Multi-variate Quickest Detection of Significant Change Process

Multi-v ariate Quic k est Detection of Significan t Change Pro cess Krzyszto f Sza jowski Institute of Mathematics and Computer Science, W roc la w Universit y of T ec hn ology , Wybrze ˙ ze Wyspia´ nskiego 27, 50-370 W roc law , Po land Krzysztof. szajowski@pwr.wr oc.pl http://www .im.pwr.wroc.pl/ ~ szajow Abstract. The pap er deals with a mathematical model of a s urveillance system based on a net of sensors. The signals acquired by ea ch node of the net are Mark ovia n process, ha ve tw o different transition probabilities, whic h dep ends o n the presence or absence of a intruder nearby . The detection of the transition p robabilit y change at one no de should b e confirmed by a d etection of similar c hange at some oth er sensors. B ased on a si mple game the model of a fusi on center is then constructed. The aggregate fun ction defi ned on t he n et is the b ac kground of the defi nition of a n on-co operative stopping g ame which is a mod el of the multiv ariate disorder detectio n. Key wor ds: voting stopp ing rule, ma jority voting rule, monotone vot- ing strategy , change-point problems, qu ic kest d etection, sequentia l de- tection, simple game 1 In tro duction The aim of this consideration is to construct the mathematical model of a multi- v ariate surveillance system. It is assumed that there is net N of p no des. At eac h no de the s ta te is the signal a t moment n ∈ N which is at least one co or dinate of the vector − → x n ∈ E ⊂ ℜ m . The distribution of the signa l at each no de has t wo forms and dep ends on a pur e or a dirty environmen t of the node. The state of the system change dynamically . W e consider the discrete time observed signal as m ≥ p dimensional pro cess defined on the fixe d pro ba bilit y space ( Ω , F , P ). The observed at ea ch no de pro cess is Ma rko vian with tw o differen t tra nsition probabilities (see [17] for details). In the signal the v isual co nsequence of the transition distribution changes at moment θ i , i ∈ N is a change o f its character. T o avoid false ala rm the confirmation from other no des is nee ded. The family of subsets (co alitions) o f no des are defined in suc h a way that the decision o f all mem b er of s ome co a lition is equiv alent with the claim of the net that the disorder app eared. It is not sur e that the disorder has ha d place. The aim is to define the rules of no des and a co nstruction of the net decision ba sed on individual no des claims. V ario us approaches c an b e found in the r ecent research for description or mo deling o f such systems (see e.g. [2 3], [16]). The problem is q uite similar 2 K.Sza jowski to a pattern reco gnition with multiple algor ithm when the fusions of individual algorithms results ar e unified to a final decision. The prop os ed solutio n will b e based on a simple g ame and the stopping game defined by a simple game on the observed signa ls. It gives a centralized, Bay esia n version of the multiv aria te detection with a common fusion cent er that it has p erfect information ab out observ ations and a priori knowledge o f the statistics ab out the p oss ible distri- bution changes at ea ch no de. E ach sensor (play er) will declare to stop when it detects disorder at his region. Based o n the simple game the sensors’ decisions are ag g regated to formulate the decisio n of the fusion c ent er . T he sensor s’ stra te- gies are co nstructed a s an equilibrium s trategy in a no n-co op erative game with a lo gical function defined by a simple game (which aggre gates their decisio n). The general descr iption of such mult iv ariate s topping games has b een for - m ulated by Kurano , Y asuda and Nak agami in the cas e when the a ggreg a tion function is defined by the voting ma jor it y rule [9 ] or the monotone voting stra t- egy [24] and the obser ved sequences o f the random v ariable s are independent, ident ica lly distributed. It was F erg uson [5] who substituted the voting agg rega- tion rules by a simple game. The Markov s equences hav e bee n inv estigated by the author and Y asuda [21]. The mode l o f detectio n the disorder at each sensor are presented in the next sectio n. It allows to define the individual pay offs of the players (senso rs). Section 3 intro duces the agg regation metho d based o n a simple game of the sensors. Section 4 co nt ains deriv ation of the no n-co op erative game a nd existence theorem for e q uilibrium strategy . The final decision bas e d on the state o f the sensors is given by the fusion center and it is describ ed in Sectio n 6. The natura l direction of fur ther resea rch is formulated also in the sa me section. A c o nclusion and resume of an a lgorithm for rationa l cons truction o f the surveillance system is included in Section 7. 2 Detection of disorder at sensors F ollowing the consider a tion of Section 1, let us s uppo se that the pro cess { − → X n , n ∈ N } , N = { 0 , 1 , 2 , . . . } , is observed sequentially in such a wa y that each sensor , e.g. r (gets its co or dina tes in the vector − → X n at moment n ). B y assumption, it is a sto chastic s equence that has the Ma rko vian structure given random moment θ r , in such a wa y that the pro cess after θ r starts from state − → X n θ r − 1 . The ob jective is to detect these moments based o n the obs erv ation of − → X n · at each sensor separately . There are s ome results on the discr e te time cas e of such disorder detection which generalize the basic problem s tated by Shiryaev in [18] (see e.g. Bro dsky and Darkhovsky [2], Bo jdec ki [1], Y oshida [25], Sza jowski [20]) in v arious directions . Application of the mo del for the detection o f traffic a nomalies in netw orks has bee n discussed by T a rtako vsky et al. [22]. The version of the problem when the moment of dis order is detected with g iven precisio n will b e used here (see [17]). Multiv ariate Qu ic kest Detection 3 2.1 F orm ulation o f the probl e m The observ able r andom v ariables { − → X n } n ∈ N are consistent with the filtratio n F n (or F n = σ ( − → X 0 , − → X 1 , . . . , − → X n )). The random v ector s − → X n take v alues in ( E , B ), where E ⊂ ℜ m . On the same probability space ther e are defined unobserv able (hence not measura ble with resp ect to F n ) rando m v aria bles { θ r } m r =1 which hav e the geometric dis tributions: P ( θ r = j ) = p j − 1 r q r , q r = 1 − p r ∈ (0 , 1), j = 1 , 2 , . . . . (1) The senso r r follows the pro cess which is based on switching be tw een tw o, time homog e neous and indep e nden t, Mar ko v pro ces ses { X i r n } n ∈ N , i = 0 , 1, r ∈ N with the state space ( E , B ), b oth indep endent of { θ r } m r =1 . Mor e ov er, it is assumed that the pr o cesses { X i r n } n ∈ N hav e transition densities with res p ect to the σ -finite measure µ , i.e ., for any B ∈ B we hav e P i x ( X i r 1 ∈ B ) = P ( X i r 1 ∈ B | X i r 0 = x ) = Z B f r i x ( y ) µ ( dy ) . (2) The random pro cesses { X r n } , { X 0 r n } , { X 1 r n } and the random v aria bles θ r are connected v ia the rule: conditionally on θ r = k X r n =  X 0 r n , if k > n , X 1 r n +1 − k , if k ≤ n , where { X 1 r n } is started fro m X 0 r k − 1 (but is otherwise indep endent o f X 0 r · ). F or any fix e d d ∈ { 0 , 1 , 2 , . . . } we are lo oking for the stopping time τ ∗ r ∈ T such that P x ( | θ r − τ ∗ r | ≤ d ) = sup τ ∈ S X P x ( | θ r − τ | ≤ d ) (3) where S X denotes the set of all stopping times with resp ect to the filtration {F n } n ∈ N . The parameter d determines the pr ecision level of detection and it can be differen t for to o ear ly and to o late detection. 2.2 Construction of the optimal detection strategy In [17] the co nstruction of τ ∗ by transfor mation of the problem to the o ptimal stopping problem for the Marko v pro ces s − → ξ has been made, suc h that − → ξ r n = ( − → X r n − 1 − d,n , Π n ), where − → X r n − 1 − d, n = ( − → X r n − 1 − d , . . . , − → X r n ) and Π r n is the po sterior pro cess: Π r 0 = 0 , Π r n = P x ( θ r ≤ n | F n ) , n = 1 , 2 , . . . which is des igned as infor ma tion ab out the distr ibution of the disor der instant θ r . In this equiv alen t the pro blem of the pay off function for sensor r is h r ( − → x r d +2 , α ). 4 K.Sza jowski 3 The aggregated decision via the c o op erativ e game There a re v arious methods combining the decisions of several c lassifiers or sen- sors. E ach e nsemble member contributes to some deg ree to the decision a t an y po int of the sequentially delivered states. The fusion a lgorithm takes into account all the dec is ion o utputs from ea ch ensemble member and co mes up with a n en- semble decis ion. When classifier outputs ar e binar y , the fusion a lgorithms include the ma jority voting [10], [11], na ¨ ıve B ayes co m binatio n [3], b ehavior knowledge space [7], probability approximation [8] and singular v alue decomp osition [12]. The ma jority vote is the simplest. The e xtension of this metho d is a simple game. 3.1 A sim ple game Let us assume that ther e ar e many no des absor bing information and make deci- sion if the diso r der has app ear ed or not. The final decision is made in the fusion center which aggr egates information from all s ensors. The nature of the system and their role is to detect intrusion in the s ystem as so on as p ossible but without false a larm. The voting decision is made according to the rules of a simple game . Let us recall that a coalition is a subset o f the players. Let C = { C : C ⊂ N } denote the class of all coalitio ns. Definition 1. ( s e e [15], [5]) A simple ga me is c o alition game having t he char- acteristic function, φ ( · ) : C → { 0 , 1 } . Let us denote W = { C ⊂ N : φ ( C ) = 1 } and L = { C ⊂ N : φ ( C ) = 0 } . The coalitions in W are called the w inning co a litions, a nd those from L are called the losing coalitions . Assumptions 2 By assumption the char acteristic function satisfies t he pr op er- ties: 1. N ∈ W ; 2. ∅ ∈ L ; 3. (the m onotonicity): T ⊂ S ∈ L implies T ∈ L . 3.2 The aggregated decis ion rule When the simple game is defined a nd the play er s can vote presence or abse nce , x i = 1 or x i = 0, i ∈ N , of the intruder then the aggr egated decision is given by the logical function π ( x 1 , x 2 , . . . , x p ) = X C ∈ W Y i ∈ C x i Y i / ∈ C (1 − x i ) . (4) F or the logical function π we hav e (cf [24]) π ( x 1 , . . . , x p ) = x i · π ( x 1 , . . . , i ˘ 1 , . . . , x p ) + x i · π ( x 1 , . . . , i ˘ 0 , . . . , x p ) . Multiv ariate Qu ic kest Detection 5 4 A non-co op erative stopping game F ollowing the re s ults of the a uthor and Y asuda [2 1] the mult ilater al stopping of a Marko v chain pr oblem can be desc rib ed in the terms of the notation used in the non-co oper ative game theory (see [14], [4], [1 3], [15]). Le t ( − → X n , F n , P x ), n = 0 , 1 , 2 , . . . , N , b e a ho mo geneous Markov chain with state s pace ( E , B ). The horizon can be finite or infinite. The players are a ble to o bserve the Mar ko v chain sequentially . Each play er ha s their utilit y function f i : E → ℜ , i = 1 , 2 , . . . , p , such that E x | f i ( − → X 1 ) | < ∞ . If pro cess is not sto pp ed at moment n , then e a ch play er, based on F n , can declare independently their willingness to stop the observ ation of the pr o cess. Definition 3. (s e e [24]) An individual st opping str ate gy of the player i (ISS) is the se qu en c e of r andom variables { σ i n } N n =1 , wher e σ i n : Ω → { 0 , 1 } , such that σ i n is F n -me asur able. The interpretation of the stra teg y is following. If σ i n = 1 then player i declares that they w ould like to s top the pro cess and accept the realization o f X n . Denote σ i = ( σ i 1 , σ i 2 , . . . , σ i N ) and let S i be the set of ISSs of player i , i = 1 , 2 , . . . , p . Define S = S 1 × S 2 × . . . × S p . The elemen t σ = ( σ 1 , σ 2 , . . . , σ p ) T ∈ S will b e called the stopping strategy (SS). The stopping stra tegy σ ∈ S is a ra ndo m matrix. The rows of the matrix a re the ISSs. The columns are the decisio ns of the play ers at successive mo ment s. The factual sto pping of the observ ation pro cess, a nd the play ers realizatio n of the pay offs is defined b y the stopping stra tegy exploiting p -v a riate logical function. Let π : { 0 , 1 } p → { 0 , 1 } . In this stopping game mo del the stopping strateg y is the list of decla rations of the individual play er s. The aggre g ate function π conv erts the declaratio ns to an effective stopping time. Definition 4. A stopping time t π ( σ ) gener ate d by the SS σ ∈ S and the aggr e- gate fun ction π is define d by t π ( σ ) = inf { 1 ≤ n ≤ N : π ( σ 1 n , σ 2 n , . . . , σ p n ) = 1 } (inf ( ∅ ) = ∞ ) . Sinc e π is fix e d during the analysis we skip index π and write t ( σ ) = t π ( σ ) . W e hav e { ω ∈ Ω : t π ( σ ) = n } = T n − 1 k =1 { ω ∈ Ω : π ( σ 1 k , σ 2 k , . . . , σ p k ) = 0 } ∩ { ω ∈ Ω : π ( σ 1 n , σ 2 n , . . . , σ p n ) = 1 } ∈ F n , then the random v ar iable t π ( σ ) is stopping time with resp ect to { F n } N n =1 . F or a n y stopping time t π ( σ ) and i ∈ { 1 , 2 , . . . , p } , let f i ( X t π ( σ ) ) =  f i ( X n ) if t π ( σ ) = n , lim sup n →∞ f i ( X n ) if t π ( σ ) = ∞ (cf [1 9], [2 1]). If play ers us e SS σ ∈ S and the individual preferences a re co n- verted to the effective stopping time by the ag gregate rule π , then play er i gets f i ( X t π ( σ ) ). 6 K.Sza jowski Let ∗ σ = ( ∗ σ 1 , ∗ σ 2 , . . . , ∗ σ p ) T be fixed SS. Denote ∗ σ ( i ) = ( ∗ σ 1 , . . . , ∗ σ i − 1 , σ i , ∗ σ i +1 , . . . , ∗ σ p ) T . Definition 5. (cf. [21]) L et the aggr e gate rule π b e fixe d. The str ate gy ∗ σ = ( ∗ σ 1 , ∗ σ 2 , . . . , ∗ σ p ) T ∈ S is an e quilibrium st r ate gy with r esp e ct t o π if for e ach i ∈ { 1 , 2 , . . . , p } and any σ i ∈ S i we have E x f i ( − → X t π ( ∗ σ ) ) ≥ E x f i ( − → X t π ( ∗ σ ( i )) ) . (5) The set of SS S , the vector o f the utilit y functions f = ( f 1 , f 2 , . . . , f p ) a nd the monotone rule π define the no n-co op erative game G = ( S , f , π ). The construction of the equilibrium stra teg y ∗ σ ∈ S in G is provided in [2 1]. F o r completeness this construction will b e recalled her e. Let us define an individua l stopping set on the state space. This set des crib es the I SS of the play er. With each ISS o f player i the sequence o f s to pping even ts D i n = { ω : σ i n = 1 } combines. F or ea ch aggr e gate rule π there exists the corr esp onding set v a lue function Π : F → F such that π ( σ 1 n , σ 2 n , . . . , σ p n ) = π { I D 1 n , I D 2 n , . . . , I D p n } = I Π ( D 1 n ,D 2 n ,...,D p n ) . F or solutio n of the considered g ame the imp ortant class of ISS and the s to pping event s can be defined by subsets C i ∈ B of the sta te spa c e E . A given set C i ∈ B will b e called the stopping set for player i a t moment n if D i n = { ω : X n ∈ C i } is the s topping even t. F or the logical function π we have π ( x 1 , . . . , x p ) = x i · π ( x 1 , . . . , i ˘ 1 , . . . , x p ) + x i · π ( x 1 , . . . , i ˘ 0 , . . . , x p ) . It implies that for D i ∈ F Π ( D 1 , . . . , D p ) = { D i ∩ Π ( D 1 , . . . , i ˘ Ω , . . . , D p ) } ∪{ D i ∩ Π ( D 1 , . . . , i ˘ ∅ , . . . , D p ) } . (6) Let f i , g i be the rea l v alued, integrable (i.e. E x | f i ( X 1 ) | < ∞ ) function defined on E . F or fixe d D j n , j = 1 , 2 , . . . , p , j 6 = i , and C i ∈ B define ψ ( C i ) = E x h f i ( X 1 ) I i D 1 ( D i 1 ) + g i ( X 1 ) I i D 1 ( D i 1 ) i where i D 1 ( A ) = Π ( D 1 1 , . . . , D i − 1 1 , A, D i +1 1 , . . . , D p 1 ) and D i 1 = { ω : X n ∈ C i } . Let a + = max { 0 , a } a nd a − = min { 0 , − a } . Lemma 1. L et f i , g i , b e inte gr able and let C j ∈ B , j = 1 , 2 , . . . , p , j 6 = i , b e fixe d. Then the set ∗ C i = { x ∈ E : f i ( x ) − g i ( x ) ≥ 0 } ∈ B is su ch that ψ ( ∗ C i ) = sup C i ∈B ψ ( C i ) and ψ ( ∗ C i ) = E x ( f i ( X 1 ) − g i ( X 1 )) + I i D 1 ( Ω ) (7) − E x ( f i ( X 1 ) − g i ( X 1 )) − I i D 1 ( Ω ) + E x g i ( X 1 ) . Based on Lemma 1 we derive the recur sive for mulae defining the equilibr ium po int and the e q uilibrium pay off for the finite horizon ga me. Multiv ariate Qu ic kest Detection 7 4.1 The finite h o rizon game Let hor izon N b e finite. If the equilibrium strateg y ∗ σ exists, then we denote v i,N ( x ) = E x f i ( X t ( ∗ σ ) ) the equilibrium pay off of i -th play er when X 0 = x . F or the backw ard induction we intro duce a useful notation. Let S i n = {{ σ i k } , k = n, . . . , N } b e the set of ISS for moments n ≤ k ≤ N and S n = S 1 n × S 2 n × . . . × S p n . The SS for moments not earlier than n is n σ = ( n σ 1 , n σ 2 , . . . , n σ p ) ∈ S n , where n σ i = ( σ i n , σ i n +1 , . . . , σ i N ). Denote t n = t n ( σ ) = t ( n σ ) = inf { n ≤ k ≤ N : π ( σ 1 k , σ 2 k , . . . , σ p k ) = 1 } to b e the s to pping time not earlier than n . Definition 6. The stopping st r ate gy n ∗ σ = ( n ∗ σ 1 , n ∗ σ 2 , . . . , n ∗ σ p ) is an e quilib- rium in S n if E x f i ( X t n ( ∗ σ ) ) ≥ E x f i ( X t n ( ∗ σ ( i )) ) P x − a.e. for every i ∈ { 1 , 2 , . . . , p } , wher e n ∗ σ ( i ) = ( n ∗ σ 1 , . . . , n ∗ σ i − 1 , n σ i , n ∗ σ i +1 , . . . , n ∗ σ p ) . Denote v i,N − n +1 ( X n − 1 ) = E x [ f i ( X t n ( ∗ σ ) ) | F n − 1 ] = E X n − 1 f i ( X t n ( ∗ σ ) ) . A t moment n = N the players hav e to declar e to stop a nd v i, 0 ( x ) = f i ( x ). Let us assume that the pro cess is not stopp ed up to moment n, the players are using the equilibrium s trategies ∗ σ i k , i = 1 , 2 , . . . , p, at moments k = n + 1 , . . . , N . Cho os e play er i and a ssume that o ther play ers are using the equilibrium s tr ategies ∗ σ j n , j 6 = i , and player i is using str ategy σ i n defined b y stopping set C i . Then the exp ected pay off ϕ N − n ( X n − 1 , C i ) of player i in the ga me sta r ting at moment n , when the state of the Markov chain at moment n − 1 is X n − 1 , is equa l to ϕ N − n ( X n − 1 , C i ) = E X n − 1 h f i ( X n ) I i ∗ D n ( D i n ) + v i,N − n ( X n ) I i ∗ D n ( D i n ) i , where i ∗ D n ( A ) = Π ( ∗ D 1 n , . . . , ∗ D i − 1 n , A, ∗ D i +1 n , . . . , ∗ D p n ). By Lemma 1 the conditional expec ted gain ϕ N − n ( X N − n , C i ) attains the maximum o n the stopping set ∗ C i n = { x ∈ E : f i ( x ) − v i,N − n ( x ) ≥ 0 } a nd v i,N − n +1 ( X n − 1 ) = E x [( f i ( X n ) − v i,N − n ( X n )) + I i ∗ D n ( Ω ) | F n − 1 ] − E x [( f i ( X n ) − v i,N − n ( X n )) − I i ∗ D n ( ∅ ) | F n − 1 ] + E x [ v i,N − n ( X n ) | F n − 1 ] (1) P x − a.e.. It allows to formulate the following constructio n of the equilibrium strategy and the equilibrium v alue for the game G . Theorem 1. In the game G with fin ite horizon N we have the fol lowing solution. 8 K.Sza jowski (i) The e quilibrium value v i ( x ) , i = 1 , 2 , . . . , p , of the game G c an b e c alculate d r e cursively as fol lows: 1. v i, 0 ( x ) = f i ( x ) ; 2. F or n = 1 , 2 , . . . , N we have P x − a.e. v i,n ( x ) = E x [( f i ( X N − n +1 ) − v i,n − 1 ( X N − n +1 )) + I i ∗ D N − n +1 ( Ω ) | F N − n ] − E x [( f i ( X N − n +1 ) − v i,n − 1 ( X N − n +1 )) − I i ∗ D N − n +1 ( ∅ ) | F N − n ] + E x [ v i,n − 1 ( X N − n +1 ) | F N − n ] , for i = 1 , 2 , . . . , p . (ii) The e quilibrium str ate gy ∗ σ ∈ S is define d by the SS of t he players ∗ σ i n , wher e ∗ σ i n = 1 if X n ∈ ∗ C i n , and ∗ C i n = { x ∈ E : f i ( x ) − v i,N − n ( x ) ≥ 0 } , n = 0 , 1 , . . . , N . We have v i ( x ) = v i,N ( x ) , and E x f i ( X t ( ∗ σ ) ) = v i,N ( x ) , i = 1 , 2 , . . . , p . 5 Infinite horizon game In this class of games the equilibrium str ategy is presented in Definition 5 but in class of SS S ∗ f = { σ ∈ S ∗ : E x f − i ( X t ( σ ) ) < ∞ for every x ∈ E , i = 1 , 2 , . . . , p } . Let ∗ σ ∈ S ∗ f be an equilibrium s trategy . Denote v i ( x ) = E x f i ( X t ( ∗ σ ) ) . Let us a ssume that ( n +1) ∗ σ ∈ S ∗ f ,n +1 is co nstructed and it is an equilibr ium strategy . If play ers j = 1 , 2 , . . . , p , j 6 = i , a pply at moment n the eq uilibrium strategies ∗ σ j n , play er i the stra tegy σ i n defined by sto pping set C i and ( n +1) ∗ σ at moments n + 1 , n + 2 , . . . , then the exp ected pay off o f the play er i , when history of the pro cess up to moment n − 1 is kno wn, is given by ϕ n ( X n − 1 , C i ) = E X n − 1 h f i ( X n ) I i ∗ D n ( D i n ) + v i ( X n ) I i ∗ D n ( D i n ) i , where i ∗ D n ( A ) = Π ( ∗ D 1 n , . . . , ∗ D i − 1 n , A, ∗ D i +1 n , . . . , ∗ D p n ), ∗ D j n = { ω ∈ Ω : ∗ σ j n = 1 } , j = 1 , 2 , . . . , p , j 6 = i , and D i n = { ω ∈ Ω : σ i n = 1 } = 1 } = { ω ∈ Ω : X n ∈ C i } . By Lemma 1 the co nditional e x pec ted gain ϕ n ( X n − 1 , C i ) attains the maximum on the s topping se t ∗ C i n = { x ∈ E : f i ( x ) ≥ v i ( x ) } a nd ϕ n ( X n − 1 , ∗ C i ) = E x [( f i ( X n ) − v i ( X n )) + I i ∗ D n ( Ω ) | F n − 1 ] − E x [( f i ( X n ) − v i ( X n )) − I i ∗ D n ( ∅ ) | F n − 1 ] + E x [ v i ( X n ) | F n − 1 ] . Let us assume that there ex is ts solution ( w 1 ( x ) , w 2 ( x ) , . . . , w p ( x )) of the equations w i ( x ) = E x ( f i ( X 1 ) − w i ( X 1 )) + I i ∗ D 1 ( Ω ) (1) − E x ( f i ( X 1 ) − w i ( X 1 )) − I i ∗ D 1 ( ∅ ) + E x w i ( X 1 ) , Multiv ariate Qu ic kest Detection 9 i = 1 , 2 , . . . , p . Co ns ider the stopping game with the following pay off function for i = 1 , 2 , . . . , p . φ i,N ( x ) =  f i ( x ) if n < N , v i ( x ) if n ≥ N . Lemma 2. L et ∗ σ ∈ S ∗ f b e an e quilibrium s t r ate gy in the infinite horizon game G . F or every N we have E x φ i,N ( X t ∗ ) = v i ( x ) . Let us a ssume tha t for i = 1 , 2 , . . . , p and every x ∈ E we hav e E x [sup n ∈ N f + i ( X n )] < ∞ . (2) Theorem 2. L et ( X n , F n , P x ) ∞ n =0 b e a homo gene ous Markov chain and the p ay- off fun ctions of the players fulfil l (2). If t ∗ = t ( ∗ σ ) , ∗ σ ∈ S ∗ f then E x f i ( X t ∗ ) = v i ( x ) . Theorem 3. L et the stopping str ate gy ∗ σ ∈ S ∗ f b e define d by t he stopping set s ∗ C i n = { x ∈ E : f i ( x ) ≥ v i ( x ) } , i = 1 , 2 , . . . , p , then ∗ σ is the e quilibrium str ate gy in the infinite stopping game G . 6 Determining the strategies of sensors Based on the mo del constructed in Sections 2 – 4 for the net of sens ors with the fusion cen ter determined by a simple game, o ne can determine the r ational decisions of each no des. The r ationality o f such a constr uction refers to the individual aspira tion for the highest sensitivity to detect the dis o rder without false alar m. The Na sh equilibrium fulfills requirement that nob o dy devia tes from the equilibrium strategy b ecause its probability o f detection will b e smaller. The role of the simple game is to define wining co alitions in such a way that the detection of intrusion to the guarded ar ea is maximal and the pro bability of false alarm is minimal. The metho d of constructing the optimum winning coalitio ns family is not the s ub ject o f the res e arch in this ar ticle. How ever, there ar e some natural metho ds of solving this pro blem. The r e search her e is fo cused on constr uc ting the solution of the no n-co op erative stopping g ame as to de ter mine the detection strategy o f the sensors . T o this end, the game ana lyzed in Section 4 with the pa yoff function o f the play ers defined by the individua l disorder problem formulated in Section 2 should b e derived. The prop osed mo del disrega rds cor relation o f the s ignals. It is also a ssumed that the fusio n center ha s per fect informa tion abo ut signals and the informa tion is av aila ble at e a ch no de. The further res earch sho uld help to q ualify these rea l needs o f such mo dels and to extend the model to mor e genera l cases. In some t yp e o f distribution of sensor s , e.g. when the distribution o f the p ollution in the given direction is o bserved, the multiple disor der mo del should work b etter than the game approach. In this case the a priori distr ibution of diso rder moment has the fo rm of sequentially dep endent ra ndo m moments and the fusio n decision 10 K.Sza jowski can b e formulated as the threshold one: stop when k ∗ disorder is detected. The metho d o f a co op erative ga me was used in [6 ] to find the bes t co alition of sensors in the problem of the ta rget lo caliz a tion. 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