Learning Stable Multivariate Baseline Models for Outbreak Detection

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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
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Motivation

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

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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

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Outline

• Linear Dynamical Systems
• Learning Stable Models
• Experimental Setup
• Results
• Conclusion

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Linear Dynamical Systems (LDS)
hidden variables (low-dimensional)
. . .
X1
X2
Xt
Xt1
Y1
Y2
Yt
Yt1
observed data (high-dimensional)
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Linear Dynamical Systems (LDS)
hidden variables (low-dimensional)
. . .
X1
X2
Xt
Xt1
Y1
Y2
Yt
Yt1
observed data (high-dimensional)
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Linear Dynamical Systems (LDS)
hidden variables (low-dimensional)
. . .
X1
X2
Xt
Xt1
Y1
Y2
Yt
Yt1
observed data (high-dimensional)

• Dynamics matrix A models temporal evolution

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Linear Dynamical Systems (LDS)
hidden variables (low-dimensional)
. . .
X1
X2
Xt
Xt1
Y1
Y2
Yt
Yt1
observed data (high-dimensional)

• Dynamics matrix A models temporal evolution
• Multivariate Gaussian noise vt , wt models
  interaction between streams

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Linear Dynamical Systems

• The Good
• Linear Dynamical Systems (aka State-Space models,
  aka Kalman Filters) are a generalization of ARMA
  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

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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 lt 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
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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

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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
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Data Simulations

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

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State Reconstruction

• Obtained by computing the residual ?t Axt
  xt12 , where xt are the estimated states
• Least squares has the best score by definition,
  since it is learned by regression on xt?xt1, but
  at the cost of instability

squared error Multi-drug data Multi-zip data
Constraint Generation 57,338 26,171
LB-1 60,669 26,431
Least Squares 56,203 16,918

• Stable methods trade off reconstruction error vs.
  stability
• Constraint Generation learns the most accurate
  models that are also stable

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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 gt t

avg sqd err (std dev) Multi-drug data Multi-zip data
Constraint Generation 59,845 (310) 45,465 (317)
LB-1 53,494 (364) 44,677 (266)
Least Squares 79,649 (648) n/a

• Stable methods yield superior results to least
  squares

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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
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Thank You! Questions?

• further questions to siddiqi_at_cs.cmu.edu