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Learning Stable Multivariate Baseline Models for Outbreak
Detection Sajid M. Siddiqi, Byron Boots,
Geoffrey J. Gordon, Artur W. DubrawskiThe Auton Lab
School of Computer Science Carnegie Mellon University
This work was partly funded by NSF grant IIS-0325581 and CDC award R01-PH000028
presented by Robin Sabhnani from the Auton Lab
Motivation
• Lots of health-related data available
• Much of this data is temporal
• Many data sources are also multivariate
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
Outline
• Linear Dynamical Systems
• Learning Stable Models
• Experimental Setup
• Results
• Conclusion
Linear Dynamical Systems (LDS)
X1
Y1
X2
Y2
. . . Xt
Yt
Xt+1
Yt+1
hidden variables (low-dimensional)
observed data (high-dimensional)
Linear Dynamical Systems (LDS)
X1
Y1
X2
Y2
. . . Xt
Yt
Xt+1
Yt+1
hidden variables (low-dimensional)
observed data (high-dimensional)
hidden variables (low-dimensional)
observed data (high-dimensional)
Linear Dynamical Systems (LDS)
X1
Y1
X2
Y2
. . . Xt
Yt
Xt+1
Yt+1
• Dynamics matrix A models temporal evolution
hidden variables (low-dimensional)
observed data (high-dimensional)
Linear Dynamical Systems (LDS)
X1
Y1
X2
Y2
. . . Xt
Yt
Xt+1
Yt+1
• Dynamics matrix A models temporal evolution
• Multivariate Gaussian noise vt , wt models
interaction between streams
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
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 |1during 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
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
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
Data Simulations
• Instability causes Least-Squares simulations to diverge
• Constraint Generation yields most realistic simulations that are also stable
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 xtxt+1, 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
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
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
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
