Markov Chain Analysis of the Sequential Probability Ratio Test for Automatic Track Maintenance

Transcription

1 Marov hain Analysis o the Seuential Probability Ratio Test or Automatic Maintenance Graham W. Pulor General Sonar Stuies Grou Thales Unerwater Systems Pty. Lt Ryalmere NSW, Australia graham.ulor@au.thalesgrou.com Abstract Wal s 47 seuential robability ratio test has been alie to the automatic trac maintenance roblem. While the execte time to elete a alse trac or to conirm a true trac is easy to comute, the time to conirm a alse trac is not. Marov chain analysis is alie to etermine the latter uantity as well as the robabilities o true trac an alse trac conirmation. The techniue alies in low robability o alse alarm situations in which Monte arlo simulation is ineasible. Keywors: ing, trac maintenance, Marov chain, Monte arlo simulation, system erormance analysis, seuential robability ratio test, mean time between ailures.. Introuction Systems or the automatic tracing o targets in clutter reuire a strategy or testing the uality o tracs. The roblem o classiying tracs as goo, ba or unsure an uating these categories as they change with time is nown as the trac maintenance roblem. Regarless o the system one aots or the actual scoring o tracs, it is natural that one shoul conirm goo tracs, elete ba tracs an reserve jugement on tracs that are tentative. maintenance can be viewe as a statistical hyothesis testing roblem. A ba trac is eine as one that is orme rom alse alarms or clutter whereas a goo trac contains etections rom a true target. (The uestion o target multilicity is not consiere herein.) Base on a moel or the receiver [], a statistical moel may then be erive or goo an ba tracs to which a lielihoo ratio test can be alie to ecie between the null hyothesis (alse trac) H an the alternative hyothesis (true trac) H. A classical trac maintenance aroach is M/N logic [], which uses a ixe interval criterion such as 3 hits out o loos to conirm a trac. Another aroach is base on the seuential robability ratio test (SPRT) o Wal [], ating bac to 47. The alication o this test to automatic trac maintenance is covere in Blacman [] an we review the techniue in section. This test is o interest because it minimises the average ecision time over the class o all tests with the same or lower tye I an II error robabilities α, β where α = Pr(choose H β = Pr(choose H H H true) true). () In articular, the SPRT oes not have a ixe observation winow length. The test has been alie or many years in aerosace areas such as trac maintenance [4], ailure etection or gyro navigation systems [] an more recently in multihyothesis tracing (MHT) systems []. More recent aroaches to automatic trac maintenance mae use o trac covariance inormation o which the robabilistic ata association metho o olgerove et al. [3] an integrate robabilistic ata association [8] are examles. The utility o covariance inormation is that it can moel not just the resence or absence o a etection but also the resence o many etections in the trac gate. An alication o the SPRT test or a PA tracer has recently aeare in [7]. In Wal s treatise on the SPRT, a metho was ut orwar or the selection o the two threshols or the lielihoo ratio test. The threshols are etermine urely as a unction o the tye I an II error robabilities an thus o not reuire the istributional roerties o the test statistic. This choice ensures that the error robabilities or the test will not excee the seciie values o α an β; but the true error robabilities may be somewhat less than these values. The SPRT itsel is covere in section. One o the contributions (section 4) o this article is to set out a metho or etermining exactly what the actual SPRT error robabilities are or a given air o threshols. The execte ecision times or the SPRT are easily comutable in the case o true trac conirmation an alse trac eletion (see []). There seems to be however no easy way to etermine the execte time to conirm a alse trac. Many surveillance systems have a reuirement on the mean time between alse tracs (MTBFT) but this is roblematic to veriy when the only tool available is Monte arlo simulation. Inee, at low alse alarm robabilities it is oten ineasible to estimate the MTBFT using Monte arlo runs. The secon contribution o this wor is to rovie a tight over-boun on the MTBFT (section ). As with the etermination o the error robabilties, the theory o inite-state Marov chains is use to erive the result. For clarity o resentation we rovie an examle o conventional Marov chain analysis in section 3 or a

2 simle trac maintenance metho (the 3-out-o- rule) beore roceeing with the novel SPRT analysis in sections 4 an. The reaer who is unamiliar with Marov chains may wish to consult reerences [] an more articularly [6]. Section 6 resents numerical simulation results an also etails the Monte arlo aroach, which we use to veriy the accuracy o the Marov chain metho. onclusions are rawn in section 7.. Seuential Probability Ratio Test We base our escrition o the SPRT on Blacman [] an escribe its alication to the trac maintenance roblem, namely the roblem o eciing whether to conirm or elete a trac base on a nown receiver oerating characteristic (RO) []. In this context we wish to istinguish between the two hyotheses: H : the trac consists o alse alarms (no target is resent); H : the trac consists o etections rom the target. We ocus attention on a single resolution cell in etection sace. All resolution cells are assume to be ineenent an to have ientical etection (signal/noise) statistics. No account is taen o issues such as target an source ynamics, measurement errors ue to noise or ositioning errors, misassociation errors or gating. At any scan, a etection may occur or no etection may result. A etection is reerre to as a hit an a non-etection as a miss. Assuming that at each scan a target may be etecte with robability P an that the robability o alse alarm is P, the log lielihoo ratio test statistic or the SPRT can be written in the orm Λ m, n) = a ( P, P ) m a ( P, P ) ( m ), () ( + n where m is the number o hits, n is the number o misses an the unctions a an a are given by P ( P ) a( P, P ) = ln P ( P ) (3) P a ( P, P ) = ln. P The trac is conirme i Λ ( m, n) T or elete i u Λ ( m, n) T l, where T l < T u are ixe arameters, resectively calle the eletion an conirmation threshols. In the event that T l < Λ( m, n) < T, the u ecision is eerre until urther ata are available. The threshols are chosen accoring to [] as β Tl = ln α β (4) Tu = ln α α = Pr( alse trac conirmation) = Pc β = Pr( true trac conirmation) = Pct. Thus it is seen that the test has a ranom ecision time. This is unlie conventional ixe-length tests such as the M/N conirmation rule, but as ointe out in [4]: Wal an Wolowitz [3] have shown that the seuential robability ratio test will reuire ewer samles, on the average, than a ixe samle test with the same alse-alarm an etection robabilities. Thus among tests o eual or higher ower in the ecision theoretic sense, Wal s test has the minimum mean ecision time. The ientiication o α as the robability o conirming a trac when a true trac is absent (a tye I error) an o β as the robability o ailing to conirm a trac when it is resent (a tye II error) is slightly misleaing. Accoring to Wal [] the true error robabilities are uer-boune by α an β, with one ineuality holing exactly. In ractice, as ointe out in [], the actual error robabilities are smaller than α an β, with the alse trac conirmation robability tyically being substantially less than α. One o the aims o this article is to rovie analytical means to etermine the exact robabilities o tye I an II errors or the SPRT. 3. Marov hain Analysis Examle As an illustrative examle o Marov chain analysis we consier the 3-out-o- (3 hits out o loos) rule or trac conirmation in this section. A more thorough treatment o this toic is containe in Blacman []. The techniue relies on moelling the trac conirmation rocess as a inite state Marov chain. This is a owerul metho because it allows uestions relating to trac maintenance statistics an robabilities to be answere using analytical techniues rather than simulation. The metho reuires us to (i) escribe the stages leaing to trac conirmation in terms o iscrete states, (ii) seciy the robabilities or transitions between these states, an (iii) seciy the initial robability istribution across the states. We start by eining a ranom variable z() that has the roerty z()= i we receive a hit on a target at scan number an z()= otherwise. We are then le to consier the length vector o shit register states x() = [z(-4), z(-3), z(-), z(-), z()] o which there are 3 ossible values. Many o these states can be lume into a single conirme state since they contain at least 3 out o hits. Some o the other states can also be aggregate ue to symmetry. Ater tabulating the ossible transitions rom each o the 3 states, it turns out that only states are reuire to eine the Marov chain. For the 3-out-o- trac conirmation rule, the states x() are ( X enotes a on t care value o either or ): [X ], [X ], [X ], [ ], [X ], [ ], [ ], [X ], [ ], [ ], [ ], {three or more s}. The th state corresons to a conirme trac. Let =P an =-P, then with this einition o states,

3 the ollowing state transitions aly rom one time to the next:. state state with robability an state with robability ;. state state 3 with robability an state 4 with robability ; 3. state 3 state with robability an state 6 with robability ; 4. state 4 state 7 with robability an state with robability ;. state state 8 with robability an state with robability ; 6. state 6 state with robability an state with robability ; 7. state 7 state with robability an state with robability ; 8. state 8 state with robability an state with robability ;. state state 3 with robability an state with robability ;. state state with robability an state with robability ;. state state 8 with robability an state with robability ;. state state with robability (the conirme trac state is absorbing). The above inormation eines the state transition robability matrix o the Marov chain P which has (i, j) element Pr( x(+)=i x()=j ): P =. The inal uantity reuire to comlete the einition o the Marov chain is the initial state robability istribution, which we assume to be [,,,,,,,,,,,], that is, initially there are no revious hits. Imortantly the Marov chain ossesses a single absorbing state. The utility o this ormulation is that it allows the ollowing uantity to be calculate: the execte time to reach the conirme trac state (state ). This is one using the next result [6, ]: Theorem : Let the transition matrix P an initial istribution be artitione as: Q P = an = r', then the execte time to reach the absorbing state is given by µ = ' ( I Q) where is a column vector o ones an I is the ientity matrix. As an examle, at P =., the execte time or trac conirmation is µ(.) = 3.33 scans, an at P =., µ(.) = Similarly, the mean time to conirm a alse trac can be comute simly by relacing the etection robability P with the alse alarm robability P. For instance at P =., the mean time to conirm a alse trac is.678e+8 scans (er resolution cell). 4. SPRT False Probability 4. Analysis Blacman [, 7.] gives a generally alicable metho or analysing trac maintenance rocesses that involve a constant increase (or a hit) or ecrease (or a miss) in trac score at each time instant. This aroach is alicable to the SPRT. The aroach rests on the einition o a - artesian Marov chain that reresents the ranom evolution o the trac score as various hit an miss events occur. The state o the Marov chain is enote X at scan number. enote by (i, j) the value o the Marov chain state corrseoning to any seuence o m=i hits, n=j misses. The inices i an j or the states are counte rom, thus s(,) corresons to the no trac state (no hits or misses). Given a seuence o m etections an n misse etections the Marov chain state X taes the value ( m +, n + ). The robability o occuying a articular state at time is written i, j ( ) = Pr( X ( i, j)). These ieas are eicte in Fig.. In orer to etermine the robability o an event such as trac conirmation we must seciy (i) the initial robability istribution across the states; an (ii) the uating euations or the Marov chain, taing eletion an conirmation rocesses into account. The uating metho suggeste in Blacman section 7. is lacing in imlementation etail an oes not seem to yiel a sensible result (the robabilities o not sum to one at all times ). Instea a way was sought that ensures that the robability istribution across the states is normalise at all times. This metho will be reerre to as the rile metho. Letting enote the robability o a hit an = the robability o a miss, the roagation euations or the rile metho are, or =,..., K:, () =, i, j () =, i, j, j ( + ) =, j ( ), j =, K, () i, ( + ) = i, ( ), i =, K, ( + ) = ( ) + ( ), i, j =, K,. i, j i, j i, j The uating on (i, j) rocees along iagonal lines or riles such that i+j=+. The irst three o these

4 euations are as in Blacman []. The irst euation seciies the initial robability istribution o the Marov chain. The ourth euation is reuire to uate the robability o states that o not lie on either the i= or j= axes. This euation can be erive in a straightorwar manner rom robability theory (conitioning on the mutually exclusive events o hit an miss). Notice that the time interval K or moelling the evolution o the Marov chain has been assume to be inite. =4 =3 = = 4 i-axis Time onirm (4,) X elete j-axis Figure : Marov chain reresentation or the Rile Metho or analysing the SPRT. The state X lives on a - gri o oints (i, j). States that are reachable at time are connecte in a line that avances as time increases. A ranom samle seuence consisting o {miss, hit, hit, miss, hit, hit} is shown that results in trac conirmation at time 6. The robability is enote or a hit an or a miss with +=. In roagating the Marov chain euations, we tae the ollowing aroach. The -th rile o the state robability istribution is generate using () to give i, j ( ) or i+j=+. This has the roerty that the sum o its elements (along the rile) is unity. All receing riles are set to zero. Next we etermine the states in the -th rile that lie in the eletion region an accumulate their robabilities in the trac eletion robability _el. We then set the state robabilities i, j ( ) to zero or the elete states. The states in the -th rile that lie in the conirmation region are etermine an we also accumulate their robabilities in the trac conirmation robability _con. We then set the state robabilities i, j ( ) to zero or the conirme states. The time is incremente an we rocee to the next rile. 4. Imlementation The rile metho has been imlemente in a Matlab unction taing as inut the ollowing uantities: P - inut robability o etection P - inut robability o alse alarm Ph - inut robability o a hit (eual to either P or P) Tl - lower threshol on log lielihoo Tu - uer threshol on log lielihoo K - maximum number o time samles allowe to gather tracer statistics; an oututs the trac eletion robability (_el) an the trac conirmation robability (_con). The utility o the rogram is that it can be alie to etermine the eletion an conirmation robabilities o true an alse tracs or given inut P, P an SPRT threshols. The unction oes not use Monte arlo simulation, but a maximum number o time samles K must be chosen. As a uality measure, the rogram islays the sum o _con an _el, which shoul be aroximately eual to. The value o K shoul be chosen large enough to mae this hol. At low values o the inut robability o alse alarm (e.g..e ), it may not be easible to mae K large enough so that _con + _el = ; in such cases, a value o K shoul be chosen so that the sum o _el an _con oes not increase signiicantly as K is increase.. SPRT onirmation Time. Analysis The ormulae or the execte ecision times or the SPRT in [, 6.] are arametrise in terms o the outut P an P (enote by β, α resectively) as well as on the inut P an P an SPRT threshols Tl an T. Accoring to [] the execte time to ecie on u H is E[ H αtu + ( α) Tl ] = P a ( P, P ) a ( P, P while the execte time to ecie on H is: E[ H ( β ) Tu + βtl ] = P a ( P, P ) a ( P, P where the symbols are as eine in section. The execte ecision time means, in the case o H, the execte time to conirm a true trac, an, in the case o H, the execte time to elete a alse trac. These ormulae are aroximations an, as we saw in the receing section, one cannot arbitrarily ix α an β since these uantities een on the inut P an P as well as on the SPRT threshols. The time to conirm a true trac is clearly o tactical imortance, while the time to elete a alse trac has imlications or the comutational loa in the system. In many systems however there is an aitional reuirement on the execte time to conirm a alse trac, which is imortant since it is the MTBFT. As note earlier, existing escritions o the SPRT metho rovie no obvious way to comute the execte time to conirm a alse trac Tc. ; ) ; ) (6) (7)

5 The lac o such inormation maes it iicult to esign a trac maintenance test using the SPRT that can be roven to satisy MTBFT reuirements. This iiculty becomes acute at low alse alarm robabilities since an ineasibly large number o Monte arlo runs woul be reuire to veriy the algorithm erormance. For instance, suose the robability o conirming a alse trac is.e-6, that is, this event occurs once in every,, observations on average. To estimate the MTBFT, we may nee to observe aroun occurrences o this event, thus we might reuire,, runs or each set o system tuning arameters. Tuning a system in this manner is clearly not racticable. We will show in this section that it is not necessary to now the error robabilities α, β an that a etermination o Tc is ossible base only on the inut P, P an the threshols Tu an Tl. Our metho is analytical but aroximate, an yiels a reasonably tight over-boun on the mean alse trac time. The metho has been resente in an algorithmic manner since it was not convenient to erive a general ormula or Tc. However, unlie simulation base aroaches, it can be alie at any oint on the receiver oerating characteristic, or instance at low alse alarm robabilities. The analysis in this section is similar to that in the receing section in that we use a - artesian Marov chain moel. Whereas in the receing section we roagate the state robability istribution or the Marov chain, in this section, we see to etermine the actual transition robability matrix P. In orer to achieve this, we must irstly etermine an inexing scheme or the states; seconly we must etermine the transition structure that exists between the states (i.e., the connections); inally we seciy the transition robabilities between the connecte states. The state inexing scheme aote or this roblem is shown in Fig.. States are numbere rom the bottom let corner, thus state (, ). Subseuent states are numbere seuentially along the riles or =,, 3, etc. We lum all conirme states into a single state an all elete states into a single elete state. There are an ininite number o states in the Marov chain, however we set a maximum time K as beore to mae the number inite (but ossibly uite large). For examle, with P=., P=., Tu=4 an ignoring eletion or now, the corresonence between artesian states an state inexes woul be: (, ), (, ), 3 (, ), 4 (, 3), {(, ), (3,), (,4),...}. =4 =3 = = 4 i-axis onirm (4,) X elete j-axis Figure : Ste in the etermination o the transition robability matrix or the rile metho: state numbering. Any state in the conirmation region is labelle, any state in the eletion region is labelle ; all other states are numbere seuentially along the iagonal time riles. The theory o inite-state Marov chains [, 6] allows us to comute the execte time to reach the absorbing set o states. For the SPRT the absorbing set consists o two unconnecte absorbing states, namely, elete an conirme. Alication o absorbing Marov chain theory woul then yiel the time to enter either the elete or conirme state, which is not what is reuire. The theory o regular an ergoic Marov chains [, IV- V] is not alicable since the chain is not ergoic. This imlies that the mean assage time matrix [,.] cannot be comute or this rocess. Since we see to etermine the execte time only to conirm a alse trac, we instea treat the elete trac states as transient rather than absorbing an ee them bac into the no trac state (number ). This concet is illustrate in Fig. 3. We can then aly the result rom Theorem to obtain an estimate o the mean conirmation time. As a conseuence o this ormulation, the time to reach the single absorbing (conirme) state will be an uer boun on the time to conirm a trac (whether true or alse). Exeriment conirms that, in the cases consiere, the boun on the true trac conirmation time excees that obtaine by simulation by less than.6 scans. It remains to etermine the actual transition robabilities. This can be accomlishe algorithmically by assigning horizontal transitions in Fig. 3 to (the hit robability), vertical transitions to = (the miss robability), an transitions rom the conirme an elete states to one.

6 onirm elete Figure 3: Ste in the etermination o the transition robability matrix or the rile metho: transition structure. Any state in the ban between elete an conirme trac regions can transit one unit to the right (or a miss) or one unit u (or a hit). The conirme state is absorbing. The elete trac state is assume to be transient an lins irectly to the notrac state number. The estimation o the conirmation time or a alse trac Tc can be chece by increasing the time limit Kt until there is no signiicant change in the value o Tc.. Imlementation The algorithm to calculate the execte alse trac conirmation time can easily be coe in Matlab. The unction taes as inut the ollowing uantities: P - inut robability o etection P - inut robability o alse alarm Ph - inut robability o a hit (eual to either P or P) Tl - lower threshol on log lielihoo Tu - uer threshol on log lielihoo Kt - maximum number o time samles or moelling o Marov chain; an oututs the execte time to conirm the trac τ using the seciie hit robability. This can be alie either to true (Tct) or alse tracs (Tc) by assigning Ph to P or P resectively. 6. Perormance Evaluation 6. Monte arlo Simulation Monte arlo trials can be use to estimate metrics relating to the erormance o the SPRT. This is a irect simulation aroach that erorms a seuence o ineenent ranom trials an accumulates statistics on the number o conirme an elete tracs an their urations. Monte arlo simulation comutes erormance statistics by brute-orce an it is limite by the number o trials that can be one. For instance it can be alie to comute the mean conirmation time or true tracs an the robability o conirming a true trac, but it is ineective or estimating alse trac conirmation statistics or ractical values o the alse alarm robability. unction [ Tct, Tt, Pct, Pt ] = mc_sim ( P, P, Tl, Tu, K ) Set =, Tcts = zeros ( x K ), Tts = zeros ( x K ) Let a = log {P ( P ) / [ P ( P )] } Let a = log {( P ) / ( P ) } while K Set time =, log_li =, t =, tc = while True let u = ranom samle rom U[, ] i u < P log_li = log_li + ( a a ) else log_li = log_li a en i i log_li Tl t = time brea out o while loo else i log_li Tu tc = time brea out o while loo en i time = time + en while i tc Tcts ( ) = tc en i i t Tts ( ) = t en i = + en while Tct = mean o non-zero entries o Tcts Tt = mean o non-zero entries o Tts Pct = ( number non-zero entries o Tcts ) / K Pt = ( number non-zero entries o Tts ) / K return The above seuo-coe imlements a Monte arlo simulation o the SPRT or true trac conirmation. The rogram inuts are the etection an alse alarm robabilities (P, P), the SPRT threshols (Tl, Tu) an the number o trials K. The rogram oututs estimates o the execte times an robabilities or conirmation an eletion o true tracs. In the next section this coe is use to valiate some o results rom the Marov chain analysis an to comare with conventional SPRT ormulae. 6. Numerical Examle A esign examle is rovie to emonstrate the erormance o the Marov chain analysis techniue an to veriy its erormance against conventional SPRT moelling rom [] an the Monte arlo simulation aroach. The erormance metrics o interest are (i) time to conirm a true trac (Tct) ; (ii) robability o conirming a true trac (Pct = β) ; (iii) time to conirm a alse trac (Tc) ; robability o conirming a alse trac (Pc = α). The aitional metric o time to elete a alse trac given by euation (6) is not resente, although it is straightorwar to comute once Pct an Pc are nown. The esign entails the

7 etermination o the two SPRT threshols T an l T to u meet a system reuirement o conirme alse trac in scans (er resolution cell) over a range o receiver oerating oints. For a surveillance system with, e.g., resolution cells an a revisit time o secon, this euates to alse trac every.78 hours on average (ignoring the eects o target ynamics an sensor measurement / ositioning errors). Metho / Metric Tct Pct Tc Pc onventional Monte arlo Marov hain Table : Perormance metrics sulie by the 3 methos. Insection o Table shows what the three methos have to oer in terms o arameter estimates: all three tests give the time to conirm a true trac; the Monte arlo simulation an Marov chain aroaches both yiel the robability o conirming a true trac; but only the Marov chain techniue gives estimates o conirmation time an robability or alse tracs. We have not resente alse trac eletion metrics or this examle. Note that in the conventional aroach base on (4) one seciies esign values o Pc an Pct to obtain threshols Tl, Tu so that the actual Pc an Pct are aroximately obtaine. However the actual Pc is oten much lower. The erormance metrics or each metho were comute on a RO gri o 3 oints (P, P) {.4,.,.6,.7,.8,.} {.,.,.,.,.}. oncerning test methoology, the Monte arlo aroach was imlemente with K = runs, which was aeuate or ecimal oints on the Tct estimate. Trial an error was use to establish SPRT threshols (Tl = 3, Tu = 8) to meet the Tc reuirement. For these threshols the conventional aroach yiels Pc=.4E 8, Pct=.. The Marov chain metho use arameters K to ensure _con + _el = (see section 4.), an Kt to ensure convergence o Tct (as in section.). omutational loa o the Marov chain metho is etermine by K, Kt, an maximum loa occurs or low P / high P arts o the RO. For instance at P=.4, P=., K= is reuire or convergence o the rile metho or Pc to.e an Kt 36 is reuire or convergence o Tc to 4 signiicant igures. This value o Kt results in a Marov chain with 44 states or Tl = 3, Tu = 8. Fig. 4 islays the results or the robability to conirm a true trac (Pct). The estimates or the Marov chain an Monte arlo methos agree to 3 signiicant igures on all 3 RO airs. The orm o the erormance surace is airly lat in P but it exhibits a large trough aroun P =.8. The results or the robability o conirming a alse trac (Pc), only available with the Marov chain metho, aear in Fig.. Note the z-axis units are to be multilie by. The surace is somewhat ine but islays the execte uwar tren or low P an high P. Figure 4: Probability o conirming a true trac as a unction o P an P with SPRT threshols Tl= 3, Tu=8 obtaine with both Marov chain analysis an Monte arlo simulation. Figure : Probability o conirming a alse trac as a unction o P an P with threshols Tl= 3, Tu=8. Estimates or the mean time to conirm a alse trac (Tc), obtaine via the Marov chain aroach, are tabulate in Table. The results, which have converge to 4 signiicant igures, emonstrate that the Tc= reuirement is met throughout most o the RO excet or the high P / low P region. To increase the Tc, one woul nee to increase the uer threshol Tu, which woul also result in a longer time to elete alse tracs. Finally the results or the execte time to conirm a true trac (Tct) are given in Table 3. This erormance metric is calculable via all three methos an thereore aors a means o comarison. The estimate Tct values obtaine by the conventional aroach were greater than those o the Monte arlo metho in the to let triangle o the RO an less elsewhere. The Marov chain estimates o Tct are uniormly greater

8 than those o the other two aroaches, but the overboun is seen to be tight. P / P Table : Execte time to conirm a alse trac as a unction o etection an alse alarm robabilities with SPRT threshols Tl= 3, Tu=8 (Marov chain analysis). Units are scans er resolution cell. P / P Table 3: Execte time in scans to conirm a true trac as a unction o etection an alse alarm robabilities with SPRT threshols Tl= 3, Tu=8 or the three methos: conventional, Monte arlo simulation an Marov chain analysis (to to bottom in each box). 7. onclusions & Further Wor A metho or analysing the seuential robability ratio test or automatic trac maintenance has been resente. The metho harnesses inite state Marov chain theory to estimate: (i) the robabilities o trac conirmation events; (ii) the execte times or such events. The so-calle "rile metho" is an extension o the Marov chain aroach reorte in [] an allows statistics or alse trac conirmation to be estimate at low alse alarm rates. Numerical evaluation conirme the accuracy o the robability estimates comare with Monte arlo simulation. The actual Pc values were generally an orer o magnitue smaller than that reicte by the conventional SPRT aroach. The execte time estimates were seen to over-boun tightly those o the conventional an Monte arlo aroaches. Further wor is reuire to etermine analytical orms or the Marov chain results base on the aroach given here. An imortant extension concerns the inclusion o sensor measurement errors an target ynamics into the Marov chain analysis. Reerences [] S. Blacman, Multile Target ing with Raar Alications, Artech House, MA, 86. [] S. Blacman, R. emster, an T. Broia, Multiles Hyothesis onirmation or Inrare Surveillance Systems,, IEEE Trans. AES, vol., no. 3,. 8 84, July. 3. [3] S. B. olegrove, A. W. avis, an J. K. Aylie, Initiation an Nearest Neighbours Incororate into Probabilistic ata Association, J. Elec. an Electronic Eng., Australia, vol. 6, no. 3,. 8, Set. 86. [4] W. Fleses an G. Van Keu, Aative ontrol an ing with the ELRA Phase Array Raar Exerimental System, Proc. IEEE Int. Raar on., VA,. 8 3, 8. [] L. Isaacson & R. W. Masen, Marov hains: Theory an Alications, J. Wiley Inc, 8. [6] J. G. Kemeny & J. L. Snell, Finite Marov hains, Sringer-Verlag, New Yor, 76. [7] X. R. Li, N. Li, an V. P. Jilov, SPRT-Base onirmation an Rejection, Proc. Int. on. Ino. Fusion,. 8. [8]. Musici, R. J. Evans, an S. Stanovic, Integrate Probabilistic ata Association, IEEE Trans. Auto. ontrol, Vol. 3, no. 6,. 37 4, June 4. [] P. M. Newbol an Y.. Ho, etection o hanges in the haracteristics o a Gauss-Marov Process, IEEE Trans. AES, vol. 4, no., , Set. 68. [] G. Pulor, eveloments in Non-Linear Eualisation, Ph.. thesis, eartment o Systems Engineering, Australian National University,. [] H. L. Van Trees, etection, Estimation an Moulation Theory Part I, J. Wiley, 68. [] A. Wal, Seuential Analysis, J. Wiley, N.Y., 47. [3] A. Wal an J. Wolowitz, Otimum haracter o the Seuential Probability Ratio Test, Ann. Math. Statist., vol.,. 36, 48. [4] N.. Wallace, esign o Truncate Seuential Tests or Raily Faing Raar Targets, IEEE Trans. AES, vol. 4, no. 3, , May 68.