Temporal Graph Pattern Mining
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The document discusses temporal graph pattern mining in clinical pathways to analyze relationships between clinical activities using directed acyclic dependency graphs. It details the process of constructing aggregated graphs from activity records, and presents methods for regularization and optimization in the context of temporal phenotyping from electronic health records. The approach aims to improve prediction accuracies for clinical events, exemplified by a case study on congestive heart failure hospitalization predictions.
Temporal Graph Pattern Mining
- 1. TEMPORAL GRAPH PATTERN MINING PRESENTED BY EUGENEYANG EY120@GEORGETOWN.EDU
- 2. MOTIVATION • Find the relationship between clinical activities • Encode these information as feature
- 3. DEPENDENCY PATTERNS IN CLINICAL PATHWAYS Lin, Fu-ren, et al. "Mining time dependency patterns in clinical pathways." International Journal of Medical Informatics 62.1 (2001): 11-25.
- 4. BASIC SETUP • An activity record(process log data) of the process 𝑃𝑖 can be written as a 4- tuple 𝑃𝑖, 𝐴𝑖𝑗, 𝑇𝑠,𝑖𝑗, 𝑇𝑒,𝑖𝑗 • If activity 𝐴𝑖𝑗 starts right after 𝐴𝑖𝑘 without any other activities in between, we say 𝑨𝒊𝒋 is depends on 𝑨𝒊𝒌 • We can then construct a directed acyclic dependency graph 𝐺 𝑝 = (𝑉𝑝, 𝐸 𝑝) where 𝑉𝑝 is the activities with pseudo start/end node and 𝐸 𝑝 is the Boolean dependency between every 2 activities.
- 6. AGGREGATING THE INFORMATION • A Large Graph 𝐿𝐺 𝑘 is set of aggregated graphs with at most 𝑘 nodes • Algorithm 1. Create activity nodes by counting the activity with support over certain threshold (i.e. creating 𝐿𝐺1). 2. By looking at dependency between every 2 activities, create 𝐿𝐺2 by adding pairs with certain minimum support. 3. Construct 𝐿𝐺 𝑘 by adding edges that have not included in 𝐿𝐺 𝑘−1 from 𝐿𝐺2
- 9. EXAMPLE ON REAL DATA • Adding some runtime activities • E.g. blood pressure • Using training/testing to test the validity and stability of this method 𝑆𝑖𝑚 𝑃𝑖, 𝑃𝑗 = 2 × 𝐸𝑖 ∪ 𝐸𝑗 E𝑖 + 𝐸𝑗 𝑃𝑟𝑒𝑑𝑖𝑐𝑖𝑜𝑛 𝐴𝑐𝑐𝑢𝑟𝑎𝑐𝑦 = 1 𝑛 𝑖=1 𝑛 max 𝑃 𝑗∈𝑇𝑟𝑎𝑖𝑛𝑖𝑛𝑔 𝑆𝑖𝑚(𝑃𝑖, 𝑃𝑗)
- 12. TEMPORAL PHENOTYPING WITH GRAPH BASED FRAMEWORK Liu, Chuanren, et al. "Temporal phenotyping from longitudinal electronic health records: A graph based framework." Proceedings of the 21th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. ACM, 2015.
- 13. BASIC SETUP • Motivation: try to embed relationship between events as features • Construct a weighted directed temporal graph of event sequence • E.g. diagnosis, medication, lab test, etc. • Weight Function 𝑊𝑖𝑗 𝑛 = 1 𝐿 𝑛 1≤𝑝≤𝑞≤𝐿 𝑛 𝑥 𝑛𝑝 = 𝑖 ∧ 𝑥 𝑛𝑞 = 𝑗 𝜅(𝑡 𝑛𝑞 − 𝑡 𝑛𝑝) where 𝜅 ∙ is a non-increasing function(here we choose as Exponential distribution function if input is greater than constant Δ)
- 15. BASIS DECOMPOSITION OF GRAPH • Decompose graph weight matrix into aggregation of basis • Use index of basis as feature
- 16. BASIS DECOMPOSITION OF GRAPH • For weight matrix 𝑊 𝑛 , 𝑊 𝑛 = 𝑘=1 𝐾 𝐴 𝑛𝑘 𝐵 𝑘 where 𝐴 ∈ ℝ 𝑁×𝐾 • We minimize the estimation error 𝒥 𝐴, 𝐵 = 1 2 𝑛=1 𝑁 𝑊 𝑛 − 𝑘=1 𝐾 𝐴 𝑛𝑘 𝐵 𝑘 𝐹 2 where ∙ 𝐹 is the matrix Frobenius norm.
- 17. REGULARIZATION • Like most of the ML minimization problem, we can we can add regularization term to objective function 𝒥 𝐴, 𝐵 = 1 2 𝑛=1 𝑁 𝑊 𝑛 − 𝑘=1 𝐾 𝐴 𝑛𝑘 𝐵 𝑘 𝐹 2 + 𝜆Ω 𝐴 where regularization function Ω 𝐴 ≥ 0 and some constant 𝜆 ≥ 0
- 18. REGULARIZATION • Similarity based regularization Ω 𝐴 = 1 2 𝑛1,𝑛2 1 2 𝐴 𝑛1 − 𝐴 𝑛2 2 𝑆 𝑛1 𝑛2 = 1 2 𝑡𝑟 𝐴′ 𝐿𝐴 where 𝑆 is the similarity symmetric matrix and 𝐿 = 𝐷 − 𝑆 • Model based regularization Ω 𝐴 = − 1 ℒ 𝑛∈ℒ log Pr 𝐴 𝑛, 𝑌𝑛 ℋ where ℒ is the training set and 𝑌𝑛 is the correspondent label
- 19. REGULARIZATION • We can easily embed a discriminative model under this setup, e.g. logistic regression Pr 𝐴 𝑛, 𝑌𝑛 ℋ = 1 1 + exp(−𝑌𝑛 𝑓(𝐴 𝑛)) ℋ: 𝐴 𝑛 ⟼ 𝑓 𝐴 𝑛 = AnΘ + 𝜃 • Or hinge loss(SVM) Ω 𝐴 𝑛 = 1 ℒ 𝑛∈ℒ max(0, 1 − 𝑌𝑛 𝑓(𝐴 𝑛))
- 20. OPTIMIZATION • Similarity based regularization 𝐴 ⟵ 𝑝𝑟𝑜𝑗𝑠𝑝𝑙𝑥 𝐴 − 𝛼 𝜕𝒥 𝜕𝐴 min 𝐵𝑖𝑗 ∗ 1 2 𝑛=1 𝑁 𝑊𝑖𝑗 𝑛 − 𝑘=1 𝐾 𝐴 𝑛𝑘 𝐵𝑖𝑗 𝑘 2
- 21. OPTIMIZATION • Model based regularization • The loss function space could be convex but not differentiable(e.g. SVM) • Apply proximal gradient optimization
- 25. REAL EXAMPLE • Congestive Heart Failure(CHF) • One-year hospitalization prediction after CHF confirmation • Prediction of CHF
- 30. QUESTIONS?