Adaptive Sequential Bayesian Change Point Detection Ryan Turner

Whistler, BC December 12, 2009 Joint work with Yunus Saatci and Carl Edward Rasmussen

Turner (Engineering, Cambridge)

Adaptive Sequential Bayesian Change Point Detection

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Motivation

Handle nonstationarity in time series Avoid making point estimates of (changing) parameters Modular framework Tractability Online Probabilistic predictions Minimal hand tuning

Turner (Engineering, Cambridge)

Adaptive Sequential Bayesian Change Point Detection

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Ingredients The time since the last change point, namely the run length rt (τ ) The underlying predictive model (UPM) p(xt |x(t−τ ):t−1 =: xt , θm ) for any τ ∈ [1, . . . , (t − 1)], at time t The hazard function H(r|θh ) The hyper-parameters θ := {θh , θm } 8

Observations

6 4 2 0 −2 −4

Run Length

−6 200 150 100 50 0

0

100

200

300

400

500

600

700

800

900

1000

Time

Figure: Sample drawn from BOCPD. Turner (Engineering, Cambridge)

Adaptive Sequential Bayesian Change Point Detection

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Previous Work

Test based approaches Retrospective Bayesian approaches Bayesian Online Change Point Detection (BOCPD) (e.g., Adams & MacKay 2007) BOCPD sensitive to hyper-parameters

Turner (Engineering, Cambridge)

Adaptive Sequential Bayesian Change Point Detection

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The BOCPD Algorithm The goal in BOCPD is to calculate the posterior run length at time t, i.e., p(rt |x1:t ), sequentially. p(xt+1 |x1:t ) =

X

p(xt+1 |x1:t , rt )p(rt |x1:t ) =

rt

X

(r)

p(xt+1 |xt )p(rt |x1:t ) ,

rt

(1) γt := p(rt , x1:t ) =

X

p(rt , rt−1 , x1:t )

rt−1

=

X rt−1

(r)

p(rt |rt−1 ) p(xt |rt−1 , xt ) p(rt−1 , x1:t−1 ) . {z }| {z } | {z } | hazard

likelihood (UPM)

(2)

γt−1

This defines a forward message passing scheme p(rt |x1:t ) ∝ γt .

Turner (Engineering, Cambridge)

Adaptive Sequential Bayesian Change Point Detection

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Learning

Learn by maximizing (log) marginal likelihood, the evidence Done by decomposing into the one-step-ahead predictive likelihoods log p(x1:T |θ) =

T X

log p(xt |x1:t−1 , θ)

(3)

t=1

Compute derivatives using forward propagation (r) ∂ ∂θm p(xt |rt−1 , xt , θm ) hazard function ∂θ∂h p(rt |rt−1 , θh )

The derivatives of the UPM The derivatives of the

Turner (Engineering, Cambridge)

Adaptive Sequential Bayesian Change Point Detection

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Improvements

Pruning Naive implementation is O(T 2 ) Eliminate low probability messages for O(T )

Modularity Any hazard function H(t) ∈ [0, 1] Any model that provides a posterior predictive Gaussian process regression, Bayesian linear regression, and Kernel Density Estimation

Caching Repetitive predictions under given run length (r) Use intelligent caching p(xt |rt−1 , xt )

Turner (Engineering, Cambridge)

Adaptive Sequential Bayesian Change Point Detection

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Well Log Data We used the logistic hazard, H(t) = hσ(at + b), and used an IID Gaussian UPM, with the aim of detecting changes in mean and variance. After learning the parameters our method has a better predictive likelihood than Adams & MacKay 2007. 2

NMR

0 −2 −4 0

500

1000

1500

2000

2500

3000

3500

4000

500

1000

1500

2000

2500

3000

3500

4000

Run Length

50 100 150 200 250 300

Measurements

Figure: The BOCPD run length distribution on the well log data. Turner (Engineering, Cambridge)

Adaptive Sequential Bayesian Change Point Detection

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Industry Portfolios Tried the “30 industry portfolios” data set (from Ken French repository). Change points found coincide with significant events: the climax of the Internet bubble, the burst of the Internet bubble, and the 2004 presidential election. Dot−com bubble burst September 11 Asia crisis, Dot−com bubble

US presidential election Major rate cut

Northern Rock bank run Lehman collapse

Run Length (trading days)

50 100 150 200 250 300 350 400 450 1999

2000

2001

2002

2003

2004

2005

2006

2007

2008

Date (years)

Figure: The BOCPD run length distribution between 1998 and 2008. Turner (Engineering, Cambridge)

Adaptive Sequential Bayesian Change Point Detection

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Results

Table: A summary of comparing the negative log predictive likelihoods (NLL) (nats/observation) on test data. We also include the 95% error bars on the NLL and the p-value that the joint model/learned hypers has a higher NLL using a one sided t-test.

Well Log Method NLL error bars TIM 1.53 0.0449 fixed hypers 0.313 0.0267 0.0293 learned hypers 0.247 Industry Portfolios TIM 42.6 0.246 indep. 39.64 0.217 joint 39.54 0.213

Turner (Engineering, Cambridge)

p-value <1e-10 6e-04 NA <1e-10 0.271 NA

Adaptive Sequential Bayesian Change Point Detection

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Summary

Extended work of Adams and MacKay 2007 Made more general through hyperparameter learning Increases predictive performance on real-world datasets Extended modularity to non-trivial UPMs Improved efficiency using pruning and caching

Turner (Engineering, Cambridge)

Adaptive Sequential Bayesian Change Point Detection

11

Adaptive Sequential Bayesian Change Point Detection

Dec 12, 2009 - The underlying predictive model (UPM) p(xt|x(t−τ):t−1 =: x. (τ) t. ,θm) for any τ ∈ [1,..., (t − 1)], at time t. The hazard function H(r|θh).

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