it is temporally ordered b/w time 𝑢0 = 0 ≤ 𝑢1 and 𝑢𝑂+1 = 𝑈 ≥ 𝑢𝑂 – Note that there are 𝑁 types of event types/labels and 𝑂 events in the dataset

Objective: To learn statistical and causal relationships between event types in the form of graphical models using event datasets

Prescription refill Hospital admission Home health visit

IBM Research

Proximal Graphical Event Model (PGEM)

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• PGEM = 𝐻, 𝑋, Λ ; graph + set of (time) windows on each edge

and conditional intensity parameters

• Assumption: The intensity of an event label (node) depends on

whether or not its parents have happened at least once in their respective recent histories

B A C

waa wac wbc wab

Formally, denoting a node 𝑌’s parents as 𝑽:

• 𝐻 = 𝑀, 𝐹 where 𝑀 is the event label set
• There is a window for every edge, 𝑋 = 𝑥𝑦: ∀𝑌 ∈ 𝑀 , where 𝑥𝑦 = 𝑥𝑨𝑦: ∀𝑎 ∈ 𝑽
• There is an intensity parameter for every node 𝑌 and for every instantiation 𝒗 of

its parent occurrences, Λ = 𝜇𝑦|𝒗

𝑥𝑦 : ∀𝑌 ∈ 𝑀

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Parameter and Structure Learning

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Learning problem: Given an event dataset 𝑬, learn PGEM = 𝐻, 𝑋, Λ

• Log-likelihood:

– 𝑂 𝑦; 𝒗 : # of times 𝑌 is observed and the condition 𝒗 is true in the relevant windows – 𝐸 𝒗 : duration over the entire time period where the condition 𝒗 is true

• For a given graph, finding the optimal (MLE) conditional

intensities when given the windows is easy, but finding the

• ptimal windows is hard!
• Contribution 1: Analysis and proof that reduces the window

search to a finite set that is algorithmically constructed.

• Contribution 2: A method to search over graph structures,

with some theoretical results on efficient search and consistency justification

IBM Research

Results: Synthetic Datasets

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Wed Dec 5, 5:00 – 7:00 pm, Room 210 & 230 AB #6