Modeling and Mitigating Errors in Belief Propagation for Distributed Detection

This is a sel -a chi ed e sion o an o iginal a icle. This e sion may di e om he o iginal in pagina ion and ypog aphic de ails. Au ho (s): Ti le: Yea : Ve sion: Copy igh : Righ s: Righ s u l: Please ci e he o iginal e sion: CC BY 4.0 h ps://c ea i ecommons.o g/licenses/by/4.0/ Modeling and Mi iga ing E o s in Belie P opaga ion o Dis ibu ed De ec ion © Au ho s, 2021 Published e sion Abdi, Younes; Ris aniemi, Tapani Abdi, Y., & Ris aniemi, T. (2021). Modeling and Mi iga ing E o s in Belie P opaga ion o Dis ibu ed De ec ion. IEEE T ansac ions on Communica ions, 69(5), 3286-3297. h ps://doi.o g/10.1109/TCOMM.2021.3056679 2021 3286 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 69, NO. 5, MAY 2021 Modeling and Mi iga ing E o s in Belie P opaga ion o Dis ibu ed De ec ion Younes Abdi ,Membe , IEEE, and Tapani Ris aniemi ,Senio Membe , IEEE Abs ac — We s udy he beha io o he belie -p opaga ion (BP) algo i hm a ec ed by e oneous da a exchange in a wi eless senso ne wo k (WSN). The WSN conduc s a dis ibu ed mul i- dimensional hypo hesis es o e bina y andom a iables. The join s a is ical beha io o he senso obse a ions is modeled by a Ma ko andom ield whose pa ame e s a e used o build he BP messages exchanged be ween he sensing nodes. Th ough linea iza ion o he BP message-upda e ule, we analyze he beha io o he esul ing e oneous decision a iables and de i e closed- o m ela ionships ha desc ibe he impac o s ochas ic e o s on he pe o mance o he BP algo i hm. We hen de elop a decen alized dis ibu ed op imiza ion amewo k o enhance he sys em pe o mance by mi iga ing he impac o e o s ia a dis ibu ed linea da a- usion scheme. Finally, we compa e he esul s o he p oposed analysis wi h he exis ing wo ks and isualize, ia compu e simula ions, he pe o mance gain ob ained by he p oposed op imiza ion. Index Te ms— Dis ibu ed sys ems, coope a i e communica- ions, likelihood- a io es , communica ion e o s, compu a ion e o s, blind signal p ocessing, message-passing algo i hms, lin- ea da a- usion, ac o g aphs. I. INTRODUCTION DESIGN o s a is ical in e ence sys ems o en in ol es analysis and modeling o he collec i e beha io o a g oup o andom a iables and hei in e ac ions. Consequen ly, ac o g aphs, which a e commonly used o cap u e he in e dependencies be ween co ela ed andom a i- ables, p o ide a powe ul amewo k o de eloping e ec i e low-complexi y in e ence algo i hms in a ious ields such as wi eless communica ions, image p ocessing, combina o ial op imiza ion, and machine lea ning, see e.g., [1]–[3]. Belie p opaga ion (BP) [4] is a well-known s a is ical in e ence algo- i hm ha wo ks based on pa allel message-passing be ween he nodes in a ac o g aph. BP is some imes e e ed o as he sum-p oduc algo i hm. When wo king wi h he BP algo i hm, we should bea in mind ha digi al compu a ion and digi al communica ion a e bo h e o -p one p ocesses in gene al. The messages exchanged be ween he nodes in a wi eless ne wo k can always be ad e sely a ec ed by e o s caused by un eliable ha dwa e Manusc ip ecei ed Ma ch 12, 2020; e ised Sep embe 6, 2020 and Janua y 25, 2021; accep ed Janua y 25, 2021. Da e o publica ion Feb ua y 3, 2021; da e o cu en e sion May 18, 2021. The associa e edi o coo dina ing he e iew o his a icle and app o ing i o publica ion was A. Cohen. (Co esponding au ho : Younes Abdi.) Younes Abdi is wi h he Facul y o In o ma ion Technology, Uni e si y o Jy äskylä, 40014 Jy äskylä, Finland (e-mail: younes.[email p o ec ed]). Tapani Ris aniemi, deceased, was wi h he Facul y o In o ma ion Technology, Uni e si y o Jy äskylä, 40014 Jy äskylä, Finland (e-mail: apani. is aniemi@jyu. i). Colo e sions o one o mo e igu es in his a icle a e a ailable a h ps://doi.o g/10.1109/TCOMM.2021.3056679. Digi al Objec Iden i ie 10.1109/TCOMM.2021.3056679 componen s, quan iza ion p ocesses, app oxima e ep esen a- ions, wi eless channel impai men s, e c. E en hough he BP algo i hm has been ex ensi ely s udied in he li e a- u e, we ha e a he limi ed knowledge abou how s ochas ic e o s in messages a ec he belie s ob ained and how hese e oneous belie s in luence he esul o s a is ical in e ence schemes implemen ed by he BP algo i hm. This e i o y is di icul o explo e mainly due o he nonlinea i ies in he BP message-passing i e a ion. In [5], we ha e de eloped a sys ema ic amewo k o analyzing he beha io o BP and op imizing i s pe o mance in a dis ibu ed de ec ion scena io. In pa icula , we ha e shown ha he decision a iables buil by he BP algo i hm a e, app oxima ely, linea combina ions o he local likeli- hoods in he ne wo k. Consequen ly, we ha e de i ed in [5] closed- o m ela ionships o he sys em pe o mance me ics and o mula ed a dis ibu ed op imiza ion scheme o achie e a nea -op imal de ec ion pe o mance. Mo eo e , we ha e discussed he ela ionship be ween he BP and he max-p oduc algo i hms in [6] whe e we ex end he p oposed ame- wo k in [5] o op imize he pe o mance o he max-p oduc algo i hm in a dis ibu ed de ec ion scena io. In his pape , we u he ex end ha amewo k o gain insigh in o he impac o compu a ion and communica ion e o s, in a BP i e a ion, on he esul ing decision a iables and o e ec i ely mi iga e ha impac . Examples o BP being used in dis ibu ed de ec ion can be ound in [7]–[10]. Accumula ion o message e o s and hei ad e se e ec on he pe o mance o BP is analyzed in [11] whe e he message e o s a e modeled as unco ela ed andom a iables o ind p obabilis ic gua an ees on he magni ude o e o s a ec ing he belie s. The wo k in [11] is inspi ed by obse ing he beha io and s abili y o digi al il e s, in he p esence o quan iza ion e ec s, which can be analyzed eliably by assuming unco ela ed beha io in he co esponding andom e o s [12]. Such a modeling app oach is in line wi h he on Neumann model o noisy ci cui s [13], which conside s ansien aul s in logic ga es and wi es as message and node compu a ion noise ha is bo h spa ially and empo ally independen [14]. The beha io o BP implemen ed on noisy ha dwa e is in es iga ed in [15] whe e i is obse ed ha unde he so-called con ac ing mapping condi ion [16], he dis ance be ween successi e messages in a noise- ee BP dec eases by he numbe o i e a ions. Consequen ly, in he p esence o ha dwa e (o compu a ion) noise, he aul y messages ha iola e his end can be de ec ed and disca ded (censo ed) om he BP i e a ions. Such an app oach is e med censo ing This wo k is licensed unde a C ea i e Commons A ibu ion 4.0 License. Fo mo e in o ma ion, see h ps://c ea i ecommons.o g/licenses/by/4.0/ ABDI AND RISTANIEMI: MODELING AND MITIGATING ERRORS IN BELIEF PROPAGATION FOR DISTRIBUTED DETECTION 3287 BP in [15] and is shown o pe o m well when he ha dwa e noise dis ibu ion has a la ge mass a ze o and non-negligible masses a some poin s su icien ly away om ze o. As an al e na i e app oach, he so-called a e aging BP (ABP) is also p oposed in [15]. In his me hod, as he name implies, an a e age o he messages up o he las i e a ion is sa ed and hen used, ins ead o he ac ual messages, o build he belie s. This me hod is p oposed and i s con e gence is es ablished o gene al ze o-mean compu a ion noise dis ibu ions. Again, he on Neumann model is used in [15] o analyze he beha io o message e o s. In his pape , we use he ac ha he BP algo i hm and he linea da a- usion scheme a e elegan ly ela ed o each o he in he con ex o dis ibu ed de ec ion. Fo una ely, he e al eady exis s a ich collec ion o scien i ic wo ks in he li e a u e ha in es iga e low-complexi y de ec o s uc u es based on linea usion in a ious design scena ios [17]–[21]. In many o hese wo ks, he da a-exchange p ocess wi hin he senso ne wo k is assumed ad e sely a ec ed by non-ideali ies in he unde lying communica ion links. Hence, dealing wi h e oneous da a is a amilia challenge aced when designing wi eless senso ne wo ks (WSN). We use his knowledge o cope wi h he impac o message e o s on dis ibu ed de ec ion sys ems ealized by BP. I is a common p ac ice o e e o he channels o e which he da a exchange be ween he sensing nodes is con- duc ed as epo ing channels o dis inguish hem om he channels o e which he a ge signal is de ec ed, which a e e e ed o as lis ening channels. The impo ance o he p esen wo k can be highligh ed by no ing ha dis ibu ed de ec ion sys ems can be highly sensi i e o he epo ing e o s. This phenomenon is illus a ed in [22, Sec. IV-C] by a simple example ha shows ha he o e all de ec ion pe o mance canno go beyond he limi s dic a ed by he epo ing channel condi ions i espec i e o he signal- o- noise- a io (SNR) le els o he a ge signal expe ienced a he lis ening channels. This e ec is u he s udied and quan i ied o se e al ha d- and so -decision usion schemes in [23]–[25] whe e i is shown ha he epo ing chan- nel e o s can ha e a signi ican impac on he de ec ion pe o mance. In he exis ing li e a u e, he epo ing channels ha e been commonly conside ed nonideal o accoun o ealis ic da a-exchange p ocesses be ween he ne wo k nodes [17], [20], [21], [26]–[30]. A majo applica ion scena io o dis- ibu ed de ec ion sys ems is spec um sensing in cogni i e adio ne wo ks (CRN) whe e communica ion be ween he sensing nodes is ypically conduc ed wi hou ha ing access o dedica ed spec um bands. This means ha he da a exchange be ween hose nodes could ace s ingen cons ain s in e ms o ansmi powe and bandwid h. Consequen ly, many wo ks on dis ibu ed de ec ion in CRNs ocus on epo ing links wi h bandwid h o powe cons ains, see e.g., [27]. These cons ain s a e ypically aken in o accoun , in he sys em modeling and op imiza ion, by non-ideal epo ing links ha in oduce non-ze o bi e o p obabili ies (BEP) in digi al [21] and unco ela ed noise in analog epo ing schemes [20]. Since he link noise le el and BEP a e mono onically ela ed o each o he [21, Eq. (12)], simila app oaches can be used in bo h analog and digi al cases o mi iga e he impac o epo ing e o s, see [17], [21]. In his pape , we a e ocused on a BP-based decen alized dis ibu ed de ec ion scheme. We iew message e o s in he BP i e a ion as epo ing e o s and app oxima e he messages by a linea exp ession o s udy he impac o e oneous da a-exchange on he BP algo i hm and o cla i y how i a ec s he pe o mance o he esul ing dis ibu ed de ec ion. We de i e app oxima e exp essions ha measu e he s eng h o he cumula i e e o s ha a ec he BP-based decision a iables. These exp essions a e in he o m o mean-squa ed e o (MSE) le els. We compa e he MSE le els ob ained wi h he one in [11] o gain insigh in o he beha io o BP and o see how compu a ion and communica ion e o s p opaga e h oughou he unde lying ac o g aph. Ou analysis closely p edic s he ex en o he de ia ion o he e oneous decision a iables, ob ained by an e oneous BP i e a ion, om hei ac ual alues. This is a signi ican imp o emen o e Ihle ’s bound in [11]. Mo eo e , based on he p oposed linea app oxima ion, we show ha ABP is e ec i e in alle ia ing message e o s and alls sho o mi iga ing he impac o e oneous local likelihood a ios (LLRs) on he esul ing decision a iables. We also show, unde p ac ical assump ions, ha he decision a iables buil by an e oneous BP a e dis u bed by a sum o independen e o componen s whose collec i e impac can be modeled, app oxima ely, by Gaussian andom a iables. Consequen ly, we es ablish he p obabili y dis ibu ion o he esul ing e oneous decision a iables, de i e he pe o mance me ics o he BP-based dis ibu ed de ec ion in closed o m, and p opose a wo-s age op imal linea usion scheme o cope wi h he impac o e o s on he sys em pe o mance. We hen de elop a blind adap a ion algo i hm o ealize he p oposed wo-s age op imiza ion when he s a is ics desc ibing he adio en i onmen a e no a ailable ap io i. The p oposed blind adap a ion modi ies he pa ame e s o he BP and he decision h eshold a each node, in acco dance wi h he e o s a is ics and channel condi ions, o mi iga e he impac o e o s and o enhance he de ec ion pe o mance. To summa- ize, we ex end he wo ks in [5] and [6] by he ollowing con ibu ions: •We analyze he beha io o he BP algo i hm in he p esence o message (and likelihood) e o s and de i e i s pe o mance me ics in closed o m in a dis ibu ed de ec ion subjec o hose e o s. •We build a dis ibu ed op imiza ion amewo k o he sys em ha akes in o accoun and e ec i ely mi iga es he impac o e oneous da a exchange in BP. Mo eo e , •We ex end he wo k in [11] by p oposing a igh e e o bound ha mo e accu a ely desc ibes he impac o message e o s on he decision a iables buil by he BP algo i hm. •We ex end he wo k in [15] by analyzing he beha io o ABP. Ou wo k sheds ligh on ABP’s e ec i eness and sho comings. He e is an o e iew o he pape o ganiza ion: In Sec. II, we b ie ly explain he use o linea usion and BP in dis ibu ed 3288 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 69, NO. 5, MAY 2021 de ec ion and p o ide he ela ed o mula ions. In Sec. III, we discuss e o s in BP and model hei impac on he decision a iables ob ained. In Sec. IV, we iew BP as a dis ibu ed linea usion and o mula e he p oposed op imiza- ion amewo k. In Sec. V, we conduc compu e simula ions o e i y ou analysis and o illus a e how e ec i ely he p oposed me hod mi iga es he impac o e o s in a WSN wi h aul y de ices. Finally, we p o ide ou concluding ema ks in Sec. VI. II. LINEAR FUSION AND BELIEF PROPAGATION FOR DISTRIBUTED DETECTION We conside Nbina y andom a iables, ep esen ed by x=[x1,...,x N]T, whose s a us a e es ima ed based on N obse a ions deno ed Y=[y1,...,yN]made by a ne wo k o Nsensing nodes. Each node, say node i, which in ends o es ima e he s a us o xi, collec s Kobse a ion samples, deno ed by yi=[yi(1),...,y i(K)]T, and exchanges in o - ma ion wi h o he nodes in he ne wo k o ealize oge he a mul idimensional hypo hesis es as ˆ x=max xp(x|Y)= maxxp(Y|x)p(x). This es can be conduc ed wi h low implemen a ion complexi y in wo al e na i e ways ha a e explained in he ollowing. A. Linea Da a-Fusion Linea usion has been ex ensi ely used in he con ex o spec um sensing whe e he aim is o de ec he p esence o absence o a a ge signal by e alua ing noisy obse a- ions made h oughou a WSN. Fo b e i y, we explain he uni- a ia e case whe e we ha e a single bina y a iable x∈ {0,1}. The op imal app oach o such a de ec ion is known o be he so-called likelihood- a io es (LRT) [31], which is conduc ed by e alua ing he LLR, i.e., by ˆx=1{λLRT −τ} whe e λLRT ln p(Y|x=1) p(Y|x=0)= N  i=1 γi(1) whe e γiln p(yi|x=1) p(yi|x=0)=sT iyi−1 2si2(2) whe e γiis e e ed o as he local LLR a node i.By 1{·} we ep esen he indica o unc ion ha e u ns one i i s a gumen is posi i e and e u ns ze o o he wise. τis a de ec ion h eshold selec ed ia a a ge alse-ala m a e. Eq. (2) indica es ha , he LRT is a ma ched- il e ing p ocess, which equi es he a ge signal si o be known ap io ia he sensing nodes. Mo eo e , o Gaussian obse a ions, γi in (2) ollows a Gaussian dis ibu ion. In p ac ice, he local sensing p ocess is ealized by ene gy de ec ion, due o i s ease o implemen a ion and because i s s uc u e does no equi e he a ge signal o be known. Ene gy de ec ion is ealized by γi1 Kyi2and he senso ou comes a e combined linea ly o build a global es s a is ic [17]–[21], i.e., λLF  N  i=1 wiγi=wTγ(3) whe e w[w1,...,w N]Tand γ[γ1,...,γ N]T. Then, λLF is compa ed agains τ o conduc he hypo hesis es , i.e., ˆx=1{λLF −τ}.wcan be se o maximize he de ec ion p obabili y. Acco ding o he cen al limi heo em (CLT) [32], when he numbe o signal samples Kis la ge enough [17]–[21], he ou come o ene gy de ec ion ollows a Gaussian dis ibu ion and we can model he es summa y λLF, gi en he s a us o x, as a Gaussian andom a iable. Con- sequen ly, he de ec o pe o mance can be op imized by he well-known Neyman-Pea son app oach [31]. This op imiza ion is o mula ed as w∗=a gmin w Q−1(α)wTΣ0w−wTδ wTΣ1w(4) whe e δμ1−μ0while μbE[γ|x=b]and Σb co (γ|x=b)and Q−1(·)deno es he in e se o he Q- unc ion. αdeno es he a ge alse-ala m p obabili y a which he de ec ion p obabili y is maximized. This non-con ex p ob- lem is sol ed in [17], [19], [20]. F om hese wo ks, we know ha he pe o mance o linea usion is close o he LRT pe o mance. Al e na i ely, we can maximize he so-called de lec ion coe icien o he de ec o . This app oach, which has a low compu a ional complexi y and leads o a good pe o mance le el, is ealized by w∗=a gmax wΔ2(w),s. ., w=1 (5) whe e Δ2(w)(E[λLF|x=1]−E[λLF|x=0]) 2 Va [λLF|x=0] =wTδ2 wTΣ0w(6) Consequen ly, by using he Rayleigh-Ri z inequali y [17], w∗is ob ained in closed o m as w∗=Σ−1 0δ/ Σ−1 0δ . Ex ension o he linea de ec ion s uc u e in (3) o N a iables is discussed in [18] in he con ex o mul iband spec um sensing. B. Belie P opaga ion We model he senso ne wo k s uc u e conce ned by an MRF de ined on an undi ec ed g aph G=(V,E).In his model, he se o e ices Vco esponds o he se o ne wo k nodes while each edge (i, j)∈E ep esen s a possible connec ion be ween nodes iand j. Each node, say node i, is associa ed wi h a andom a iable xiand he edge (i, j) models a possible co ela ion be ween xiand xj. This model i s well in o he commonly-used ad-hoc ne wo k con igu a- ions in which majo ne wo k unc ionali ies a e conduc ed h ough pai wise i.e., one-hop, links be ween he nodes loca ed close o each o he . This design me hod is based on he common assump ion ha nodes loca ed close enough o each o he o one-hop communica ion, expe ience some le els o co ela ion be ween hei senso ou comes. By using he MRF, we w i e p(x|Y)as a p oduc o uni a ia e and bi a ia e unc ions, i.e., p(x|Y)∝ n∈V φn(xn) (i,j)∈E ψij (xi,x j)(7) No e ha ∝in (7) e e s o a no maliza ion ha ensu es xp(x|Y)=1and includes bu is no limi ed o 1/p(Y). When including he bi a ia e e ms in he p oduc , each edge in he ac o g aph is included in he p oduc only once. This is ealized by doing he mul iplica ion on i<jwhile i∈N j. ABDI AND RISTANIEMI: MODELING AND MITIGATING ERRORS IN BELIEF PROPAGATION FOR DISTRIBUTED DETECTION 3289 We use Nj o deno e he se o neighbo s o node jin he g aph, i.e., Nj{k:(k,j)∈E}. By using (7), we o mula e he message ecei ed a node j om node kas μ(l) k→j(xj)∝ xk φk(xk)ψkj (xk,x j) n∈Nj k μ(l−1) n→k(xk)(8) whe e by Nj kNk {j}we deno e all nodes connec ed o node kexcep o node j. We deno e by b(l) j(xj) he belie , abou he s a us o xj, o med a node j, which is ob ained ia mul iplying he po en ial a node jby he messages ecei ed om all i s neighbo s, i.e., b(l) j(xj)∝φj(xj) k∈Nj μ(l) k→j(xj)(9) The belie s a e used as es ima es o he desi ed ma ginal dis ibu ions, i.e., b(l) j(xj)≈p(xj|Y). By adop ing he commonly-used exponen ial model [4] o ep esen he ap io i p obabili y measu e de ined on x,weha e p(x)∝exp ⎛ ⎝ n∈V θnxn+ (i,j)∈E Jij xixj⎞ ⎠(10) Fo no a ional con enience, we use bipola bina y a iables, i.e., xj∈{−1,+1}in ou o mula ions o BP. Fo a gi en x, we assume he local obse a ions o be mu ually independen . Consequen ly, as explained in [5, Sec. I-B], we ha e p(x|Y)∝ n∈V p(yn|xn)eθnxn (i,j)∈E eJij xixj(11) Hence, by using (11), he BP messages a e buil as μ(l) k→j(xj)∝ xk p(yk|xk)eθkxkeJkj xkxj n∈Nj k μ(l−1) n→k(xk) (12) and he belie s a i e a ion la e exp essed as b(l) j(xj)∝p(yj|xj)eθjxj k∈Nj μ(l) k→j(xj)(13) In he log domain, (12) and (13) con e , espec i ely, as cla - i ied in Appendix A, o m(l) k→j=S⎛ ⎝Jkj,γ k+ n∈Nj k m(l−1) n→k⎞ ⎠(14) λ(l) j=γj+ k∈Nj m(l) k→j(15) whe e λ(l) jln b(l) j(xj=+1) b(l) j(xj=−1) (16) m(l) k→jln μ(l) k→j(xj=+1) μ(l) k→j(xj=−1) (17) deno e, espec i ely, he es ima ed likelihood a io a node j and he message sen o node j om node kwhile S(a, b) ln 1+ea+b ea+eband γkln p(yk|xk=+1) p(yk|xk=−1) =sT kyk−1 2sk2.In his model, yk=1 2(xk+1)sk+nkdeno es he signal ecei ed a node k. Hence, xk=−1indica es ha he a ge signal skis absen lea ing he he spec um ee whe e node k ope a es. I xk=+1, hen he co esponding spec um band is occupied. Jkj ’s a e calcula ed as in Eq. (16) in [5] by p ocessing a window o Tsensing ou comes. No e ha θk in (14) is me ged in o γkwi hou ha ing any impac on he es o he analysis. A e l∗i e a ions, λ(l∗) jis compa ed, as a decision a iable, agains a de ec ion h eshold τja node j o decide he s a us o xj, i.e., ˆxj=1{λ(l∗) j−τj}. By a linea app oxima ion o (14), we ha e [5] m(l) k→j≈cjk ⎛ ⎝γk+ n∈Nj k m(l−1) n→k⎞ ⎠(18) whe e cjk (e2Jkj −1) (1+eJkj )2. This app oxima ion is ob ained by he i s -o de Taylo se ies expansion, i.e., S(a, b)≈Sb(a, 0)b whe e Sb(a, b)=∂S(a, b)/∂b. By using (18) we see ha liml→∞ λ(l) j≈λjwhe e λjγj+ k∈Nj cjkγk+ k∈Nj n∈Nj k cjkcknγn + k∈Nj n∈Nj k m∈Nk n cjkckncnmγm+... (19) The e o e, his app oxima ion e eals ha , gi en enough ime, all he local likelihood a ios obse ed in he ne wo k a e almos linea ly combined a node j o calcula e i s decision a iable λj. We ha e shown in [5] ha , he con e gence o his linea message-passing algo i hm is gua an eed when |cj,k|<1 maxn|Nn|−1,∀(j, k)∈E. The linea combina ion in (19) can be exp essed as λj=N i=1 ajiγi,whichis compac ly s a ed in ma ix o m as λ=Aγ (20) whe e λ[λ1,...,λ N]Tand A[a1,...,aN]Twhile aj[aj1,...,a jN]T. He e we de i e he ela ionship be ween Aand cjk’s in (19) as A≈I+ ∞  n=1 Cn−D∞  n=1 Cn(21) whe e C[cjk]N×Nand D(X)deno es a diagonal ma ix whose main diagonal is equal o ha o X. The p oo is p o ided in Appendix B. I is now clea ha o ha e con e gence in he message-passing i e a ion (18), he spec al adius o Chas o be less han one. This c i e ion may be used o impose bounds on cjk’s o gua an ee he con e gence o he algo i hm. Al e - na i ely, he con e gence can be gua an eed, wi hou dealing wi h he complexi ies o inding he spec al adius, by using he con ac ing mapping condi ion as we ha e discussed in [5]. We use (21) in he ollowing sec ion o de i e an es ima ion o he e o s eng h a ec ing he decision a iables buil by an e oneous BP. III. ERRORS IN BELIEF PROPAGATION Eq. (14) shows ha a each BP i e a ion each node c ea es i s messages in e ms o i s local LLR alue as well as he messages ecei ed om he neighbo ing nodes a he p e ious i e a ion. In ou sys em model, we assume ha he local LLRs and he BP messages a e e oneous. As in [11] and [15], 3290 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 69, NO. 5, MAY 2021 we use he on Neumann app oach o modeling he join s a is ical beha io o e o s. A. E o Model and Analysis Since he messages a e mul iplied oge he o build he belie s, we o mula e hem as mul iplica i e pe u ba ions a ec ing ue (i.e., e o - ee) message alues, i.e., ˜μ(l) k→j(xj)=μ(l) k→j(xj)ε(l) k→j(xj)(22) whe e ˜μ(l) k→j(xj)deno es he e oneous message sen o node j om node ka i e a ion lwhile ε(l) k→j(xj)deno es he co esponding e o , which is conside ed in his pape as a s ochas ic p ocess. Eq. (22) di e s om he model used in [11] in he sense ha he e o model in ha wo k measu es he di e ence be ween he messages a i e a ion lwi h hei coun e pa s a he ixed poin o he message-passing i e a ion. In o he wo ds, he e o model in [11] measu es he de ia ion o he messages a each i e a ion om hei inal alue eached by BP a e con e gence. The s ochas ic e o we discuss he e is b ie ly s udied in [11] unde he no ion o addi ional e o . By exp essing he messages in he he log domain, we ha e ˜m(l) k→jln ˜μ(l) k→j(xj=+1) ˜μ(l) k→j(xj=−1) =m(l) k→j+ν(l) k→j(23) whe e ν(l) k→jln ε(l) k→j(xj=+1) ε(l) k→j(xj=−1) (24) Based on he on Neumann model, we assume ha i k=n, hen E[ln ε(l) k→j(x)lnε(l) n→j(x)] = 0 o all x. Consequen ly, we ha e E[νk→jνn→j]=0. To measu e he collec i e impac o e o s on he belie o node j,weuse E(l) j(xj)˜ b(l) j(xj) b(∗) j(xj)(25) whe e ˜ b(l) j(xj)deno es he belie a node j esul ing om a BP i e a ion wi h e oneous messages as in (22) while b(∗) j(xj) deno es he belie o node ja a ixed poin eached by an e o - ee BP i e a ion. We use (∗)ins ead o (l) o indica e he messages and belie s a a ixed poin o he e o - ee BP. By assuming unco ela ed s ochas ic beha io o he mes- sage e o s, an uppe bound on cumula i e e o s a ec ing he belie s can be ob ained. Speci ically, assuming Va ν(l) k→j≤ (ln u)2 o all k,j,l, an uppe bound on he esul ing cumu- la i e s eng h o e o s a node jis de i ed in [11] as, Eln dE(l) j2≤ k∈Njσ(l) kj 2 (26) whe e σ(1) kj =lnd(ψkj )2and σ(l+1) kj 2=ln d(ψkj )2ω(l) kj +1 d(ψkj)2+ω(l) kj 2 +(lnu)2(27) while ln ω(l) kj 2= n∈Nj kσ(l) nk2 (28) whe e dE(l) jsup a,b     E(l) j(a) E(l) j(b)(29) d(ψkj)2sup a,b,c,d ψkj(a, b) ψkj(c, d)(30) We use he uppe bound in (26) in he log domain based on he ac ha (see (16) and (25)) ˜ λ(l) jln ˜ b(l) j(+1) ˜ b(l) j(−1) =λ(∗) j+lnE(l) j(+1) E(l) j(−1) (31) which leads o E ˜ λ(l) j−λ(∗) j 2=Eln E(l) j(+1) −ln E(l) j(−1) 2 =Eln dE(l) j2≤ k∈Njσ(l) kj 2 (32) Hence, in he de ec ion s uc u e discussed, (26) gi es an uppe bound on he MSE le el obse ed in he decision a iable a node j. B. Linea App oxima ions In ou analysis, we dis inguish be ween he message e o s and he e o s in he compu a ion o local LLRs o gain u he insigh in o he beha io o he BP algo i hm. In pa icula , we model he e oneous local LLRs as ˜γkγk+kand e e o k’s as likelihood e o s (LE) while assuming ha LEs a e unco ela ed as well, i.e., E[kn]=0 o k=n. We e e o νk→j’s as message e o s (ME) and assume ha LEs and MEs a e mu ually independen . Mo eo e , we assume ha all MEs and LEs a e independen o he messages and o he local LLRs. No e ha he bound in (32) does no ake LEs in o accoun . Taking bo h ypes o e o in o accoun , we exp ess he messages decision a iables as ˜m(l) k→j=S⎛ ⎝Jkj,˜γk+ n∈Nj k ˜m(l−1) n→k⎞ ⎠+ν(l) k→j(33) ˜ λ(l) j=˜γj+ k∈Nj ˜m(l) k→j(34) which shows ha he e o s pass h ough he same nonlinea ans o ma ion (i.e., S) as he messages do. By using (33), we can analyze he beha io o e o s. The p oposed linea BP i e a ion in he p esence o message e o s is exp essed as ˜m(l) k→j≈cjk ⎛ ⎝˜γk+ n∈Nj k ˜m(l−1) n→k⎞ ⎠+ν(l) k→j(35) Consequen ly, simila o he way (19) is de i ed, he esul ing e oneous decision a iable is o med as ˜ λ(l) j≈˜γj+ k∈Nj cjk˜γk+ k∈Nj n∈Nj k cjkckn˜γn+... + k∈Nj ν(l) k→j(36) ABDI AND RISTANIEMI: MODELING AND MITIGATING ERRORS IN BELIEF PROPAGATION FOR DISTRIBUTED DETECTION 3291 which can be eo ganized as ˜ λ(l) j≈λj+ξ(l) j(37) whe e ξ(l) j N  i=1 ajii+ k∈Nj ν(l) k→j(38) Eq. (38) shows ha he e o a ec ing he decision a iable a node jhas wo dis inc componen s. The i s componen is buil as a linea combina ion o LEs while he second one is he sum o he MEs ecei ed a node j om i s one-hop neighbo s. The i s componen is ixed whe eas he second one exhibi s a new ealiza ion a e e y i e a ion. Acco ding o (38), de ia ion om he e o - ee decision a iables, caused by e o s in he BP i e a ions, can app oxi- ma ely be measu ed by E ˜ λ(l) j−λ(∗) j 2≈Eξ(l) j 2=aT jΣaj+ Σνj(39) whe e Σco ()and Σνjco ν(l) jwhile  [1,..., N]Tand ν(l) jdeno es an |Mj|-by-1 ec o ha con ains ν(l) k→j’s o k∈M jwhe e MjNj∪{j}while ν(l) j→j0. No e ha (36) includes mo e ME e ms han jus k∈Njν(l) k→j. Howe e , hey can all be neglec ed since |cjk|<1 o all j, k. Eq. (36) shows ha when BP is used o ealize a dis ibu ed de ec ion, he e oneous local likelihoods in he ne wo k a e combined linea ly o build he decision a iables. We can e alua e he impac o he e o s on he sys em pe o mance by analyzing he s ochas ic beha io o he e oneous decision a iables ˜ λ(l) j.Gi enx, he decision a iable a node j is ob ained as a linea combina ion o independen andom a iables. Consequen ly, i s condi ional pd is de i ed as ˜ ιj|x(z|b)≈N  i=1 1 aji ˜γi|xz aji |b∗ k∈Nj νk→j(z)(40) whe e ˜γi|x(z|b)= γi|x(z|b)∗ i(z)(41) while and ∗deno e he con olu ion ope a o . Consequen ly, we ha e g j(τj, )P {˜ λj>τ j|xj= } = b∈{−1,1}N−1 px(j)|xj(b| ) ∞ τj ˜ ιj|x(z|Ej, (b)) dz (42) whe e ∈{−1,+1},x(j) [x1,x 2,...,x j−1,x j+1,...,x N]Tand Ej, (b){x(j)= b,x j= }while px(j)|xj(b| )P {x(j)=b|xj= }. Sol ing g j(τj,−1) = αgi es a h eshold alue ha ixes he alse-ala m a e a α. Simila ly, g j(τj,1) = β ixes he de ec ion a e a β. Recall ha aji’s a e ound by using cjk’s, see (21). As a common p ac ical case, when he local LLRs and he e o s ollow Gaussian dis ibu ions [17]–[21] he decision a iable ˜ λj ollows a Gaussian dis ibu ion as well and i is ully cha ac e ized by i s i s - and second-o de s a is ics. Speci ically, we ha e ∞ τj ˜ ιj|x(z|Ej, (b)) dz =Qτj−μj, (b) σj, (b)(43) whe e μj, (b)E˜ λj|Ej, (b) =E[γj|xj= ]+ i=j ajiE[γi|xi=bi](44) σ2 j, (b)Va ˜ λj|Ej, (b) =Va [γj|xj= ]+ i=j a2 jiVa [γi|xi=bi]+E|ξj|2 (45) In (44) we ha e assumed, wi hou loss o gene ali y, ze o-mean e o s. No e ha , wi hou he p oposed app oxima ion hese pe o mance measu es a e no a ailable analy ically due o he nonlinea i y o (14). In he es o he pape , we assume ha he local likelihoods, LEs, and MEs a e Gaussian andom a iables. Eq. (40) shows ha , acco ding o he CLT, e en i he local LLRs and e o s a e no Gaussian andom a iables, he s ochas ic beha io o he decision a iables can s ill be app oxima ely desc ibed by Gaussian dis ibu ions. C.Impac o A e aging In ABP, he message-passing i e a ion is he same as in BP. Howe e , ins ead o he ac ual message alues, an a e age o he messages a e used o build he decision a iables. To be mo e speci ic, in he log domain and o l≥L+1,le ¯m(l) k→j1 L+1 l  =l−L ˜m( ) k→j(46) The decision a iable a node jis calcula ed by ¯ λ(l) jγj+ k∈Nj ¯m(l) k→j(47) Simila o ou discussion ega ding (19), we can show ha when he message-passing i e a ion is e o - ee, ¯ λ(∗) j liml→∞ ¯ λ(l) j=λj. Hence, we can see ha he a e aging p ocess does no al e he ixed poin s achie ed by he e o - ee linea BP. This obse a ion is in line wi h he con e gence analysis p o ided in [15]. The impac o a e aging on LEs and MEs can be cla i ied by no ing ha ¯ λ(l) j=λj+¯ ξ(l) j(48) whe e, assuming L o be la ge enough, we ha e ¯ ξ(l) j= N  i=1 ajii+ k∈Nj ¯ν(l) k→j≈ N  i=1 ajii(49) since ¯ν(l) k→j1 L+1 l =l−Lν( ) k→j≈0. We can s a e (49) in he o m o MSE as E ¯ λ(l) j−λ(∗) j 2≈aT jΣaj+1 L+1 Σνj(50) Assuming L o be la ge enough and MEs o ha e ze o mean, (49) shows ha he esul ing decision a iable buil by 3292 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 69, NO. 5, MAY 2021 ABP in (47) is almos clea ed o MEs. Howe e , he a e aging p ocess has almos no impac on LEs. No e ha in ABP he message-passing i e a ion is he same as in BP and he a e aging is only pe o med when compu ing he decision a iables. Mo eo e , in ABP, ins ead o s o ing he messages in pas i e a ions sepa a ely, we only need o s o e he sum o he messages up o he cu en i e a ion. As a consequence, he numbe o addi ional memo y cells equi ed can be kep cons an [15]. We will use ABP in Sec. IV-B o build an o line lea ning-op imiza ion s uc u e o he linea BP in he p esence o e o s. IV. MITIGATING ERRORS BY LINEAR FUSION In his sec ion, we i s p opose a wo-s age linea usion scheme o ob ain a nea -op imal de ec ion pe o mance by supp essing he impac o he e o s. Then, we ealize he p oposed op imiza ion in a blind decen alized se ing whe e he equi ed s a is ics a e no a ailable a p io i. A. Linea Fusion Fi s , since |cjk|<1, we u he app oxima e he decision a iable λjin (19) as λj≈ k∈Mj cjkγk(51) Due o he symme y o he da a- usion p ocess in (19), he app oxima ion in (51) is an e ec i e app oach o building a dis ibu ed compu ing amewo k o sys em pe o mance op imiza ion. In his amewo k, each node in e ac s only wi h i s immedia e neighbo s. We ha e cla i ied his symme y in [5, Sec. III-B]. By aking in o accoun he e o s while analyzing he linea BP, (19) and (36) lead o ˜ λj≈ k∈Mj cjk (γk+k)+  k∈Nj νk→j(52) We see ha he dis u bance on he decision a iable caused by LEs is buil , app oxima ely, as a linea combina ion o k’s wi h cjk’s ac ing as weigh s in his combina ion. The e o e, we use cjk’s as design pa ame e s o mi iga e he impac o k’s. Mo eo e , MEs a e combined in (52) linea ly and in his combina ion, all weigh s a e one. We p opose o ex end his combina ion by using a modi ied e sion o (34) as ˆ λ(l) j˜γj+ k∈Nj wjk ˜m(l) k→j(53) This modi ica ion in he s uc u e o he decision a iable does no a ec he con e gence o he p oposed linea BP since i does no al e he message-passing i e a ion. Now, based on an app oxima ion simila o he one in (52), we ha e ˆ λj≈ k∈Mj wjkcjk (γk+k)+  k∈Nj wjkνk→j(54) Since ˆ λjis a Gaussian andom a iable, we only need i s mean and a iance o cha ac e ize i s s a is ical beha io . Speci ically, o b∈{−1,1},weha e P {ˆ λj>τ j|xj=b}=Qτj−E[ˆ λj|xj=b] Va [ˆ λj|xj=b](55) whe e E[ˆ λj|xj=b]≈ T jμb(56) Va [ˆ λj|xj=b]≈ T jΣγj|b+Σj j+wT jΣνjwj(57) whe e jwj◦cjin which ◦deno es he Hadama d p oduc while Σγj|b=co (γj|xj=b)and Σj=co (j). Mo eo e , wj,cj,γj,andja e |Mj|-by-1 ec o s con aining wji’s, cji’s, γi’s, and i’s o i∈M j, espec i ely. Eq. (55) gi es he sys em alse-ala m p obabili y o b=−1and he de ec ion p obabili y o b=1. The alse-ala m p obabili y can be se o P(j) =αby τj=Q−1(α)Va [ˆ λj|xj=−1] + E[ˆ λj|xj=−1] (58) and hen by using (55) – (57), wjand cjcan join ly be op imized in a Neyman-Pea son se ing. To a oid he challenges o his op imiza ion, we maximize he de lec ion coe icien o he de ec o . We al eady know ha he esul ing de ec o pe o ms well when he decision a i- ables ollow Gaussian dis ibu ions. In his manne , we mi i- ga e he join impac o LEs and MEs wi h low compu a ional complexi y. The p oposed op imiza ion is conduc ed in wo consecu- i e s ages based on he ac ha we can decompose he cons uc ion o ˆ λjin o wo consecu i e usion p ocesses. Tha is, we i s op imize cjk’s by conside ing he impac o k’s on γk’s. Then, we conside he esul ing scaled LLRs, i.e., cjkγk’s, as new s a is ics o be linea ly combined, while being weigh ed by wjk’s and dis o ed by νk→j’s, o make he decision a iable a node j. Mo e speci ically, i s , we op imize cjin a hypo he ical linea de ec o wi h i s decision a iable de ined as ˆ λ jcT jγj+j(59) The coe icien s esul ing om his op imiza ion scale up he mo e eliable local LLRs, wi h espec o he ones buil unde low SNR egimes, o supp ess he e ec o LEs. We deno e he esul ing usion weigh s by c∗ j. Then, we use c∗ jwi hin he s uc u e o he ac ual de ec o o op imize wj o mi iga e he impac o MEs. Tha is, we conside he ollowing linea de ec o a node j ˆ λ jwT jχj+νj(60) whe e χjc∗ j◦(γj+j)con ains χjk’s o k∈M jwhile χjk =c∗ jk(γk+k). The ec o νjcon ains νk→j’s wi h k∈M j. In his s uc u e, he elemen s o χjk’s a e seen as he ac ual local LLRs ha a e combined o build he decision a iable a node jwhile he combina ion akes in o accoun he join deg ading e ec o MEs and LEs. Based on he ma e ial p o ided in Sec. II-A, he i s s age o he p oposed op imiza ion is o mally s a ed as c∗ j=a gmax cjΔ j(cj),s. ., cj=1 (61) whe e Δ j(cj)= cT jδj2 cT jΣγj|−1+Σjcj (62) whe e δjE[γj|xj=1]−E[γj|xj=−1]. The esul ing c∗ jis hen used o ealize he second s age o he p oposed ABDI AND RISTANIEMI: MODELING AND MITIGATING ERRORS IN BELIEF PROPAGATION FOR DISTRIBUTED DETECTION 3293 op imiza ion by sol ing w∗ j=a gmax wjΔ j(wj),s. ., wj=1 (63) whe e Δ j(wj)= wT jˆ δj2 wT jΣχj|−1+Σνjwj (64) whe e ˆ δj=c∗ j◦δjand Σχj|−1=co (χj|xj=−1) = c∗ jc∗T j◦Σγj|−1+Σj.Ha ingc∗ jand w∗ j, he de ec ion h eshold τjis de i ed as τj=Q−1(α)Va [λ j|xj=−1] + E[λ j|xj=−1] o ix he sys em alse-ala m a e a α. The con e gence condi ion |cj,k|<1 maxn|Nn|−1,∀(j, k)∈ Ecan be ealized by a simple no maliza ion o c∗ j,k’s since he objec i e unc ion in (61) does no change by no malizing i s a gumen . Th ough he p oposed wo-s age op imiza ion, we enhance he de ec ion pe o mance a node jby supp essing he join impac o MEs and LEs wi h low compu a ional complexi y. The s a is ics equi ed in his op imiza ion a e collec ed om he one-hop neighbo s o node j. This makes he p oposed me hod a iable app oach in ad-hoc ne wo k con igu a ions whe e majo ne wo k unc ionali ies a e conduc ed h ough one-hop links be ween he ne wo k nodes. B. O line Lea ning and Adap a ion To ealize he p oposed op imiza ion, we need he mean and co a iance o he local e oneous LLRs. In a blind se ing whe e he e is no p io in o ma ion a ailable ega ding he adio en i onmen , we ha e o es ima e hose pa ame e s based on he de ec ion ou comes. The main challenge he e is ha he s a e o xjis equi ed a node jwhile he only in o ma ion a ailable in p ac ice is he de ec ion ou come ˆxj. Hence, node jhas o es ima e he condi ional s a is ics equi ed in (61) and (63) based on ˆxj. The p oblem wi h such an adap a ion mechanism is ha i makes he de ec ion ou come ˆxjdepend on hose es ima es. This dependence c ea es an inhe en de e io a ing loop by eeding he de ec ion e o s back in o he sys em s uc u e h ough e oneous es ima es o he equi ed s a is ics. To o e come his challenge, we p opose an ex ended e sion o he blind lea ning-adap a ion loop in [5] ha accommoda es he p oposed e o -mi iga ing s uc u e. The pseudo-code o his adap a ion is p o ided in Algo i hm 1 whe e he ask o each node is speci ied in a dis ibu ed compu ing amewo k. Algo i hm 1 ope a es on a window o s o ed sensing ou comes and in ol es a seconda y BP ha is un much less equen ly han he a e a which he dis ibu ed de ec ion is pe o med. The ou comes o his o line BP a e used in he es ima ion o he equi ed unknown s a is ics. In his adap a ion, he desi ed op imiza ions a e ealized i e a i ely while each node in e ac s only wi h i s one-hop neighbo s. Consequen ly, Algo i hm 1 can be well inco po a ed in a decen alized ne wo k con igu- a ion. In he sequel, we p opose a blind adap a ion s uc u e in which we use κ o deno e he i e a ion index. No e ha we use las he i e a ion index in he main BP h ough which Algo i hm 1 Blind Adap a ion o Fusion Weigh s in E oneous Linea Belie P opaga ion Inpu : ˜ γT,¯ γT,τ(0),κmax,η Ou pu : Nea -op imal cjand wj o j=1,...,N 1. Le κ←0and ini ialize ˆ x(0) by compa ing ˜ γTagains τ(0); 2. while κ≤κmax 3. o node j∈{1,2,...,N} 4. Calcula e E[¯γi|ˆx(κ) j]and co (¯γi,¯γk|ˆx(κ) j) o all i, k ∈ Mj; 5. Sol e (61) o ind c(κ) jand τ(κ) j; 6. Se c∗ jby an η- es on c(κ) j; 7. end 8. Use c∗ j’s and τ(κ) j’s o un linea ABP on ˜ γT o ind ˆ x(κ+1); 9. κ←κ+1; 10. end 11. o node j∈{1,2,...,N} 12. Use c∗ jand ˆ x(κmax) o calcula e ˆ δj,Σχj|0and Σνj; 13. Sol e (63) o ind w∗ j; 14. end 15. Ou pu c∗ jand w∗ j o j∈1,2,...N; he dis ibu ed de ec ion is ealized. The o line adap a ion upda es he usion weigh s in he p oposed linea BP by p ocessing Ts o ed samples o ˜ γ. This window o e oneous local likelihoods is deno ed by ˜ γTand con ains samples o ˜ γ( ) o =1,2,...,T. Recall ha , ˜ γ=γ+whe e  deno es he ec o o LEs. The o line de ec ion ou comes a i e a ion κa e deno ed by ˆ x(κ)[ˆx(κ) 1,...,ˆx(κ) N]while he esul ing usion weigh s and de ec ion h esholds a e deno ed c(κ) jand τ(κ) j espec i ely. ˆ x(κ)deno es a window o s o ed sensing ou comes ˆ x(κ)( ) o =1,2,...,T. Fo simplici y, we do no show he ime index when dealing wi h ˜ γT,and ˆ x(κ). Due o e o s caused by he wi eless links be ween he sensing nodes, node jdoes no ha e access o ˜γk( ),k∈N j. Speci ically, wha node j ecei es om node kis ˜γk( )+νk→j whe e νk→jdeno es he co esponding link e o . Wi hou loss o gene ali y, we a ibu e MEs o wi eless link e o s. To alle ia e he link e o s, be o e s a ing he adap a ion p ocess node j ecei es Lcopies o ˜γk( ) om node kand calcula es an a e age o ob ain ¯γk( )˜γk( )+¯νk→jwhe e ¯νk→jdeno es he a e age o Lindependen ealiza ions o νk→j. The desi ed s a is ics a e hen calcula ed by p ocessing ¯γk’s, which app oxima e ˜γk’s. We use ¯ γT o con ain he samples o ¯γk( ) o =1,2,...,T o k=1,2,...,N. In a ealis ic de ec ion scena io, he da a exchanged be ween he nodes in he p oposed o line adap a ion is impai ed by bo h ypes o e o s. Since in he i s linea usion (61) we ake in o accoun he impac o LEs only, we need o isola e his op imiza ion om he MEs. To his end, we es ima e he desi ed s a is ics by using linea ABP. As we saw in Sec. III-C, MEs do no a ec he ABP ou comes signi ican ly. The e o e, he esul ing o line decision a iables a e almos clea ed o