Optimum CUSUM Tests for Detecting Changes in Continuous Time Processes

Transcription

1 Optimum CUSUM Tests for Detecting Changes in Continuous Time Processes George V. Moustakides INRIA, Rennes, France

2 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

3 The change detection problem We are observing sequentially a process {ξ t } with the following statistics: ξ t ~ P for 0 6 t 6 τ ~ P 0 for τ <t Goal: Detect the change time τ as soon as possible 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

4 The observation process {ξ t } is available sequentially. This can be expressed through the filtration: F t = σ{ξ s : 0 < s6 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

5 Overview of existing results P τ : the probability measure induced, when the change takes place at time τ E τ [.] : the corresponding expectation P : P 0 : 0 P P 0 all data under nominal regime 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 and Min-max τ t G.V. Moustakides, INRIA, Rennes, France. 11-th Annual Applied Probability Day, Columbia University 5

6 Baysian approach (Shiryayev 1978) The change time τ is random with exponential prior. For any stopping time T define the criterion: J(T) = ce[ (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

7 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). Compute the statistics: S n = (S n-1 +1) f 0 (ξ n ). 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

8 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

9 Define the CUSUM process y t as follows: where y t = u t m t dp 0 u (F dp t ) t = log( ) m t = inf 06s 6t 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

10 u t m t TC G.V. Moustakides, INRIA, Rennes, France. 11-th Annual Applied Probability Day, Columbia University 10

11 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: dw t 0 6 t 6 τ α t dt + dw t τ <t {w t } standard Brownian Motion {α t } adapted to the history F t = σ{ξ s : 06s6t} d ξt ={ 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

12 To {ξ t } we correspond the following process {u t } 2 du t = α t dξ t 0.5α t dt. dp 0 We would like: u (F dp t ) t = log( ). We need the following conditions: 0 t 1. P [ 0 α s ds < ] = P [ α2 s ds < ] = t 2. A Novikov condition 3. P [ 0 α 2 s ds = ] = P [ α 2 s ds = ] = G.V. Moustakides, INRIA, Rennes, France. 11-th Annual Applied Probability Day, Columbia University 12

13 From Condition 1&2 we have validity of Girsanov s theorem: dp 0 dp (F t ) = e u t dp τ dp (F t ) = e u t -u τ The Kullback-Leibler Divergence can then be written as: dp τ (F dp t ) E [ log( ) τ F ] τ τ t τ t = E [ τ α s dw s α ds F ] 2 s τ τ t = E τ [0.5 α2 s ds F τ ], 0 6 τ 6 t G.V. Moustakides, INRIA, Rennes, France. 11-th Annual Applied Probability Day, Columbia University 13

14 The original Lorden criterion J(T) = sup τ essup E τ [ (T - τ) + F τ ] using the Kullback-Leibler Divergence can be modified as J(T) = sup τ essup E [ τ 0.5 T 1l α 2 t dt F ] {T> τ} τ τ The two criteria are equivalent in the case α2 t = constant i.e. Brownian motion with constant drift. G.V. Moustakides, INRIA, Rennes, France. 11-th Annual Applied Probability Day, Columbia University 14

15 Similarly dp (F dp t ) 0 [ log( )] E 0 t 0 t [ = E - α s dw s α ds ] 2 s 0 t 2 = E [0.5 α s ds ] This suggest replacing the constraint E [ T ] > γ with E [0.5 T α 2 t dt ] > γ 0 G.V. Moustakides, INRIA, Rennes, France. 11-th Annual Applied Probability Day, Columbia University 15

16 Summarizing: J(T) = sup τ essup E [ τ 0.5 T 1l α 2 t dt F ] {T> τ} τ τ Optimization problem: inf T J(T); subject to: E [0.5 0 T 2 α t dt ] > γ G.V. Moustakides, INRIA, Rennes, France. 11-th Annual Applied Probability Day, Columbia University 16

17 CUSUM tests for Ito processes The CUSUM statistics y t for Ito processes takes the form 2 du t = α t dξ t 0.5α t dt m t = inf 06s 6t u s y t = u t m t and the optimum CUSUM test is where ν such that: T C = inf t { t: y t > ν } E [0.5 0 T α t dt ] = γ 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 C 2

18 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 u t = m t or y t = u t - m t = 0 =/ 0 only when 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

19 If f(y) is a twice continuously differentiable function with 0 f (0) = 0, using standard Ito calculus, we can write d f(y t ) = f (y t )(du t - dm t )+ 0.5α t f (y t )dt = f (y t )du t + 0.5α t f (y t )dt Theorem 1: T C is a.s. finite, furthermore TC 2 τ E τ [ 0.5 α t dt F τ ]=[g(ν) - g(y τ )] 1l {T> τ} 1l {T> τ} C E [ C 0.5 α t dt F τ ]=[h(ν) - h(y τ )] 1l {T> τ} C τ T 2 1l {T> τ} g(y) = y + e -y -1 h(y) = e y - y -1 C C G.V. Moustakides, INRIA, Rennes, France. 11-th Annual Applied Probability Day, Columbia University 19

20 The functions g(y), h(y) are increasing, strictly convex, with g(0) = h(0) = 0. We can therefore conclude τ T 2 J(T C ) = sup τ essup E [ τ 0.5 α t dt F ] C τ 1l {T> τ} C = sup τ essup[g(ν) - g(y τ )]1l {T> τ} C = g(ν) - g(0) = g(ν) = ν + e -ν -1 Similarly E [0.5 0 T C 2 α t dt ] = h(ν) - h(0) = h(ν) = γ e ν - ν -1 = γ G.V. Moustakides, INRIA, Rennes, France. 11-th Annual Applied Probability Day, Columbia University 20

21 For any stopping time T, using again standard Ito calculus, we have the following corollary of Theorem 1 Corollary: E [ τ 0.5 T 1l α 2 {T> τ} t dt F τ ]=E τ [g(y T )-g(y τ ) F τ ] τ 1l {T> τ} E [ 0.5 T 1l α 2 {T> τ} t dt F τ ]=E [h(y T )-h(y τ ) F τ ] τ 1l {T> τ} Remark 1: The false alarm constraint can be written as E [0.5 0 T 2 α t dt ]= E [h(y T )-h(0)] = E [h(y T )] > γ G.V. Moustakides, INRIA, Rennes, France. 11-th Annual Applied Probability Day, Columbia University 21

22 Remark 2: The modified performance measure J(T) can be suitably lower bounded as follows J(T) = sup τ essup E [ τ 0.5 T 1l α 2 t dt F ] {T> τ} τ τ > E [e y Tg(y T )] E [e y T ] In the case of CUSUM the lower bound coincides with the corresponding performance measure J(T C ) Remark 3: We can limit ourselves to stopping times that satisfy the false alarm constraint with equality, i.e E [h(y T )] = γ = h(ν) G.V. Moustakides, INRIA, Rennes, France. 11-th Annual Applied Probability Day, Columbia University 22

23 Theorem 2: Any stopping T that satisfies the false alarm constraint with equality has a performance measure J(T) that is no less than J(T C )= g(ν). Proof: Let T satisfy the false alarm constraint with equality, i.e. E [h(y T )] = γ = h(ν) we then like to show that: J(T) > g(ν). J(T) > E [e y Tg(y T )] Since it is sufficient to show E [e y T ] E [e y T{g(y T ) - g(ν)} + h(ν) - h(y T )] > 0 G.V. Moustakides, INRIA, Rennes, France. 11-th Annual Applied Probability Day, Columbia University 23

24 If we define the function p(y) = e y {g(y) - g(ν)} + h(ν) - h(y) then the previous inequality becomes: E [p(y T )] > 0 p(y) 0 ν We observe that p(y) > 0 therefore we also have E [p(y T )] > 0 with equality iff y T = ν, i.e. the CUSUM test. G.V. Moustakides, INRIA, Rennes, France. 11-th Annual Applied Probability Day, Columbia University 24 y

25 Extensions Can our result be extended to the discrete time case? ξn ={ w n α n-1 + w n 0 6 n 6 τ τ <n {w n } an i.i.d. Gaussian process {α n } adapted to the history F n = σ{ξ k : 06k6n} Not Straightforward! E[ 0.5 Σ 2 1l α k F ] {T> τ} τ T k=τ? = E[g(y T )-g(y τ ) F τ ] 1l {T> τ} Similar problem exists for SPRT. G.V. Moustakides, INRIA, Rennes, France. 11-th Annual Applied Probability Day, Columbia University 25

26 Straightforward extension for scalar processes ={ d ξ α t dt + σ t dw t t β t dt + σ t dw t or vector processes ={ d Ξ A t dt + Σ t dw t t B t dt + Σ t dw t 0 6 t 6 τ τ <t 0 6 t 6 τ τ <t G.V. Moustakides, INRIA, Rennes, France. 11-th Annual Applied Probability Day, Columbia University 26

27 EnD G.V. Moustakides, INRIA, Rennes, France. 11-th Annual Applied Probability Day, Columbia University 27