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Detecting changes in Detecting changes in the rate the rate of a of a Poisson process Poisson process George V. Moustakides

Outline � Overview of the change detection problem � CUSUM test and Lorden’s criterion � The Poisson disorder problem � CUSUM average run length for Poisson processes � CUSUM optimality in the sense of Lorden G.V. Moustakides: Detecting changes in the rate of a Poisson process 2

Change detection - Overview Available sequentially an observation process { ξ t } with the following statistics: for 0 6 t 6 τ ~ P ∞ ξ t P 0 ~ for τ < t Detect change as soon as possible � Change time τ : � Random with known prior (Bayesian) � Deterministic but unknown (Non-Bayesian) � Known statistics P ∞ , P 0 . G.V. Moustakides: Detecting changes in the rate of a Poisson process 3

We are interested in sequential schemes . With every new observation the test must decide � Stop and issue an alarm � Continue sampling Decision at time t uses available information F t = σ { ξ s : 0 6 s 6 t } . up to time t . Sequential test � stopping time T adapted to the filtration {F t } . G.V. Moustakides: Detecting changes in the rate of a Poisson process 4

P ∞ P 0 τ 0 t P τ : the probability measure induced, when change takes place at time τ E τ [ . ] : the corresponding expectation P ∞ : all data under nominal regime P 0 : all data under alternative regime Parameters to be considered � The detection delay T - τ � Frequency of false alarms G.V. Moustakides: Detecting changes in the rate of a Poisson process 5

Bayesian approach (Shiryayev 1978) Change time τ random with exponential prior. J ( T ) = c E [ ( T - τ ) + ] + P [ T < τ ] Optimization problem: inf T J ( T ) π t = P [ τ 6 t | F t ] ; T S = inf t { t : π t > ν } � Discrete time: i.i.d. observations (Shiryayev 1978, Poor 1998) � Continuous time: Brownian Motion (Shiryayev 1978, Beibel 2000, Karatzas 2003) G.V. Moustakides: Detecting changes in the rate of a Poisson process 6

Non-Bayesian setup (Pollak 1985) The change time τ is deterministic & unknown. J ( T ) = sup τ E τ [ ( T - τ ) | T > τ ] Optimization problem: inf T J ( T ) subject to: E ∞ [ T ] > γ Discrete time: i.i.d. detect change in the pdf from f ∞ ( ξ ) to f 0 ( ξ ). Roberts (1966) proposed f 0 ( ξ t ) S t = ( S t -1 + 1) f ∞ ( ξ t ) T SRP = inf t { t : S t > ν } (Mei 2006) G.V. Moustakides: Detecting changes in the rate of a Poisson process 7

CUSUM test and Lorden’s criterion Discrete time, i.i.d. observations. Pdf before and after the change: f ∞ ( ξ n ) , f 0 ( ξ n ) Since change time τ is unknown t f 0 ( ξ n ) log ( ) > ν Σ sup sup 0 6 τ 6 t f ∞ ( ξ n ) n = τ +1 t τ f 0 ( ξ n ) f 0 ( ξ n ) log ( ) log ( ) > ν Σ Σ - inf 0 6 τ 6 t f ∞ ( ξ n ) f ∞ ( ξ n ) n =1 n =1 > ν u t m t – G.V. Moustakides: Detecting changes in the rate of a Poisson process 8

d P 0 u t = log ( ) ( F t ) d P ∞ m t = inf 0 6 s 6 t u s CUSUM process: y t = u t – m t > 0 The CUSUM stopping time (Page 1954): T C = inf t { t : y t > ν } G.V. Moustakides: Detecting changes in the rate of a Poisson process 9

ν ν ν u t m t T C ML estimate of τ G.V. Moustakides: Detecting changes in the rate of a Poisson process 10

Non-Bayesian setup (Lorden 1971). Change time τ is deterministic and unknown. J ( T ) = sup τ essup E τ [ ( T - τ ) + | F τ ] Optimization problem: inf T J ( T ) subject to: E ∞ [ T ] > γ � Discrete time: i.i.d. observations (Moustakides 1986, Ritov 1990, Poor 1998) � Continuous time: BM (Shiryayev 1996, Beibel 1996); Ito processes (Moustakides 2004) G.V. Moustakides: Detecting changes in the rate of a Poisson process 11

The Poisson disorder problem Let {N t } homogeneous Poisson, with rate λ satisfying: λ ∞ , 0 6 t 6 τ λ = { λ 0 , τ < t Bayesian Approach � Linear delay: Galchuk & Rozovski 1971, Davis 1976, Peskir & Shiryayev 2002. � Exponential delay: Bayraktar & Dayanik 2003, B & D & Karatzas 2004 and 2005 (adaptive) G.V. Moustakides: Detecting changes in the rate of a Poisson process 12

CUSUM & average run length u t = ( λ ∞ - λ 0 ) t + log( λ 0 / λ ∞ ) N t m t = inf 0 6 s 6 t u s y t = u t – m t T C = inf t { t : y t > ν } We are interested in computing E [ T C ] when N t is Poisson with rate λ . Existing formula (Taylor 1975) for: u t = at + b W t ; W t is standard Wiener G.V. Moustakides: Detecting changes in the rate of a Poisson process 13

a> 0; b< 0 ν u t b m t a U 1 U 2 T C U 3 U 4 U 5 Find f ( y ) such that f ( y 0 )= E [ T C ]. Study the paths of f ( y t ) f ( y )= f (0); y 6 0 G.V. Moustakides: Detecting changes in the rate of a Poisson process 14

G.V. Moustakides: Detecting changes in the rate of a Poisson process 15

We end up with a DDE and the following boundary conditions: af 0 ( y ) + λ [ f ( y + b ) - f ( y )] = -1; y ∈ [0, ν ) f ( y ) = f (0) for y 6 0; f ( ν )=0 f ( y 0 ) = E [ T C ] ν u t m t T C G.V. Moustakides: Detecting changes in the rate of a Poisson process 16

( Existing formula (Taylor 1975) for: u t = at + b W t ; W t standard Wiener b 2 f 00 ( y ) + af 0 ( y ) = -1; y ∈ [0, ν ] 2 f 0 (0)=0; f ( ν )=0 ) G.V. Moustakides: Detecting changes in the rate of a Poisson process 17

af 0 ( y ) + λ [ f ( y + b ) - f ( y )] = -1; y ∈ [0, ν ) f ( y ) = f (0) for y 6 0; f ( ν )=0 Because b< 0 it is a forward DDE G.V. Moustakides: Detecting changes in the rate of a Poisson process 18

a< 0; b> 0 af 0 ( y ) + λ [ f ( y + b ) - f ( y )] = -1; y ∈ [0, ν ) f 0 (0)=0; f ( y )=0 for y > ν Backward DDE G.V. Moustakides: Detecting changes in the rate of a Poisson process 19

where p is defined as with G.V. Moustakides: Detecting changes in the rate of a Poisson process 20

Average over 10000 repetitions: λ ∞ = 2, λ 0 = 1, ( a =1, b =-log2), ν = 5.5 Formula Simulation E 0 [ T C ] 15.3832 15.3605 E ∞ [ T C ] 779.9669 771.1219 λ ∞ = 1, λ 0 = 2, ( a =-1, b =log2), ν = 5.5 Formula Simulation E 0 [ T C ] 12.2885 12.2673 E ∞ [ T C ] 981.9811 986.7159 G.V. Moustakides: Detecting changes in the rate of a Poisson process 21

Optimality of CUSUM J ( T ) = sup τ essup E τ [ ( T - τ ) + | F τ ] subject to: E ∞ [ T ] > γ inf T J ( T ) ; If T is such that E ∞ [ T ] > E ∞ [ T C ] = γ then J ( T ) > J ( T C ) G.V. Moustakides: Detecting changes in the rate of a Poisson process 22

h ( y ) = E ∞ [ T C | y 0 = y ] g ( y ) = E 0 [ T C | y 0 = y ] essup E τ [( T C - τ ) + | F τ ]=sup y g ( y )= g (0) T C is an equilizer rule therefore J ( T C ) = g (0) For the false alarm we have E ∞ [ T C ] = h (0) G.V. Moustakides: Detecting changes in the rate of a Poisson process 23

We would like to show: If E ∞ [ T ] > E ∞ [ T C ] then J ( T ) > J ( T C ) Lemma Sufficient: If E ∞ [ T ] > h (0) then J ( T ) > g (0) G.V. Moustakides: Detecting changes in the rate of a Poisson process 24

We will show that this is true for any T G.V. Moustakides: Detecting changes in the rate of a Poisson process 25

Consider the function f ( y ) defined as follows f ( y )= e y [ g (0) - g ( y )] - [ h (0) - h ( y )] then ? > > 0 ¥ > > 0 ¥ G.V. Moustakides: Detecting changes in the rate of a Poisson process 26

Conclusion � We considered the Poisson disorder problem of detecting changes in the rate of a homogeneous Poisson process, in the sense of Lorden. � We obtained closed form expressions for the average run length of the CUSUM stopping time. � We used these formulas to prove optimality of the CUSUM test in the sense of Lorden. G.V. Moustakides: Detecting changes in the rate of a Poisson process 27

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