International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP) (12th : 2015)

Message-passing sequential detection of multiple structural damages Liao, Yizheng; Rajagopal, Ram Jul 31, 2015

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Message-passing Sequential Detection of Multiple StructuralDamagesYizheng LiaoGraduate Student, Stanford Sustainable Systems Lab, Dept. of Civil and EnvironmentalEngineering, Stanford University, Stanford, USARam RajagopalAssistant Professor, Stanford Sustainable Systems Lab, Dept. of Civil and EnvironmentalEngineering, Stanford University, Stanford, USAABSTRACT: This paper introduces a multiple structural damage detection algorithm for structural healthmonitoring. We propose a sequential damage detection algorithm that uses the Bayesian inferences todiagnose multiple damages and their relationships. The proposed algorithm is purely data driven and doesnot require the prior knowledge of the structural properties. The sequential detectors are implemented bya computationally efficient message passing protocol that enables distributed and simultaneous damagedetection. Also, the detectors achieve minimum detection delay with a desired false alarm rate. Theperformances of our algorithm are validated on the ASCE benchmark structure.1. INTRODUCTIONStructural health monitoring (SHM) involves usinghardware and software to ensure the safety of civilstructures by diagnosing the damage efficiently andreliably. In a typical application of SHM, a sen-sor network is deployed for real-time monitoring,processing new data samples as they arrive, andmaking decisions about whether damages have oc-curred. The monitoring process contains three sub-components: (i) collecting structural responses viathe sensor network; (ii) extracting the damage sen-sitive features; and (iii) detecting damages via sta-tistical inferences. There have been various suc-cessful efforts to the structural data acquisition,such as Lynch et al. (2002), Wang et al. (2007), andNagayama and Spencer Jr (2007). The reliable androbust algorithms for damage detection should beapplicable for complex structures with various ma-terials and complicated geometry. Moreover, thealgorithms are expected to be insensitive to the un-certainties caused by the environmental and loadingconditions but to be sensitive to the damages. Fur-thermore, for life-threatening damages, the detec-tors are required to have to high accuracy and lowlatency.There have been many existing works on the sta-tistical detection of structural damages. The algo-rithms in Nair et al. (2006) and Noh et al. (2009)fit the structural responses with the autoregres-sive moving average (ARMA) model and then per-form hypothesis testing on the model coefficients.Nair and Kiremidjian (2007) models the ARMAcoefficients as a Gaussian mixture. The damageis diagnosed by measuring the distance betweenthe Gaussian mixtures fitted by the undamagedand damaged data. Peter Carden and Brownjohn(2008) also uses the ARMA coefficients as the fea-tures. The damages are detected by an unsuper-vised classifier. All of the works above are cen-tralized batch processing, which has large detectiondelay and is inefficient for a large-scale monitor-ing system. There have been several papers usingthe Bayesian inferences to detect damages, suchas Vanik et al. (2000) and Ching and Beck (2004).However, these works require to have prior knowl-edge of the model parameters. Recently works,i.e. Noh et al. (2013) and its extension Liao et al.112th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015(2014), use the Bayesian models to detect the dam-age sequentially and do not require the model pa-rameters. A disadvantage of these algorithms is thatonly the samples from a specific sensor are utilizedfor damage detection.In this paper, we propose a damage detection al-gorithm that uses all time-series based features ina network to detect damages in a Bayesian setting.The algorithm purely relays on the data and no priorknowledge of structures are needed. The sequentialdetectors can detect multiple damages with mini-mum detection delay for a desired false alarm ratesimultaneously. Also, the detectors can be imple-mented in a message passing and distributed fash-ion. Moreover, the proposed detectors identify notonly single damage but also multiple multi-stagedamages and their relationships, such as when thefirst damage occurs and when all the possible dam-ages have happened. This algorithm is motivatedby Amini and Nguyen (2013), which only focuseson singe-stage damage and the earliest damage de-tection.The paper is organized as follows. Section 2defines the damage sensitive features and formu-lates the damage detection problem in the Bayesianstatistics. In addition, the detectors and themessage-passing algorithm are discussed. Sec-tion 3 validates the proposed algorithm on theASCE benchmark structure and discusses the per-formances. Section 4 draws summaries and con-clusions.2. ALGORITHMThe proposed damage detection algorithm includesthree steps: (i) collecting structural responses; (ii)extracting damage sensitive features (DSFs) fromstructural responses; and (iii) detecting damagesvia message passing protocol. For (i), the struc-tural responses are sequentially obtained from mul-tiple sensors and are normalized. In the secondstep, the autoregressive (AR) model is applied toextract the DSFs. The first three AR coefficients aresensitive to damages (Noh et al. (2009),Noh et al.(2011)) and follow a Gaussian mixture model(Nair and Kiremidjian (2007), Noh et al. (2013)).In (iii), the sequential detectors identify the dam-ages based on the posterior probabilities of dam-ages, which is computed based on the prior distri-bution and the distributions of DSF before and afterthe occurrence of damage. Usually, the computa-tion of the posterior probability requires to central-ize the DSFs from all the sensors. In the proposedalgorithm, we introduce the message passing (MP)method to compute the posterior probability in adistributed approach.2.1. Feature ExtractionThe DSF extraction consists of two steps: (i) nor-malization and (ii) AR model fitting. The discretetime acceleration signal from sensor j, Yj(n), is di-vided into chunks with a size N. Let Y ij(n) denotethe the ith chunk of the signal Yj(n). The normal-ized signal Y˜ ij(n) is obtained as follows:Y˜ ij(n) =Y ij(n)−µ ijσ ij,where µ ij and σ ij denote the mean and the stan-dard deviation of the ith chunk. For notation conve-nience, we will use Y ij(n) as Y˜ ij(n) in the followingtext. In addition, we will use the ith time index asan alternative term for the ith chunk.After normalizing the signal, the chunk data arefitted with a single-variant AR model of order p,Y ij(n) =p∑k=1θkY ij(n− k)+ ε ij(n), (1)where θk is the kth AR coefficient and ε ij(n) is theresidual. The selection of AR model includes re-moving the trends, choosing the optimal model or-der p, and checking the assumptions of the residu-als ε . The model selection process will be discussedwith more details in Section 3.2.2. Damage DetectionFor the monitoring structure, we assume there ared potential damage locations. Each damage loca-tion is assigned with a random variable λi ∈ N, fori ∈ [d] := {1,2, . . . ,d}, independently. The damagevariable λi represents the time at when the dam-age occurs and follows a prior distribution pii(λi).The examples of damage locations include floors,walls, braces, balls and etc.. We will use the nota-tion λ⋆ = (λi, i∈ [d]) as a collection of all damages.212th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Let the sequence of DSFs up to the time n be rep-resented as Xnj := (X1j ,X2j , . . . ,Xnj ) where X ij denotethe ith DSF of sensor j. Then, given M sensors inthe network, Xn⋆ := (Xnj , j ∈ [M]) denotes the DSFsof all sensors up to the time n.Depending on the installation location, the sig-nal acquired by a sensor is affected by one or moredamages. Let index set S j = {i1, i2, . . . , ir} ⊆ [d]contains the indices of all damage variables that af-fect the measurement of sensor j. Then, λS j is theset of the damage variables that are associated withsensor j. For example, if sensor 1 measures the re-sponse affected by λ2 and λ3, then S1 = {2,3} andλS1 = {λ2,λ3}. By the definition of λ⋆, we knowλ⋆ ≡(⋃λS j , j ∈ [M]). Therefore, in the Bayesianformulation, the joint distribution of λ⋆ and Xn⋆ isP(λ⋆,Xn⋆) = ∏i∈[d]pii(λi) ∏j∈[M]P(Xnj |λS j), (2)where P(Xnj |λS j) is the likelihood probability anddepends on the damage variables. The distribu-tions of DSFs are affected by the damages. How-ever, conditioning on the damage variable λS j , theDSF Xnj is independently and identically distributed(i.i.d) with the same distribution. How to find theconditional distribution will be discussed in detailslater.In the damage detection problem, our primary in-terest is the estimation of the damages λ⋆. There aretwo groups of detectors, as shown below:φmin := φmin(λ⋆) := λminS :=mini∈Sλi, (3)φmax := φmax(λ⋆) := λmaxS :=maxi∈Sλi, (4)for some subset S ⊆ [d]. The first category, whichis referred as the minimum detector, focuses on thedetection of the earliest damage. Examples includethe detection of a single damage S= { j}, the earli-est among two damages S= {i, j}, and the earliestamong the entire network S = [d]. For the secondcategory, which is referred as the maximum detec-tor, we are interested in how many damages haveoccurred or whether all the damages in S have oc-curred or not. Examples include the detection oftwo damages S = {i, j} that have occurred and allthe damages in the network S= [d] have occurred.We assume τ to be the damage detection rule forφ . Based on Xn⋆, the rule sets τ = n if one or moredamages are detected. Thus the random variable τis a stopping time (Rajagopal et al. (2008), Durrett(2010)). τ is chosen based on two metrics: proba-bility of false alarm P(τ ≤ φ) and the detection de-lay E(τ −φ)+. Here, we use the Neyman-Pearsoncriteria to choose the stop rules set that has falsealarm at most α:∆φ (α) := {τ : P(τ ≤ φ)≤ α}. (5)Among ∆φ (α), we want to consider the rule thathas the minimum detection delay. We use τS to de-note the stopping rule associated with λS. We pro-pose a stop rule that stops at the first time the pos-terior probability is larger than a given threshold,τminS = inf{n : P(λminS ≤ n|Xn⋆)≥ 1−α}, (6)τmaxS = inf{n : P(λmaxS ≤ n|Xn⋆)≥ 1−α}, (7)where n ∈ N and α is the maximum allowablefalse alarm rate. As proven in Amini and Nguyen(2013), τS is in ∆φ (α) when φ = λS. Thus, τS hasthe minimum detection delay.Let Ai denote the domain set of λi and AS denotethe product Ai1 ×Ai2 × ·· · ×Air . For each sensor,there is a function α j(λS j) : AS j → R+. Let’s callthe variable set λS j the local domain and the func-tion α j the local kernel. In our setup, we want theproducts of the local kernels, ∏Mj=1 α j(λS j), to bethe joint distribution in (2). Therefore, α j is eitherP(Xnj |λS j) or P(Xnj |λS j)∏λi∈λS pii(λi) with S ⊆ S j.To compute the posterior probability of the localdomain, we want to marginalize the joint distribu-tion and eliminate the irrelative random variables,as follows:β j(λS j) := P(λS j ,Xn⋆) = ∑λScj∈AScjP(λ⋆,Xn⋆), (8)where Scj denotes the complement of S j relative tothe entire set [d]. We call the function β j(λS j) theobjective function of sensor j. Based on the Bayes’theorem, the posterior probability P(λS j |Xn⋆) ∝β j(λS j).As discussed in Aji and McEliece (2000) andAmini and Nguyen (2013), if a sensor network can312th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015be formulated as a polytree, then we can use thegeneralized distributive law (GDL) or the beliefpropagation algorithm to produce exact values ofthe objective function β (λS). In the polytree, ifthere exists any edge (probabilistic linkage) be-tween sensor i and sensor j, then the message fromi to j, at time n, ismni j(λSi∩S j)= ∑λSi\S j∈ASi\S jαi(λSi) ∏r∈∂ i\{ j}mnr j(λSr∩Si),(9)for λSi∩S j ∈ ASi∩S j , where ∂ i denotes the index setof all the neighbors that have an edge with sen-sor i. At time n, the dimension of the message is(n+1)|Si∩S j|. The message sent from sensor i is theproduct of its local kernel with all messages it hasreceived from its neighbors other than sensor j withfiltering out the irrelative information (by marginal-ization). Therefore, by passing the messages, wesend the inferences of one sensor to all other in thenetwork. Once the message passing is completed,on each sensor, the objective function β (λS) can becomputed asβ j(λS j) = α j(λS j) ∏r∈∂ jmnr j(λSr∩S j) ∀ j ∈ [M].(10)Then, the posterior probability P(λS j |Xn⋆), whichrelay on all the DSFs that have been collectedacross the network, can be computed directly.In SHM, we usually define the prior pii(λi) as ageometric distribution with parameter ρi ∈ (0,1),i.e. pii(k) := (1− ρi)k−1ρi. Therefore, at time n,the domain of the local kernel is Ai ≡ [n+ 1] forall i ∈ [d]. When λi ≤ n, it means that the damagehas occurred. When λi = n+ 1, it means that thedamage will happen in the future. We define pii(n+1) =∑∞k=n+1 pii(k). Therefore, for the MP setup, theprior is defined asp˜ini (k) :={pii(k) for k ∈ [n]pii[n]c = ∑∞k=n+1 pii(k) for k = n+1.where [n]c := N\[n] = {n+1,n+2, . . .}.Assume that the network can be organized as apolytree, we propose theMP algorithm at each timestep n as follows:1. Choose one sensor in the network as the rootof the tree2. Initialize messages mni j ∈ R(n+1)|Si∩S j | to the allones tensor for all edges. Compute p˜ini (k) fork ∈ [n+1], i ∈ [d].3. Compute and pass messages mni j from sensori to sensor j according to (9). Pass messagesfrom the leaves to their parents. Continue theprocess from the bottom of the tree to the toptill the root sensor is reached.4. Repeat Step 3 but start from the root. Passmessages from the root to its children. Con-tinue the process from the top of the tree to thebottom till all the leaves are reached. Whencompute mni j based on (9), use the messagesreceived in Step 3.5. Compute β j(λS j) based on (10) for λS j ∈A1 × A2 × ·· · × A|S j| and Ai ≡ [n + 1] forall j. Then normalize β j(λS j) such that∑λS j∈A1×A2×···×A|S j | β j(λS j) = 1.After the normalization, β j(λS j) becomes to theposterior probability, i.e. P(λS j |Xn⋆) = β j(λS j).This process is repeated as a new DSF is available.If S j contains the indicates of the damage vari-ables of sensor j and the subset S⊆ S j contains theindices of the damage variables that we want to de-tect, then at time n, the posterior probability of theminimum detector can be computed asP(min(λS)≤ n|Xn⋆)= 1− ∑λS j\S∈AS j\Sβ j(λS = n+1,λS j\S), (11)where λS = n+ 1 means that λi = n+ 1 for everyi ∈ S and As = [n+1] for every s ∈ S j\S. The pos-terior probability of the maximum detector can becomputed asP(max(λS)≤ n|Xn⋆) = ∑λS∈BS∑λS j\S∈AS j\Sβ j(λS j),(12)where Bs = [n] for every s ∈ S. For the single dam-age detection, the maximum detector and the min-imum detector are equivalent. Since the computa-tion of β j(λS j) only depends on the local kernel andthe received messages, the detectors on each sensorcan be computed independently. This allows multi-ple damages to be diagnosed simultaneously and ina distributed fashion.412th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 20153. SIMULATIONS AND RESULTSFigure 1: Diagram of the ASCE benchmark structure(Johnson et al. (2004))Figure 2: Sensor location and direction of acceler-ation measurements (Nair and Kiremidjian (2007);Johnson et al. (2000))To validate the proposed damage detection algo-rithm, we apply it to the ASCE benchmark structure(Johnson et al. (2004)). The benchmark structure isa four story, two bay by two bay steel braced frame,as shown in Fig. 1. 16 sensors are installed to mea-sure the accelerations. The locations are shown inFig. 2. All the sensors with odd series number aremeasured the acceleration signals on the x-axis andthe rest sensors measure the responses on the y-axis. The ASCE benchmark structure provides anumerical Matlab simulator which allows us to col-lect the acceleration signals with different degreesof freedom, mass distribution and excitations. Inthis paper, we use a 12 degrees of freedom struc-ture with symmetric mass on each floor. In addi-tion, we collect the acceleration signals of ambientvibrations.In this paper, we use the Burg algorithm to esti-mate the AR coefficients. The 1st AR coefficient,θ1, is picked as the DSF. Nair and Kiremidjian(2007) suggests that the AR coefficients are sen-sitive to the chunk size N. To choose the optimalchunk size, we analyze the mean and the standarddeviation of the first three AR coefficients with thesize grows from 3000 to 7000. We find that whenN = 7000, the AR coefficients have the minimumstandard deviation. Therefore, we use N = 7000 asthe chunk size.A widely used criteria for choosing the opti-mal model order is the Akaike information crite-ria (AIC) (Brockwell and Davis (2002),Nair et al.(2006),Noh et al. (2013)). Fig. 3 shows the varia-tion of the AIC values with the AR model order forthe undamaged signals from different sensors. Wecan observe that an AR model with order p = 8 isappropriate for this data set. The residuals are val-idated to be i.i.d with normal distribution and haveno trend.2 4 6 8 10 12 14−1.4−1.0−0.6Model OrderAICSensor 2Sensor 6Sensor 10Sensor 14Figure 3: AIC value against AR model order for theundamaged signalsThe ASCE simulator includes non-damage pat-tern and six pre-defined damage patterns, whichinclude both major damages and minor damages.However, these patterns cannot demonstrate simul-512th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015taneous damages. In order to get the responses as-sociated with multiple simultaneous damages, wemodify the original simulator and introduce the fol-lowing damage patterns (DPs):• DP0: no damage• DP1: all the braces of the 1st floor are removed• DP2: all the braces of the 3rd floor are re-moved• DP3: all the braces of the 1st and 3rd floors areremoved.In the ASCE benchmark structure, all the massesare loaded symmetrically. In addition, after remov-ing the braces, the structure remains symmetric.Therefore, rather than using the data collected byall the sensors, in this study, we only use the re-sponses collected by sensor 2,6,10 and 14. Be-cause the damage patterns are introduced to eachfloor, we assign one damage variable to each floor,i.e. λi for the ith floor. As the damages only happenon the 1st and 3rd floors, we assume that only λ1and λ3 will be the active damage variables in thistest.After assigning the damage variables, we needto investigate how the damage variables affect thesensors. In other word, we need to decide the localdomain and the local kernel of each sensor. Fig. 4shows the box plots of the DSFs with different dam-age patterns. From this figure, we can observe thatλ1 has significant effects on sensor 2 and sensor 6.λ3 affects the DSFs of all the sensors. Although λ2and λ4 are not triggered by the defined DPs, we stillassume it may affect the sensors installed one floorabove and below. Based on these observations, weform the local domains and kernels in Table. 1. Thepolytree graph is shown in Fig. 5. The edge be-tween sensors is both the probabilistic linkage andthe communication link.As discussed above, the likelihood probabilitydepends on the damage variables. Since only λ1and λ3 are active, given λ1 = n1,λ3 = n2, we definethe likelihood probability as follows:P(Xnj |λ1,λ3) =k1−1∏k=1g j(X kj )k2−1∏k=k1f j(X kj )n∏k=k2f 3j (X kj )(13)where k1 is min(n1,n2), k2 is max(n1,n2), f j(X kj )is f 1j (X kj )I(n1 < n2)+ f 2j (X kj )I(n2 < n1), and I(.)is the indicator function. For sensor j, the func-tion g j is the probability density function (PDF)of DP0 and f 1j , f2j , and f3j are the PDFs of DP1-DP3 respectively. The DSF Xnj is i.i.d with oneof them. We assume that all the densities areGaussian densities with different known parame-ters (Nair and Kiremidjian (2007)). In field ap-plications, the distribution parameters can be esti-mated empirically by utilizing historical or simu-lation data. (13) shows that the likelihood prob-ability depends on not only the damage patternsbut also the order of the damage patterns. There-fore, our proposed detectors can apply to multi-stage damages. To keep the summation of the den-sity over the local domain to be one, we need tospread the density function over the inactive vari-ables, i.e. P(Xnj |λ1,λ2 = k,λ3) = 1n+1P(Xnj |λ1,λ3)for all k ∈ [n+1].0.750.80.850.90 1 2 3DPSensor 21st AR coefficient0.90.9511.050 1 2 3DPSensor 61st AR coefficient0.90.9511.050 1 2 3DPSensor 101st AR coefficient1. 1 2 3DPSensor 141st AR coefficientFigure 4: Box plot of the 1st AR coefficient, θ1, fordifferent damage patternsTable 1: Local domains and local kernels of the ASCEbenchmark structuresensor local domain local kernel2 {λ1,λ2,λ3} pi1(λ1)P(Xn2|λ1,λ2,λ3)6 {λ1,λ2,λ3,λ4} pi2(λ2)P(Xn6|λ1,λ2,λ3,λ4)10 {λ2,λ3,λ4} pi3(λ3)P(Xn10|λ2,λ3,λ4)14 {λ3,λ4} pi4(λ4)P(Xn14|λ3,λ4)612th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 20152610 14Figure 5: The polytree graph for the sensor networkinstalled on the benchmark structure. The local domainassociated with each sensor is shown in Table. 1.Fig. 6 shows the plots of the expected delayagainst the maximum allowable probability of falsealarm. The plots are generated byMonte Carlo sim-ulation over 500 replications. All prior distributionsare geometric distributions with ρ j = 0.1. Table 2summarizes the detection rules we test on each sen-sor.Table 2: Detectors on each sensorssensor detector2 min(λ1,λ2,λ3), min(λ1,λ3),max(λ1,λ3), max(λ1), max(λ3)6 min(λ1,λ2,λ3,λ4), min(λ1,λ3),max(λ1,λ3), max(λ1), max(λ3)10 min(λ2,λ3,λ4), max(λ3)14 min(λ3,λ4), max(λ3)To compare the performances of the MP algo-rithm, we introduce another method that uses thesame detection rules but does not exchange mes-sages with the neighbors, i.e. τLOCALS := inf{n ∈N : P(λS ≤ n|Xnj) ≥ 1−α}. We call this methodthe LOCAL method. In. Fig. 6, we can observethat the MP algorithm is generally similar or betterthan the LOCAL algorithm. Specifically, for sensor2 and sensor 6, the minimum detectors of both al-gorithms have similar performances. The LOCALmethod is better than the MP algorithm when α islarge. Then α is small, the MP method is betterthan the LCOAL method. For sensor 2, the detec-tion delays of the LOCAL maximum detectors aremuch larger than those of the MP maximum detec-tors. For sensor 10, the LOCAL maximum detectorfails when α < 10−4. However, the MP maximum0 20 4000.511.5  0 20 4000.511.50 20 4001230 20 4002460 20 400. 20 4000.511.50 20 4000.511.50 20 4000.511.50 20 4002460 20 400. 20 40012340 20 40024680 20 40012340 20 400246MPLOCALFigure 6: Plots of the slope 1− logα E [τS −φ |τS ≥ φ ]against − logα . The plots in the 1st row are the de-tectors of sensor 2. The plots in the 2nd row are thedetectors of sensor 6. The first two plots in the 3rd roware the detectors of sensor 10 and the last two plots arethe detectors of sensor 14. The plots from left to rightfollow the order of the detectors in Table. 2.detector has consistent performance. For sensor 14,the LOCAL detectors have very high delay whenα is large. Meanwhile, the detection delay of MPdecreases as α → 0. In summary, the MP algo-rithm has reliable performance compared with theLOCAL detectors for different detectors. It meansthe global information helps to reduce the detectiondelay. Also, the MP detectors are more robust tothe false alarm.4. CONCLUSIONIn this paper, we propose a sequential damage de-tection algorithm that detects the damages based onthe AR model coefficients. The message passingprotocol computes the posterior probabilities con-ditioning on all the observed DSFs in the networkefficiently and allows the detectors to identify mul-tiple damages simultaneously and in a distributedfashion. In addition, the proposed detectors canachieve minimum detection delay with the desiredfalse alarm rate. The numerical validation on theASCE benchmark structure shows the robustness to712th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015various false alarm rates and the consistence withdifferent types of detectors. Moreover, the numer-ical results indicate that the global information im-proves the performance of the damage detection.To further improve this algorithm, it can be ap-plied to other types of damages, such as removingone brace of each floor. In Fig. 6, we can observethat when α → 0, the MP detectors converge tosome constants. It is worth exploring the asymp-totic property of the detectors. Last, we need toapply the algorithm to more field data and differentstructures for further validation.5. REFERENCESAji, S. M. and McEliece, R. J. (2000). “The generalizeddistributive law.” Information Theory, IEEE Transac-tions on, 46(2), 325–343.Amini, A. A. and Nguyen, X. 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