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Learning Stable Multivariate Baseline Models for Outbreak Detection

Learning Stable Multivariate Baseline Models for Outbreak Detection. Sajid M. Siddiqi, Byron Boots, Geoffrey J. Gordon, Artur W. Dubrawski The Auton Lab School of Computer Science Carnegie Mellon University. presented by Robin Sabhnani from the Auton Lab.

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Learning Stable Multivariate Baseline Models for Outbreak Detection

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  1. Learning Stable Multivariate Baseline Models for Outbreak Detection Sajid M. Siddiqi, Byron Boots, Geoffrey J. Gordon, Artur W. Dubrawski The Auton Lab School of Computer Science Carnegie Mellon University presented by Robin Sabhnani from the Auton Lab This work was partly funded by NSF grant IIS-0325581 and CDC award R01-PH000028

  2. Motivation • Lots of health-related data available • Much of this data is temporal • Many data sources are also multivariate

  3. Motivation • When detecting anomalies, the crucial information could be hidden in the dynamics of the data as well as the interaction between different data streams • Our goal: Learn good models for simulating baseline data for use in training algorithms as well as detecting anomalies • Linear Dynamical Systems are a good choice

  4. Outline • Linear Dynamical Systems • Learning Stable Models • Experimental Setup • Results • Conclusion

  5. Linear Dynamical Systems (LDS) hidden variables (low-dimensional) . . . X1 X2 Xt Xt+1 Y1 Y2 Yt Yt+1 observed data (high-dimensional)

  6. Linear Dynamical Systems (LDS) hidden variables (low-dimensional) . . . X1 X2 Xt Xt+1 Y1 Y2 Yt Yt+1 observed data (high-dimensional)

  7. Linear Dynamical Systems (LDS) hidden variables (low-dimensional) • Dynamics matrix A models temporal evolution . . . X1 X2 Xt Xt+1 Y1 Y2 Yt Yt+1 observed data (high-dimensional)

  8. Linear Dynamical Systems (LDS) hidden variables (low-dimensional) • Dynamics matrix A models temporal evolution • Multivariate Gaussian noise vt , wt models interaction between streams . . . X1 X2 Xt Xt+1 Y1 Y2 Yt Yt+1 observed data (high-dimensional)

  9. Linear Dynamical Systems • The Good: • Linear Dynamical Systems (aka State-Space models, aka Kalman Filters) are a generalization ofARMA models and can represent a wide range of time series • LDS parameters can be learned from data • The Bad: • LDSs learned from data are often unstable • Simulation from an unstable LDS degenerates

  10. Stability • Stability of an LDS depends on its dynamics matrix A • Let {1,…,n} be the eigenvalues of A in decreasing order of magnitude • A is stable if |1| <= 1 • Constraining|1| during learning is hard • We devise an iterative optimization method* that beats previous approaches in efficiency and accuracy *“A Constraint Generation Approach to Learning Stable Linear Dynamical Systems”, S. Siddiqi, B. Boots, G. Gordon, NIPS 2007

  11. Stability • Learning stable LDS models allows us to: • Compress large temporal multivariate datasets • Generate realistic data sequences • Predict the future given some data • Deviations from predicted data indicate anomalies

  12. Experimental Setup • Data: • OTC drug sales data for 22 categories in 29 Pittsburgh zip codes over 60 days • track all zipcodes for cough/cold category (multi-zipcode data) • track all drug-categories for city of pittsburgh (multi-drug data) • Experiments: • Learn a LDS model using first 15 days, and • Simulate a sequence (qualitative task) • Reconstruct state sequence (quantitative task) • Predict future occurrences (quantitative task) • Algorithms: • Constraint Generation (our method), • LB-1* (state of the art stability algorithm), • Least Squares (naïve, no stability guarantees) *“Subspace Identification with guaranteed stability using subspace identification”, S. Lacy and D. Bernstein, ACC 2002

  13. Data Simulations • Instability causes Least-Squares simulations to diverge • Constraint Generation yields most realistic simulations that are also stable

  14. State Reconstruction • Obtained by computing the residual t ||Axt – xt+1||2,where {xt} are the estimated states • Least squares has the best score by definition, since it is learned by regression on xtxt+1, but at the cost of instability • Stable methods trade off reconstruction error vs. stability • Constraint Generation learns the most accurate models that are also stable

  15. Prediction (preliminary results) • Average prediction error obtained by tracking (filtering) up to time t,then simulating upto time t’ and calculating the sum of squared error, and averaging this over all t and t’ > t • Stable methods yield superior results to least squares

  16. Conclusion • Linear Dynamical Systems effective at modeling multivariate time series data • Stability crucial for accurate performance • Superior performance of stable methods in baseline generation and prediction on OTC data • Constraint Generation learns a more accurate model with more realistic simulations, most efficiently*. Further work needed on prediction accuracy metric. * “A Constraint Generation Approach to Learning Stable Linear Dynamical Systems”, S. Siddiqi, B. Boots, G. Gordon, NIPS 2007

  17. Thank You! Questions? further questions to siddiqi@cs.cmu.edu

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