Optimum CUSUM Tests for Detecting Changes in Continuous Time Processes George V. Moustakides INRIA, Rennes, France
Outline � The change detection problem � Overview of existing results � Lorden’s criterion and the CUSUM test � A modified Lorden criterion � CUSUM tests for Ito processes � Extensions G.V. Moustakides, INRIA, Rennes, France. 11-th Annual Applied Probability Day, Columbia University 2
The change detection problem We are observing sequentially a process { ξ t } with the following statistics: for 0 6 t 6 τ ~ P ∞ ξ t ~ P 0 for τ < t Goal: Detect the change time Detect the change time τ τ “ “ as soon as possible as soon as possible ” ” Goal: � Change time τ : deterministic (but unknown) or random � Probability measures P ∞ , P 0 : known Applications include: systems monitoring; quality control; financial decision making; remote sensing (radar, sonar, seismology); speech/image/video segmentation; … G.V. Moustakides, INRIA, Rennes, France. 11-th Annual Applied Probability Day, Columbia University 3
The observation process { ξ t } is available sequentially. This can be expressed through the filtration: F t = σ { ξ s : 0 < s 6 t } . � Interested in sequential detection schemes . At every time instant t we perform a test to decide whether to stop and declare an alarm or continue sampling. The test at time t must be based on the available information up to time t . � Any sequential detection scheme can be represented by a stopping time T adapted to the filtration F t (the time we stop and declare an alarm) . G.V. Moustakides, INRIA, Rennes, France. 11-th Annual Applied Probability Day, Columbia University 4
Overview of existing results P ∞ P 0 0 τ t P τ : the probability measure induced, when the change takes place at time τ E τ [ . ] : the corresponding expectation P ∞ : all data under nominal regime P 0 : all data under alternative regime Optimality criteria They must take into account two quantities: - The detection delay T - τ - The frequency of false alarms Possible approaches: Baysian Baysian and Min and Min- -max max G.V. Moustakides, INRIA, Rennes, France. 11-th Annual Applied Probability Day, Columbia University 5
Baysian approach (Shiryayev 1978) The change time τ is random with exponential prior. For any stopping time T define the criterion: J ( T ) = c E [ ( T - τ ) + ]+ P [ T < τ ] Optimization problem: inf T J ( T ) Compute the statistics: π t = P [ τ 6 t | F t ] ; and stop: T S = inf t { t : π t > ν } - Discrete time: when { ξ n } is i.i.d. and there is a change in the pdf from f ∞ ( ξ ) to f 0 ( ξ ). - Continuous time: when { ξ t } is a Brownian Motion and there is a change in the constant drift from µ ∞ to µ 0 . G.V. Moustakides, INRIA, Rennes, France. 11-th Annual Applied Probability Day, Columbia University 6
Min-max approach (Shiryayev-Roberts-Pollak) The change time τ is deterministic but unknown. For any stopping time T define the criterion: J ( T ) = sup τ E τ [ ( T - τ ) + | T > τ ] Optimization problem: inf T J ( T ) ; subject to: E ∞ [ T ] > γ Discrete time: when { ξ n } is i.i.d. and there is a change in the pdf from f ∞ ( ξ ) to f 0 ( ξ ). f 0 ( ξ n ) Compute the statistics: S n = ( S n -1 + 1) . f ∞ ( ξ n ) and stop (Yakir 1997): T SRP = inf n { n : S n > ν } G.V. Moustakides, INRIA, Rennes, France. 11-th Annual Applied Probability Day, Columbia University 7
Lorden’s criterion and the CUSUM test Alternative min-max approach (Lorden 1971): The change time τ is deterministic and unknown. For any stopping time T define the criterion: J ( T ) = sup τ essup E τ [ ( T - τ ) + | F τ ] Optimization problem: inf T J ( T ) ; subject to: E ∞ [ T ] > γ . The test closely related to Lorden’s criterion and being the most popular test for the change detection problem in practice, is the Cumulative Sum (CUSUM) test. G.V. Moustakides, INRIA, Rennes, France. 11-th Annual Applied Probability Day, Columbia University 8
Define the CUSUM process y t as follows: y t = u t – m t where d P 0 u t = log ( ) ( F t ) d P ∞ m t = inf 0 6 s 6 t u s . The CUSUM stopping time (Page 1954): T C = inf t { t : y t > ν } - Discrete time: when { ξ n } is i.i.d. before and after the change (Moustakides 1986, Ritov 1990). - Continuous time: when { ξ t } is a Brownian Motion with constant drift before and after the change (Shiryayev 1996, Beibel 1996). G.V. Moustakides, INRIA, Rennes, France. 11-th Annual Applied Probability Day, Columbia University 9
u t m t T C G.V. Moustakides, INRIA, Rennes, France. 11-th Annual Applied Probability Day, Columbia University 10
A modified Lorden criterion We intend to extend the optimality of CUSUM to detection of changes in Ito processes by modifying Lorden’s criterion using the Kullback-Leibler Divergence (KLD) . Similar extension exists for the Sequential Probability Ratio Test (SPRT), applied in hypotheses testing, since 1978 (Liptser and Shiryayev) The observation process { ξ t } satisfies the following sde: 0 6 t 6 τ d ξ t = { dw t α t dt + dw t τ < t { w t } standard Brownian Motion { α t } adapted to the history F t = σ { ξ s : 0 6 s 6 t } If α t = α ( ξ t ), then ξ t is a diffusion process for t> τ . G.V. Moustakides, INRIA, Rennes, France. 11-th Annual Applied Probability Day, Columbia University 11
To { ξ t } we correspond the following process { u t } du t = α t d ξ t – 0.5 α t dt . 2 d P 0 u t = log ( ) . ( F t ) We would like: d P ∞ We need the following conditions: ∫ t ∫ t 2 1. P 0 [ α s ds < ∞ ] = P ∞ [ α s ds < ∞ ] = 1 2 0 0 2. A “Novikov” condition ∞ ∞ ∫ ∫ 3. P 0 [ α s ds = ∞ ] = P ∞ [ α s ds = ∞ ] = 1 2 2 0 0 G.V. Moustakides, INRIA, Rennes, France. 11-th Annual Applied Probability Day, Columbia University 12
From Condition 1&2 we have validity of Girsanov’s theorem: d P 0 d P τ ( F t ) = e u t ( F t ) = e u t - u τ d P ∞ d P ∞ The Kullback-Leibler Divergence can then be written as: d P τ log ( ) E τ [ | F τ ] ( F t ) d P ∞ ∫ t ∫ t = E τ [ α s dw s + 0.5 ds | F τ ] α s 2 τ τ ∫ t = E τ [ 0.5 α s ds | F τ ] , 0 6 τ 6 t 2 τ G.V. Moustakides, INRIA, Rennes, France. 11-th Annual Applied Probability Day, Columbia University 13
The original Lorden criterion J ( T ) = sup τ essup E τ [ ( T - τ ) + | F τ ] using the Kullback-Leibler Divergence can be modified as ∫ T 2 J ( T ) = sup τ essup E τ [ 0.5 α t dt | F τ ] 1 l { T > τ } τ The two criteria are equivalent in the case 2 α t = constant i.e. Brownian motion with constant drift. G.V. Moustakides, INRIA, Rennes, France. 11-th Annual Applied Probability Day, Columbia University 14
Similarly d P ∞ log ( ) E ∞ [ ] ( F t ) d P 0 ∫ t ∫ t = E ∞ [ - α s dw s + 0.5 ds ] α s 2 0 0 ∫ t = E ∞ [ 0.5 α s ds ] 2 0 This suggest replacing the constraint E ∞ [ T ] > γ with ∫ T 2 E ∞ [ 0.5 α t dt ] > γ 0 G.V. Moustakides, INRIA, Rennes, France. 11-th Annual Applied Probability Day, Columbia University 15
Summarizing: ∫ T 2 J ( T ) = sup τ essup E τ [ 0.5 α t dt | F τ ] 1 l { T > τ } τ Optimization problem: inf T J ( T ) ; ∫ T 2 subject to: E ∞ [ 0.5 α t dt ] > γ 0 G.V. Moustakides, INRIA, Rennes, France. 11-th Annual Applied Probability Day, Columbia University 16
CUSUM tests for Ito processes The CUSUM statistics y t for Ito processes takes the form du t = α t d ξ t – 0.5 α t dt 2 m t = inf 0 6 s 6 t u s y t = u t – m t and the optimum CUSUM test is T C = inf t { t : y t > ν } T ∫ E ∞ [ 0.5 α t dt ] = γ C 2 where ν such that: 0 Since y t has continuous paths, when the CUSUM test stops we have: y T = ν . C G.V. Moustakides, INRIA, Rennes, France. 11-th Annual Applied Probability Day, Columbia University 17
u t m t � Since u t > m t we conclude y t = u t - m t > 0 � m t is nonincreasing and dm t = 0 only when / u t = m t or y t = u t - m t = 0 � If f ( y ) continuous with f (0) = 0, then f ( y t ) dm t = 0 G.V. Moustakides, INRIA, Rennes, France. 11-th Annual Applied Probability Day, Columbia University 18
If f ( y ) is a twice continuously differentiable function with 0 f (0) = 0, using standard Ito calculus, we can write 0 00 2 d f ( y t ) = f ( y t )( du t - dm t )+ 0.5 α t f ( y t ) dt 0 00 2 = f ( y t ) du t + 0.5 α t f ( y t ) dt Theorem 1: T C is a.s. finite, furthermore ∫ T 2 E τ [ 0.5 α t dt | F τ ] =[ g ( ν ) - g ( y τ )] C 1 l { T > τ } 1 l { T > τ } τ C C ∫ T 2 E ∞ [ 0.5 α t dt | F τ ] =[ h ( ν ) - h ( y τ )] C 1 l { T > τ } 1 l { T > τ } τ C C g ( y ) = y + e - y - 1 h ( y ) = e y - y - 1 G.V. Moustakides, INRIA, Rennes, France. 11-th Annual Applied Probability Day, Columbia University 19
The functions g ( y ), h ( y ) are increasing, strictly convex, with g (0) = h (0) = 0. We can therefore conclude ∫ T J ( T C ) = sup τ essup E τ [ 0.5 α t dt | F τ ] C 2 1 l { T > τ } τ C = sup τ essup[ g ( ν ) - g ( y τ )]1 l { T > τ } C = g ( ν ) - g (0) = g ( ν ) = ν + e - ν - 1 Similarly T ∫ C E ∞ [ 0.5 α t dt ] = h ( ν ) - h (0) = h ( ν ) 2 = γ 0 e ν - ν - 1 = γ G.V. Moustakides, INRIA, Rennes, France. 11-th Annual Applied Probability Day, Columbia University 20
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