Detection of two-sided alternatives in a Brownian motion model

Detection of t w o-sided alternativ es in a Bro wnian Motion mo del 1 Hadjiliadis, Olympia (1st author) Princ eton Univ ersity, Dep artm e nt of Ele ctric al Engine ering Engine ering Qu a dr angle, Olden Str e et Princ eton, N J 08544 U.S.A . E-mail: ohadjili@princ eton.e du P o or, H. Vincen t (2nd author) Princ e ton Unive rs ity, De p artment of Ele ctric al Engine e ring Engine ering Qu a dr angle, Olden Str e et Princ e ton, NJ 08544 U.S.A . E-mail: p o or@princ eton.e du In tro duction and motiv ation The need for statistical surveill ance has b een noted in m a n y differen t areas, including qualit y con trol (see for example [2]), epidemiology (see f or example [13]), medicine (see for example [4]), mac hinery monitoring, seismology , fin ance (see for example [1]) etc. In this w ork, we address th e problem of the d et ection of t wo-sided alternativ es in a Bro wnian motion mo del. This mo del is the con tinuous time equiv alen t to the discrete time Gaussian observ ation mo del. F or sto c hastic systems with linear dy n amic s and linear observ ations that are drive n by Gaussian noise, the Kalman-Bucy inno v ation pro cess is kn o wn to b e a sequence of indep endent Gaussian r a ndom v ariables. Such m o dels can b e used to study systems s ub ject to system comp onen t failures and other systems in v olving small non-linearities ([16, 12]). F ault d e tection in a na vigation s yste m, wh er e an abru p t change in the mo del p a r ame ters corresp onds to an abrup t c hange in the mean of the Kalman filter innov ations is an instance of such a situation ([10, 2 ]) . The s ign of the c hange dep ends on the signs of the gyro errors ([15]). Another instance of suc h a mo del can b e seen in sensor failure detection for the monitoring of traffic incidents on fr ee wa ys. Eac h sensor is placed in d iffe r en t lo cat ions on th e fr ee wa y and records the m e an v elocity and densit y of cars. An abrupt and systematic c h a n ge in these recordings would trigger an abr upt change in the Kalman filter innov ations in either direction dep ending on whether the sensor is consistently o veresti mating or underestimating (see [14]). Iden tification and remov al of the fault y sensor b ecomes essen tial. Th e con tinuous ve r sion of the Kalman filter innov ations in all of the ab o v e linear Gaussian mo dels is seen to b e a Brownian motion ([8]). Other applications in c lu des the d et ection of a r hythm jum p of the heartb eat d uring an ECG (see [3]) and in the d ete ction of a p ositiv e or negativ e d rift in the log of sto c k price dynamics. This p a p er is concerned w ith the quick est detectio n of tw o-sided alternativ es in the drift of a Bro wnian motion. In particular, w e fi nd the b est 2-CUSUM stopp ing r ule with resp ect to an extended Lorden criterion. Although, th e mathematical formulation is done in the cont ext of the one- dimensional case, extension to th e vec tor case that corresp onds to the Kalman inno v ations in linear systems describ ed ab o ve is straigh tforwa rd (see [7]). 1 This research was supp orted in p art by the U.S. National Science F oundation un der Grant No. C CR -02 -05214. Mathematical formulation and main results W e sequent ially observ e a p rocess { ξ t } with the follo w ing dynamics: dξ t =                    dw t t ≤ τ µ 1 dt + d w t or − µ 2 dt + dw t t ≥ τ where τ , the time of c hange, is assu med to b e deterministic but unkno wn ; w t is a stand a rd Brownian motion pro cess; µ i , the p ossible d rifts to wh ic h the pro cess can c h an ge, are assumed to b e kno wn , but the sp eci fi c drift to whic h the pro cess is changing is u n kno wn. Both µ 1 and µ 2 are assum ed to b e p ositiv e. The pr o b abilit y trip let consists of ( C [0 , ∞ ] , ∪ t> 0 F t ), where F t = σ { ξ s , 0 < s ≤ t } and the families of probability measures {P i τ } , τ ∈ [0 , ∞ ), wh e nev er the c hange is µ i , i = 1 , 2 , and P ∞ , the Wiener measure. Our goal is to detect a c hange by means of a stopping ru le T adapted to the fi lt r at ion F t . As a p erformance measure f o r this stopping rule w e pr op ose an extended Lorden criterion (see [5]) J L ( T ) = max i sup τ essup E i τ h ( T − τ ) + |F τ i . (1) This giv es rise to the follo wing min-max constrained optimization problem: inf T J L ( T ) sub ject to E ∞ [ T ] ≥ γ , (2) where the constrain t sp ecifies the minim u m allo wable mean time b et w een false alarms. In this pap er w e seek the b est 2-CUSUM stopping ru le in the sens e d escrib ed in (2). The 2-CUSUM rules ha v e b een p r oposed and used extensive ly d ue not only to the simplicit y in the calculation of their first momen t(see [9 ]) , but also to their asymptotica lly optimal charac ter (see [5], [11]). W e b egin b y defin ing the CUSUM statistics and stopping rules of int er est. Definition Let ν 1 > 0 and ν 2 > 0. Define 1. u + t = log dP 1 0 dP ∞ |F t µ 1 = ξ t − 1 2 µ 1 t ; m + t = inf s ≤ t u + s ; y + t = u + t − m + t , 2. u − t = log dP 2 0 dP ∞ |F t µ 2 = − ξ t − 1 2 µ 2 t ; m − t = inf s ≤ t u − s ; y − t = u − t − m − t , 3. T 1 ( ν 1 ) = inf { t > 0; y + t ≥ ν 1 } , and 4. T 2 ( ν 2 ) = inf { t > 0; y − t ≥ ν 2 } . The 2-CUSUM stopping rules are then of the form T ( ν 1 , ν 2 ) = T 1 ( ν 1 ) ∧ T 2 ( ν 2 ). W e also defin e the follo wing s topp ing ru les, the use of whic h w ill b ecome apparent later. Definition F or a > 0 and b > 0, w e defin e 1. U + ( a ) = inf { t > 0; u + t ≥ a } , 2. U − ( b ) = inf { t > 0; − u − t ≤ − b } , and 3. Π( a, b ) = P ( U + ( a ) < U − ( b )) . F or any 2-CUSUM stopp ing rule T we ha v e (see [5]) J L ( T ) = max { E 1 0 [ T ] , E 2 0 [ T ] } . W e n ow classify 2-CUSUM rules according to the class G = { T ( ν 1 , ν 2 ); ν 1 = ν 2 } of h a r monic mean rules and the classes C 1 = { T ( ν 1 , ν 2 ) | ν 1 > ν 2 > 0 } and C 2 = { T ( ν 1 , ν 2 ) | ν 2 > ν 1 > 0 } of non-h arm o nic mean r ules. F or simplicit y of disp la y and notation w e finally defin e the constan ts m = min { ν 1 , ν 2 } , M = max { ν 1 , ν 2 } and the functions C m ( x, y ) = f m ( x ) 2 f m ( x )+ f m ( y ) , λ x ( y ) = 1 y f x ( y )+ x , f ∗ y ( x ) = f x ( y ) = e y x − y x − 1 y 2 . W e now su m marize the main resu lt s. Theorem Let T ( ν 1 , ν 2 ) = T 1 ( ν 1 ) ∧ T 2 ( ν 2 ) b e any 2-CUSUM stopping rule and denote T ( ν 1 , ν 2 ) by T . Then, the follo wing is true und er an y of the measures P ∞ , P 1 0 and P 2 0 : 1. for all T ∈ C 1 , m = ν 2 , M = ν 1 , w e hav e E [ T ] = E [ T 2 ( m )] · " 1 − E [ T 2 ( m )] E [ T 1 ( m )] + E [ T 2 ( m )] lim n →∞ Π  1 n , m  ( M − m ) n # , and 2. for all T ∈ C 2 , m = ν 1 , M = ν 2 , w e hav e E [ T ] = E [ T 1 ( m )] · " 1 − E [ T 1 ( m )] E [ T 1 ( m )] + E [ T 2 ( m )] lim n →∞  1 − Π( m, 1 n )  ( M − m ) n # . Corollary Let T ( ν 1 , ν 2 ) = T 1 ( ν 1 ) ∧ T 2 ( ν 2 ) b e any 2-CUSUM stoppin g ru le and d en o te T ( ν 1 , ν 2 ) by T . Then, for all T ∈ C 1 , m = ν 2 , M = ν 1 and E ∞ [ T ] ≤ 2 f m ( µ 2 ) ·  1 − C m ( µ 2 , µ 1 ) f m ( µ 2 ) e − λ m ( − µ 1 )( M − m )  , (3) E ∞ [ T ] ≥ 2 f m ( µ 2 ) ·  1 − C m ( µ 2 , µ 1 ) f m ( µ 2 ) e − λ m ( µ 2 )( M − m )  , (4) E 1 0 [ T ] ≤ 2 f m ( µ 2 + 2 µ 1 ) ·  1 − C m ( µ 2 + 2 µ 1 , − µ 1 ) f m ( µ 2 + 2 µ 1 ) ( e − λ m ( µ 1 )( M − m )  , (5) E 1 0 [ T ] ≥ 2 f m ( µ 2 + 2 µ 1 ) ·  1 − C m ( µ 2 + 2 µ 1 , − µ 1 ) f m ( µ 2 + 2 µ 1 ) e − λ m ( µ 2 +2 µ 1 )( M − m )  , (6) E 2 0 [ T ] ≤ 2 f m ( − µ 2 ) ·  1 − C m ( − µ 2 , µ 1 + 2 µ 2 ) f m ( − µ 2 ) e − λ m ( − ( µ 1 +2 µ 2 ))( M − m )  , and (7) E 2 0 [ T ] ≥ 2 f m ( − µ 2 ) ·  1 − C m ( − µ 2 , µ 1 + 2 µ 2 ) f m ( − µ 2 ) e − λ m ( − µ 2 )( M − m )  . (8) Similar results hold for T ∈ C 2 . F or more details please refer to [6]. Theorem The b est T ∗ 2-CUSUM stopping rule exists and is uniqu e and w e d istinguish the follo wing cases 1. If µ 1 < µ 2 then T ∗ ∈ C 2 . 2. If µ 2 < µ 1 then T ∗ ∈ C 1 . 3. 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