Performance Analysis of Wald s SPRT with Independent but Non-Stationary Log-Likelihood Ratios

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1 4th International Conference on Information Fuion Chicago, Illinoi, US, July 5-8, 2 Performance nalyi of Wald SPRT with Independent but Non-Stationary Log-Likelihood Ratio Yu Liu and X Rong Li Department of Electrical Engineering Univerity of New Orlean New Orlean, L 748, US {lyu2, xli@unoedu btract The characteritic and behavior of Wald equential probability ratio tet are revealed by two important function operating characteritic (OC) and average ample number (SN) Thee two function have been tudied extenively under the aumption of independent and identically ditributed (iid) log-likelihood ratio, which i too tringent for many application Thi paper relaxe the requirement of identical ditribution Two inductive equation governing the OC and SN are developed Unfortunately, they have non-unique olution in the general cae They do have unique olution in two pecial cae: (a) the log-likelihood ratio converge in ditribution and (b) the loglikelihood ratio have periodic ditribution Numerical olution for thee two pecial cae are obtained They are compared with the reult of Monte Carlo imulation becaue exiting method for thi problem etting are lacking Keyword: equential probability ratio tet, operating characteritic function, average ample number, nontationary I INTRODUCTION The well-known equential probability ratio tet (SPRT) propoed by Wald [], [2] i a powerful tool of equential analyi and i widely ued in medicine, tatitic, ocial cience and engineering, uch a clinical tet, quality control, radar ignal proceing Many verion of generalization of the SPRT (referred to a GSPRT [3]) have been propoed [4] [6] to improve the performance further for more complicated practical application or to meet the requirement for more general etting than the implet cae of binary imple hypothei teting with iid obervation Their behavior wa tudied in [3], [7] Further, SPRT alo form the foundation of ome change detection technique, eg, the celebrated Page cumulative um (CUSUM) tet [8] comprehenive urvey of equential analyi a well a the challenge wa provided by Lai [9] and the ubequent comment and dicuion by the reviewer The SPRT i generally applicable to equential teting problem Theoretically peaking, it require neither the hypothee to be imple nor the obervation to be iid, provided the likelihood ratio can be computed equentially However, analyi of SPRT propertie and behavior become much more complicated with thee aumption relaxed Reearch upported in part by RO through grant W9NF and ONR-DEPSCoR through grant N The performance of the SPRT ha been tudied extenively It optimality for binary imple hypothei teting with iid obervation wa proved in [] [2] the expected ample ize under both hypothee are imultaneouly minimized among all the tet that do not exceed the given type I and type II error rate Thi optimality i remarkable and uually not achievable elewhere But when it come to the compoite hypothei problem, thi miracle i gone and analyi of the optimality become much more difficult (ee [7], [9], [3], [4] and the reference therein) lo, the optimality propertie of the SPRT without the aumption of the iid obervation have been tudied, but only a few aymptotic reult for ome pecial cae are available (ee [3], [5] [8]) Two important function operating characteritic (OC) and average ample number (SN) characterize the behavior of the SPRT Unlike the optimality problem, the exiting method propoed to evaluate thee two function are almot all baed on the aumption of an iid proce In thi cae, it ha been known [8], [9], [2] for a long time that thee two function atify the Fredholm integral equation of the econd kind (FIESK) [2], [22] nalytical olution can not be obtained except for ome pecial cae for example, Vardeman and Ray [23] provided an exact olution to the average run length (RL) function of Page CUSUM tet when the obervation are iid and exponentially ditributed It i intereting ince the RL atifie a imilar FIESK a OC and SN do In general, one ha to reort to ome numerical technique or to approximate the olution under ome aumption Wald [2] approximated the olution of OC and SN by omitting the overhoot when the tet tatitic croe one of the boundarie n iterative method wa propoed by Page [24] to numerically olve the FIESK, and the convergence property of thi method ha been examined by Kemp [25] Beide, extenive tudie have been done to compute the OC and SN for the truncated SPRT with ome pecific procee (eg, Poion and Binomial) [26] [29] If the log-likelihood ratio are iid and Gauian ditributed, Goel and Wu [3] propoed a method to convert the FIESK to a ytem of linear algebraic equation (SLE) by approximating the integral with the Gauian quadrature It can be actually applied to the non-gauian cae provided the Gauian quadrature can work well Mikhailova, et al [3] ISIF 45

2 conidered a problem with a dicontinuou core of FIESK, and the olution become much more complicated Other method, uch a finite element analyi [32] [34], can alo be employed to find the numerical olution of the FIESK Further, the bound of the OC and SN were calculated in [2], [2], [25], [35] (ee the reference therein) However, all thee tudie of OC and SN are baed on the iid aumption The tationarity of an iid proce implifie the analyi of OC and SN greatly under which the tatitical propertie of the SPRT are the ame regardle of the tart time of the tet (provided the initial value are the ame) In thi paper, we conider the cae of an independent but non-tationary proce, meaning that the log-likelihood ratio need not be identically ditributed Thi cenario i frequently encountered in application For example, if the SPRT i implemented baed on the innovation of the Kalman filter [36], the log-likelihood ratio are independent or weakly coupled, but their ditribution at different time intant are different in general Or the parameter under tet may be time varying, rendering the log-likelihood ratio not identically ditributed Once the log-likelihood ratio are not tationary, the OC and SN in general do not obey FIESK any more, which invalidate the exiting method baed on olving FIESK In thi paper, we attack thi problem in two tep: Firt, following the derivation of FIESK for the iid cae, two inductive integral equation governing the OC and SN repectively for the non-tationary cae are obtained They can be viewed a a generalization of the FIESK The governing equation are in an inductive form ince without the tationarity of the log-likelihood ratio, the OC and SN are changing wrt the tart time of the tet the tatitical propertie of the SPRT with different tart time are related but not identical Unfortunately, the uniquene of olution can not be guaranteed in general, rendering numerical olution difficult By extending the SLE method, thee two inductive integral equation can be approximated by a ytem of linear equation But it i under-determined and there are infinite many olution Thi alo reveal the uniquene problem of the olution However, the theoretical value of thee two governing equation can not be ignored ince they form the foundation to analyze the OC and SN and help undertand the behavior of the SPRT Second, for two frequently encountered pecial cae (a) the log-likelihood ratio converge in ditribution and (b) they have periodic ditribution additional condition (equation) can be impoed, rendering a unique numerical olution, which can be eaily obtained by olving a ytem of linear equation explained in Sec IV, thee two pecial cae are not uncommon in application and their olution are imple No exiting method known to the author can be employed to compare in thi problem etting, o our olution for thee two pecial cae are inpected and verified by Monte Carlo (MC) imulation Thi paper i organized a follow Firt, Wald SPRT i reviewed in Sec II The OC and SN with independent but non-tationary log-likelihood ratio are analyzed in Sec III, and numerical olution for two pecial cae are developed in Sec IV In Sec V, two illutrative example are provided and our olution are compared with the reult of MC imulation Concluion are made in Sec VI II OVERVIEW OF WLD S SPRT For a binary hypothei teting problem, H : Θ H : Θ where i the parameter under tet, if the obervation x k are collected equentially, then the SPRT compute the cumulative um S k of the log-likelihood ratio and the deciion i made when enough data are collected: where Declare H if S k B ln β α Declare H if β S k ln α Ele, continue S k+ S k + k+ () k ln f (x k H ) f (x k H ) i the log-likelihood ratio at time k, and B are the lower and upper threhold depending on the type I error probability α and type II error probability β deired by the uer, f (x k H i ) i the likelihood function Wald approximation of and B are widely ued in practical application Operating Characteritic and verage Sample Number If k are iid (conditioned on parameter), extenive reult for the OC and SN are available Denote the probability denity function (PDF) of k a f (), where the ubcript denote the ground truth of the underlying parameter Thi notation i ued throughout thi paper Note that the ground truth of need not be in either Θ or Θ Firt, define τ k min{t : S t or S t B,t > k (2) a the topping time variable The ubcript k denote the tart time of the tet Becaue of the tationarity for the iid cae, it doe not matter when the SPRT tart (ie, it tatitical propertie do not change wrt k) Without lo of generality, we aume the tet tart at time k The OC i defined a the probability that the tet tatitic finally drop below a a function of the tet initial value (ie, S ) and the ground truth : P () P {S τ S P {S τ B S (3) Notice that P ( ) denote the OC while the P { denote the probability of an event the reader hould be able to ditinguih from the context without any ambiguity The 46

3 econd equality of Eq (3) hold if and only if the SPRT terminate within finite tep almot urely: P {τ < Wald [2] ha proved it under very mild condition One of the ufficient condition [35] i Define P { k <, k (4) N () E [τ S ] a the average ample number (SN) [2], which i alo a function of and In thi paper, we do not conider the cae that the SN doe not exit B Governing Equation and Solution with iid Log- Likelihood Ratio It i known [8], [9], [2] that P () and N () atify the following Fredholm integral equation of the econd kind (FIESK) [2], [22], P () N () + f (x)dx+ P (x)f (x )dx (5) N (x)f (x )dx (6) The exitence and uniquene of the olution for general FIESK are guaranteed under mild condition, given in [37], [38] In general, people have to reort to numerical approximation to the olution III THE OC ND SN WITH INDEPENDENT BUT NON-STTIONRY LOG-LIKELIHOOD RTIOS If the equence { k i independent but not tationary, it doe matter when the SPRT tart So we define P k () P {S τk S k a the OC for thi cae with the upercript k explicitly indicating the tart time of the tet Obviouly, thi definition degenerate to Eq (3) if { k i iid Similarly, define () E [τ k k S k ] a the SN Note that the tart time k i ubtracted ince we only conider how many future ample are needed on average Then P k() and Nk () are governed by the following inductive integral equation, P k () f k+ (x)dx+ P k+ (x)f k+ (x )dx (7) () + + (x)f k+ (x )dx (8) where f k( ) denote the PDF of k, and the non-tationarity i explicitly revealed by the upercript k The idea of the derivation follow imilarly a in [9], [2] Since by definition P k () i the probability of the event that the tet tatitic croe the lower bound with the tart time k and initial value, thi event can be partitioned to two mutually excluive event: Λ {S k+ S k + k+ + k+ Λ 2 { < S k+ < B { S τk+ S k+ The firt i the event that the tet tatitic croe at time k+ while the econd i that S k+ i between the two bound (ie, the SPRT doe not top at time k+) and croe after time k+ Thi econd event can be viewed a the tet tart at time k+ with the initial value of S k+ It i clear that the probability of the firt event equal P {Λ f k+ (x) dx The probability of the econd event equal P {Λ 2 P k+ (x)f k+ P {Λ 2 S k+ xp {S k+ xdx { P Sτk+ S k+ x f k+ (x )dx (x )dx Hence Eq (7) follow The derivation for () follow imilarly Firt, define two mutually excluive event It i clear that o, Γ {S k+ {S k+ B Γ 2 { < S k+ < B P {Γ P {Γ 2 P {Γ 2 f k+ (x )dx N() k N( Γ k )P {Γ +N( Γ k 2 )P {Γ 2 ( P {Γ 2 ) ( ) B (x)p {S k+ x Γ 2 dx P {Γ 2 ( P {Γ 2 ) ( ) + + Nk+ (x)p {S k+ x,γ 2 dx P {Γ 2 P {Γ 2 ( P {Γ 2 ) ( ) + + Nk+ (x)f k+ (x )dx P {Γ 2 P {Γ (x)f k+ (x )dx Thi i Eq (8) Unlike Eq (5) (6), even the numerical olution of Eq (7) (8) are difficult to obtain (if not impoible) in general There are two major difficultie Firt, although thee two 47

4 equation are in inductive form (wrt time k), it i virtually impoible to implement the induction, be it forward or backward, ince no initial value (thi i actually exactly what we want to have) i available for the induction Second, if P k() andnk () are viewed a function with two variablek and, then the olution of thee two equation are not unique It i clear that for the iid cae, Eq (7) (8) degenerate to Eq (5) (6) ince the tart time of the tet ha no impact on OC and SN In the next ection, we try to olve thee two equation numerically for two pecial cae IV NUMERICL SOLUTIONS FOR SOME SPECIL CSES If it i further aumed that the equence { k converge in ditribution or k have periodic ditribution, additional equation can be impoed to Eq (7) (8), reulting in unique numerical olution Sytem of Linear lgebraic Equation (SLE) Method The SLE method wa propoed [3] to olve Eq (5) and (6) numerically under the Gauian aumption It approximate the integral term in the Fredholm equation by Gauian quadrature to form a ytem of linear equation Numerical value of P () and N () on every quadrature point are calculated by olving thi linear equation ytem lthough the SLE method wa developed under the Gauian aumption, actually it i generally applicable provided the integral term in Eq (5) (6) can be well approximated by Gauian quadrature Further, the fact that many denity function can be well approximated by Gauian mixture of only a few component alo broaden it applicable ituation The SLE method can be generalized to olve Eq (7) (8) Replacing the integral term with n-point Gauian quadrature yield P k+ (x)f k+ (x )dx n ω i P k+ (z i )f k+ (z i ) i where ω i and z i are the weight and point of the Gauian quadrature repectively pplying to Eq (7) thi approximation yield P k () Fk+ ( )+ n ω i P k+ (z i )f k+ (z i ) i where F k ( ) i the cumulative ditribution function (CDF) of k Let z,,z n Then a ytem of linear equation i obtained n P k (z j ) ω i P k+ (z i )f k+ (z i z j ) F k+ ( z j ) i j,2,,n which ha the equivalent matrix form [ ] [ ] P I k Mk+ Φ P k+ (9) k+ where ( M k+ M (ij) k+ ) and M (ij) k+ ω if k+ (z i z j ); Φ k+ [ F k+ ( z ),,F k+ ( z n ) ] T ; P k [ P k(z ),P k(z 2),,P k(z n) ] T Eq (9) ha n equation but 2n unknown, o it i underdetermined and there are infinite many olution Combining Eq (9) for different time k yield I M k+ P k Φ k+ I M k+2 P k+ I M k+m P k+m Φ k+2 Φ k+m () The equation for SN can be derived imilarly I M k+ I M k+2 + I M k+m +m () where m i a poitive integer; [ (z ), (z 2),, (z n) ] T ; n vector of all Eq () () are till under-determined ince there are only nm equation but n(m + ) unknown dditional n equation are needed to have unique olution We conider two pecial cae B With Sequence { k Converging in Ditribution Firt, we conider a pecial cae that the random equence { k converge to š in ditribution [39] and the PDF of k are continuou for all k; that i, for every point of F k(), lim k Fk () F() where F() i the CDF of š Thi cae i not uncommon eg, if the Kalman filter i applied to a linear timeinvariant ytem, the innovation are converging in ditribution (provided the tandard Kalman filter aumption are atified and the filter itelf i converging) In thi cae, the equence { k aymptotically become iid and thu there exit a poitive integer M uch that Hence P k () Pk+ () Nk+ () P () () N () k > M P k P k+ ˆP k > M (2) + ˆN where ˆP and ˆN are the olution of Eq (5) (6) repectively withf () f (š) Thi can be intuitively undertood: IfM i large enough, fork > M the ditribution of k become almot identical (ie, approximately iid), and then the impact of the tet tart time on the tatitical propertie of the SPRT can be ignored, which validate Eq (2) Therefore, any exiting 48

5 technique for the iid cae can be applied to obtain ˆP and ˆN Then, we chooe the m in Eq () () uch that k +m M Plugging Eq (2) into Eq () and () repectively yield the following two n(m + ) n(m + ) equation ytem for OC I M k+ P Φ k+ I M k k+2 Φ P k+ k+2 and for SN I M k+ I M k+2 I M k+m I I M k+m I P k+m + +m Φ k+m ˆP (3) ˆN (4) Note that the coefficient matrice in Eq (3) and (4) are alway invertible By Gauian elimination, the olution for P k and are m l P k+i X k+i+ + ( ) l i X k+l+ +i Y k+i+ + li+ m li+ ( ) l i ji+ l ji+ for i,,,m, where X k+ Φ k+ Y k+ X k+2 Φ k+2 Y k+2, X k+m Φ k+m Y k+m X k+m+ ˆP Y k+m+ ˆN Y k+l+ Here and in the equel the ummation i defined atifying b ( ) i, if a > b ia C With Periodic PDF of k If the ditribution of k are periodic with period T (a poitive integer), ie, then and hence P k f() k f k+t () () Pk+T () Nk+T () k () P k P k+t k (5) +T It i eaily undertood ince the SPRT tarting at time k or k+ T are tatitically equivalent, provided the initial value are the ame Inerting Eq (5) into Eq () and () repectively yield the following two equation ytem for OC I M k+ I M k+2 I and for SN I M k+ I M k+2 I P k P k+ I M k+t P I k+t By Gauian elimination, the olution are T l P k+i U k+i+ + ( ) l i +i V k+i+ + li+ T li+ Φ k+ Φ k+2 Φ k+t (6) + I M k+t N I k+t (7) ( ) l i ji+ l ji+ for i,,,t, where U k+ Φ k+ V k+ U k+2 Φ k+2 V k+2, U k+t Φ k+t V k+t U k+t+ Û V k+t+ ˆV and T l Û Z Φ k+ + ( ) l l T l ˆV Z I+ ( ) l l T Z I+( ) T j j j U k+l+ V k+l+ Φ k+l+ provided Z i invertible It i clear that when T, Eq (6) (7) degenerate to the iid cae V ILLUSTRTIVE EXMPLES Two numerical example are provided in thi ection Since the ground truth of the OC and SN are unknown, we have to reort to the Monte Carlo (MC) method In both of the following two cae, reult of MC imulation were obtained from, run 49

6 7 6 MC_im SLE 9 8 MC_im SLE 5 4 k k2 7 6 P k 3 k3 k P k k k2 k3 k MC_im SLE 5 45 MC_im SLE k 3 2 k2 25 k k2 5 k3 k 2 k3 k Figure Reult for Cae, where equence { k converge in ditribution The SLE olution and the MC imulated reult of P() k and N() k are compared The left and right column correpond to parameter in the firt and the econd group in Table I repectively The abcia of each figure repreent the initial value and the vertical axi i the OC P k () or SN N() k The different curve in the figure correpond to the different tart time k of the tet From the plot, it i clear that the MC imulated reult (blue olid line with + ) match the SLE olution (red dahed line) very well Cae : { k Converge in Ditribution If the log-likelihood ratio k ha the Gauian ditribution where f k () N(;µ k,σ 2 ), k,2, (8) µ k +ce k i the ground truth of the underlying parameter, c a contant, and σ 2 the variance It i eay to verify that { k converge in ditribution: lim k + F k() ˇF() where ˇF() i the Gauian CDF with mean and variance σ 2 Two group of parameter were imulated and the reult were compared with the olution of our algorithm (ee the parameter in Table I and the imulation reult in Fig ) and B were computed by Wald approximation with both type I and type II error rate etting to m i the parameter in Eq (3) (4) and n i the number of Gauian quadrature point It can be oberved from Fig that P k () i monotonically decreaing wrt, a expected Thi hould happen becaue of the definition of OC Since the ditribution of k converge very fat (the term ce k diminihe exponentially), the difference of OC P k() and SN Nk () curve with different tart time k are almot unobervable for k > Further, a the mean of k i converging to exponentially, for > the mean of the cumulative um S k i increaing, rendering S k le likely to drop below the lower threhold For <, the mean of S k will finally decreae, increaing the chance that S k drop below Hence, P k () for group one i much maller than for group two For (), when the initial value of the tet i cloe to the bound, it i more likely that the tet tatitic croe the bound in fewer tep It make ene 5

7 8 7 MC_im SLE 9 MC_im SLE P k 4 P k 6 3 k 5 2 k3 k5 k7 4 3 k k3 k5 k MC_im SLE 8 7 MC_im SLE k k3 k5 k k k3 k5 k Figure 2 Reult for Cae 2, where the PDF of k i periodic in k The left and right column correpond to parameter in the firt and the econd group in Table II repectively gain, the MC imulated reult (blue olid line with + ) agree very well with the SLE olution (red dahed line), which further verifie the olution of our method for the tet to take more tep if the tet tart around the middle of the two bound Table I PRMETERS FOR CSE σ 2 c B m n Group Group B Cae 2: Periodic PDF gain, k are aumed of the Gauian ditribution of Eq (8), but the mean µ k i changing periodically: µ k +co( 2πk T ) where the period T i an integer It i obviou that f k() f k+t () Two group of parameter, given in Table II, were imulated by the MC method and the reult were compared with our olution The reult are plotted in Fig 2 The difference between the MC reult and our numerical olution are tiny, a expected Some pattern of the curve eg, P k() i monotonically decreaing and () ha it peak value in the middle of and B are imilar a in cae for the ame reaon But in thi cae, the curve for k and k+t are exactly overlapped, meaning that there are only T different curve hown in Fig and Fig 2, reult of MC imulation upport our numerical olution favorably The average computational cot (meaured in econd) of our algorithm and MC imulation i given in Table III, which how our algorithm i alo computationally efficient Table II PRMETERS FOR CSE 2 σ 2 T B n Group Group

8 Table III VERGE COMPUTTIONL COST (IN SECONDS) SLE MC imulation a Cae 3 35 Cae a The imulation platform: Dell Preciion 69, CPU: Intel Xeon E5335, memory: 4GB PC53, operating ytem: Ubuntu 94, oftware: Matlab 27a VI CONCLUSIONS In thi paper we have developed two inductive equation governing the OC and SN function repectively when the log-likelihood ratio are independent but non-tationary They can be viewed a a generalization of the Fredholm integral equation for the iid cae Numerical algorithm for two pecial cae the equence { k of log-likelihood ratio converge in ditribution and k have periodic ditribution have been propoed baed on the exiting SLE method developed for olving the Fredholm integral equation The central idea of the SLE method i approximating the integral term in the Fredholm equation by the Gauian quadrature Of coure, other numerical approximation may alo be applied here Identifying other ueful pecial cae that have unique olution are under invetigation To our knowledge, no algorithm ha been propoed to compute the OC and SN in thi problem etting, and conequently our method can not be compared or verified by exiting method other than MC imulation Two numerical example of our olution have been provided, and cloe agreement with the reult of MC imulation ha been found It i worth to note that ince the RL function of the CUSUM tet atifie a imilar Fredholm equation, our method can be eaily tranplanted to compute the RL function with independent but non-tationary loglikelihood ratio REFERENCES [] Wald, Sequential tet of tatitical hypothee, nnal of Mathematical Statitic, vol 6, pp 7 86, 945 [2], Sequential nalyi New York: Wiley, 947 [3] B Eienberg, B K Ghoh, and G Simon, Propertie of generalized equential probability ratio tet, The nnal of Statitic, vol 4, no 2, pp , 976 [4] L Wei, Teting one imple hypothei againt another, The nnal of Mathematical Statitic, vol 24, no 2, pp , June 953 [5] P rmitage, Retricted equential procedure, Biometrika, vol 44, no /2, pp 9 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