research article model selection approach for distributed...

Research ArticleModel Selection Approach for Distributed Fault Detection inWireless Sensor Networks

Mrinal Nandi1 Anup Dewanji2 Bimal Roy2 and Santanu Sarkar3

1 Department of Statistics West Bengal State University Barasat India2 ASD Indian Statistical Institute 203 B T Road Kolkata 700 108 India3 Chennai Mathematical Institute Chennai India

Correspondence should be addressed to Mrinal Nandi mrinalnandi1gmailcom

Received 2 April 2013 Revised 13 November 2013 Accepted 14 November 2013 Published 20 January 2014

Academic Editor Shuai Li

Copyright copy 2014 Mrinal Nandi et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Sensor networks aim at monitoring their surroundings for event detection and object tracking But due to failure or death ofsensors false signal can be transmitted In this paper we consider the problems of distributed fault detection in wireless sensornetwork (WSN) In particular we consider how to take decision regarding fault detection in a noisy environment as a result of falsedetection or false response of event by some sensors where the sensors are placed at the center of regular hexagons and the eventcan occur at only one hexagonWe propose fault detection schemes that explicitly introduce the error probabilities into the optimalevent detection process We introduce two types of detection probabilities one for the center node where the event occurs and theother one for the adjacent nodesThis second type of detection probability is new in sensor network literatureWe develop schemesunder the model selection procedure and multiple model selection procedure and use the concept of Bayesian model averaging toidentify a set of likely fault sensors and obtain an average predictive error

1 Introduction

Traditional and existing sensor-actuator networks use wiredcommunication whereas wireless sensor networks (WSN)provide radically new communication and networkingparadigms andmyriad new applicationsThewireless sensorshave small size low battery capacity nonrenewable powersupply small processing power limited buffer capacity andlow-power radio They may measure distance directionspeed humidity wind speed soil makeup temperaturechemicals light and various other parameters

Recent advancements in wireless communications andelectronics have enabled the development of low-cost WSNA WSN usually consists of a large number of small sensornodes which are equipped with one or more sensors someprocessing circuit and a wireless transceiver One of theunique features of a WSN is random deployment in inac-cessible terrains and cooperative effort that offers unprece-dented opportunities for a broad spectrum of civilian andmilitary applications such as industrial automation military

surveillance national security and emergency health care [1ndash3] Sensor networks are also useful in detecting topologicalevents such as forest fires [4]

Sensor networks aim at monitoring their surroundingsfor event detection and object tracking [1 5] Because ofthis surveillance goal coverage is the functional basis of anysensor network In order to fulfill its designated tasks a sensornetworkmust fully cover theRegion of Interest (ROI)withoutleaving any internal sensing hole [6ndash9] So far a number ofmovement-assisted sensor placement algorithms have beenproposed An exclusive survey on these topics is presentedby Li et al [10] On the other hand sensor could die or failat runtime for various reasons such as power depletion andhardware defects So even after the ROI is fully covered by thesensors wrong information can be communicated by somesensors or sensors may fail to detect the event due to noiseor obstructions Chen et al [11] have proposed a distributedlocalized fault detection algorithm for WSN where eachsensor identifies its own status to be either good or faultyand the claim is then supported or reverted by its neighbors

Hindawi Publishing CorporationInternational Journal of Distributed Sensor NetworksVolume 2014 Article ID 148234 12 pageshttpdxdoiorg1011552014148234

2 International Journal of Distributed Sensor Networks

The proposed algorithm is analyzed using a probabilisticapproach Sharma et al [12] have characterized the differenttypes of fault and proposed a different algorithm for faultdetection considering different types of fault Some of themethods are statistical like using histogram and so forthBoth works can only detect the faulty sensors but not theevent

One of the important sensor network applications ismonitoring inaccessible environments Sensor networks areused to determine event regions and boundaries in theenvironment with a distinguishable characteristic [13ndash15]The basic idea of distributed detection [16] is to have eachof the independent sensors make a local decision (typicallya binary one ie an event occurs or not) and then combinethese decisions at a fusion sensor (the sensor which collectsthe local information and takes the decision) or at a basestation to generate a global decision

A closely related area is neural network Several worksare there in the literature on this area Li and Qin [17]find a feasible solution to a class of nonlinear inequalitiesdefined on a graph proposing a recurrent neural networkTheconvergence of the neural network and the solution feasibilityto the defined problem are both theoretically proven Theyproposed neural network features as a parallel computingmechanism and a distributed topology isomorphic to thecorresponding graph which is suitable for distributed real-time computation The proposed neural network is appliedto range-free localization of WSNs Li et al [18] show thatfeasible solution set to the same problem is often infinityand Laplacian eigenmap is used as heuristic information togain better performance in the solution A continuous-timeprojected neural network and the corresponding discrete-time projected neural network are both given to tacklethis problem iteratively The effectiveness of the proposedneural networks is tested and compared with others via itsapplications in the range-free localization of WSNs Locationinformation is useful for mobile phones There exists adilemma between the relatively high price of GPS devicesand the dependence of location information acquisition onGPS for most phones in current stage Li et al [19] formulatethe problem as an optimization problem defined on theBluetooth networkThe solution to this optimization problemis not unique Heuristic information is employed to improvethe performance of the result in the feasible set They usedrecurrent neural networks to solve the problem distributivelyin real time The convergence of the neural network andthe solution feasibility to the defined problem are boththeoretically proven The hardware implementation of theproposed neural network is also explored in this paper

Distributed algorithms are also used for a networkdynamic system Li et al [20 21] studied the decentral-ized control and kinematic control of multiple redundantmanipulators for the cooperative task execution problemThe problem is formulated as a constrained quadratic pro-gramming problem and then a recurrent neural networkwith independent modules is proposed to solve the problemin a distributed manner They proposed a novel strategycapable of solving the problem even though there exist somemanipulators unable to access the command signal directly

Another related area is the winner-take-all (WTA) com-petition which is widely observed in both inanimate andbiological media and society Many mathematical modelsare proposed to describe the phenomena discovered indifferent fields These models are capable of demonstratingthe WTA competition Li et al [22 23] make steps in thatdirection and present a simple model which produces theWTA competition by taking advantage of selective positive-negative feedback through the interaction of neurons via 119901-norm They also present a class of recurrent neural networksto solve quadratic programming problems Different frommost existing recurrent neural networks for solving quadraticprogramming problems the proposed neural networkmodelconverges in finite time and the activation function is notrequired to be a hard-limiting function for finite convergencetime The stability finite-time convergence property andthe optimality of the proposed neural network for solvingthe original quadratic programming problem are proven intheory Extensive simulations are performed to evaluate theperformance of the neural networkwith different parametersIn addition the proposed neural network is applied to solvingthe k-winner-take-all (k-WTA) problem

11 Our Motivation In this paper we are interested in oneparticular query determining event in the environment (ieROI) with a distinguishable characteristic We assume theROI to be partitioned into suitable number of congruentregular hexagonal cells (ie we can think of ROI as a regularhexagonal grid) This physical structure of ROI is not arequirement for the theoretical analysis and we can do thesimilar analysis with another structure also Suppose thatsensors are placed a priori at the center (which are knownas nodes) of every hexagon of the grid We assume that thesensors are connected to its adjacent sensor nodes in the sensethat a hexagon will be strongly covered by its center node andweakly covered by the adjacent nodes If event occurs in thehexagon where a particular sensor lies then that particularsensor can detect the event with a greater probability whereasif event occurs in any adjacent hexagon then the particularsensor can detect the event with a lesser probability Henceonly one node (center node of the event hexagon) candetect the event hexagon with greater probability say 119901

1 and

adjacent nodes (six for interior nodes and less for boundarynodes) can detect the event hexagon with lesser probabilitysay 1199012 with 119901

1gt 1199012 We assume that no other sensor can

detect the event hexagon In this paper unlike the previousworks we assume that if the event occurs then it occursat only one hexagon of the grid which will be known asevent hexagon and there is no fusion sensor All sensorscan communicate with the base station and the base stationtakes the decision about the query As an example considera network of devices that are capable of sensing mines orbombs if we assume that a fewmines or bombsmay be placedon a particular area of ROI Information from these devicescan be sent to a nearby police station or a central facilityThen an important query in this situation could be whethera particular hexagon is the event hexagon or not (ie minesor bombs are placed or not)

International Journal of Distributed Sensor Networks 3

One fundamental challenge in the event detection prob-lem for a sensor network is the detection accuracy which isdisturbed by the noise associated with the detection and thereliability of sensor nodes A sensor may fail to detect theevent due to natural obstruction or any other causes Afterdetecting the event a sensor can send false message to thebase station due to some technical reasons The sensors areusually low-end inexpensive devices and sometimes exhibitunreliable behavior For example a faulty sensor node mayissue an alarm even though it has not received any signal forevent or it cannot detect any event and vice versa Moreovera sensor may be dead in which case the sensor cannot sendany alarm

12 Previous Work and Our Contribution Lou et al [24]consider two important problems for distributed fault detec-tion in WSN (1) how to address both the noise-relatedmeasurement error and sensor fault simultaneously in faultdetection and (2) how to choose a proper neighborhood size119899 for a sensor node in fault correction such that the energycould be conserved They propose a fault detection schemethat explicitly introduces the sensor fault probability into theoptimal event detection process They show that the optimaldetection error decreases exponentially with the increase ofthe neighborhood size

Krishnamachari and Iyengar [13] propose a distributedsolution for canonical task in WSN (ie the binary detectionof interesting environmental events)They explicitly take intoaccount the possibility of sensor measurement faults anddevelop a distributed Bayesian algorithm for detecting andcorrecting such faults

Nandi et al [25] consider the problem of distributedfault detection in wireless sensor network (WSN) wherethe sensors are placed at the center of a particular square(or hexagon) of the grid covering the ROI They proposedfault detection schemes that explicitly introduce the errorprobabilities into the optimal event detection process Theydeveloped the schemes under the consideration of Neyman-Pearson hypothesis test and Bayes test They also calculatetype I and type II errors for different values of the parameters

In almost all the previous works except [25] authorsassume that event occurs over a region and there are fusionsensors that collect the information locally and take adecision Since they do not introduce the concept of basestation there is no concept of response probability Also theyassume that information is spatially correlated Unlike theprevious work in this paper we assume that if event occursthen it occurs at only one cell of the ROI and there is nofusion sensor All the sensors send information to the basestation We introduce the probability model in two differentstages firstly when a sensor detects the event and secondlywhen a sensor sends the message to the base station In theprevious works only one type of detection probability hasbeen introduced to simulate the different error probabilitiesfor some specific values of the parameters In this paperwe introduce two different detection probabilities and obtainanalytically the exact test and estimate the error probabilitiesby simulation In almost all the previous works authors

assume the ROI to be a square grid The hexagonal grid isbetter in the sense that a minimum number of sensors arerequired to cover the entire ROI [26]

In our theoretical analysis the sensor fault probabilitiesare introduced into the optimal event detection process Weapply model selection approach multiple model selectionapproach and Bayesian model averaging methods [27 28]to find a solution of the problem We develop the schemesusing the model selection technique We calculate differenterror probabilities and find some theoretical results

In all previous works the authors assume only one detec-tion probability We introduce two detection probabilities 119901

1

and 1199012 one for the center node and other for the adjacent

nodes Even if the center node may fail to detect the eventthe adjacent nodes may detect the event and vice versaWe consider these probabilities and show that in varioussituations the adjacent nodes play key role to detect the eventOne can introduce more detection probabilities and analyzethe situation in similar manner

The parameters 1199011and 119901

2 the detection probabilities of

a sensor and error probabilities (see Section 3) cannot beestimated from the real life situations but need to be estimatedbeforehand by some experimentationThe prior probabilitiesof various events also cannot be estimated but may be knownin some cases Finally we calculate the error probabilitiesnumerically for some values of the parameters of our modeland make some concluding remarks analyzing the results

2 Statement of the Problem and Assumptions

In this section we describe the problem in more specificterms and state the assumptions that we make

Sensors are deployed or manually placed over ROI toperform event detection (ie to detect whether an event ofinterest has happened or not) in ROI If sensors are deployedfrom air then using actuator-assisted sensor placement or bymovement-assisted sensor placement sensors are so placedthat sensor network covers the entire ROI This ROI ispartitioned into suitable number of regular hexagons (ie wecan think of the ROI as a regular hexagonal grid) as shown inFigure 1 Sensors are placed a priori at every center (which areknown as nodes) of the regular hexagons Sensors have twodetection probabilities The sensor network covers the entireROI and there is only one event hexagon as discussed before

Each sensor node determines its location through beaconpositioning mechanisms [29] or by exploiting the GlobalPositioning System (GPS) Through a broadcast or acknowl-edge protocol each sensor node is also able to locate theneighbors within its communication radius Sensors are alsoable to communicate with the base station Base station willtake the decision In this paper we assume that event occursat one particular hexagon of the grid which will be known asevent hexagon or event does not occur (in that case we saythat ROI is normal) All sensors can communicate with thebase station and base station takes the decision by combiningthe information received from all the sensors

There are two phases in the whole process The first oneis detection phase when the sensor at the center of a regular


Adjacent node

Adjacent node

Adjacent node

Adjacent node

Adjacent node

Adjacent node

Center node

Figure 1 Nodes placed in centers whenROI partitioned into regularhexagons

hexagon tries to detect the event The sensor at the center ofthe event hexagon can detect the event hexagon with greaterprobability 119901

1and the sensors at the adjacent nodes (see

Figure 1) can detect the event hexagon with lesser probability1199012 We also assume that there is a prior probability that a

particular hexagon is an event hexagon The next phase isresponse phase in which sensors send message to the basestation Even if the event hexagon is detected by a sensorit may not respond (ie send message to the base stationthat no event occurred in that cell and the neighboring cellsdue to some technical fault) with some probability then wesay that the sensor is a faulty sensor Conversely if eventhexagon is not detected or there is no event hexagon at all(ie ROI is normal) then also a faulty sensor can send thewrong information to the base station with some probabilityA sensor is said to be a dead sensor if the sensor does notwork A dead sensor sends no response in either cases

Each sensor sends information to the base station As thesensors may send wrong information the base station takesthe important role in identifying the event hexagon Basestation will collect all information and take a decision aboutthe event hexagon according to a rule which we have to findout Our job is to find a rule for the base station such that basestation works most efficiently

21 Notations and Assumption Our problem is to developa strategy for the base station to take decision about eventhexagon (ie which hexagon of the ROI is the event hexagonif at all) Let 119877 be the set of all nodes For 119873 isin 119877 define119861(119873) as the set of adjacent node(s) of119873 and let 119896(119873) be thenumber of adjacent node(s) of119873 Hence 0 le 119896(119873) le 6 Calla node119873 interior if 119896(119873) = 6 Let 119878

119873be the sensor which is

placed at the node 119873 and let 119867119873be the hexagon where the

node 119873 is placed (ie 119873 is the center of 119867119873) For 119873 isin 119877

let 119883119873denote the true status of the node119873 That is 119883

119873= 1

if event occurs at 119867119873 and 0 otherwise Also define 119884

119873= 0

if 119878119873detects no event and 1 if 119878

119873detects the event in 119867

119873

or1198671198731015840 for1198731015840 isin 119861(119873) Finally define 119885

119873= 0 if 119878

119873does not

respond that is the sensor informs the base station that eventdoes not occur at 119867

119873or 1198671198731015840 for 1198731015840 isin 119861(119873) and 119885

119873= 1 if

119878119873responds that is the sensor 119878

119873informs the base station

that the event has occurred in119867119873or1198671198731015840 for1198731015840 isin 119861(119873)

Nowwemake one natural assumption that once detectionphase is completed response of a sensor depends only onwhat it detects but not on whether the event has actuallyoccurred or not that is 119875(119885

119873= 119896 | 119884

119873 119883119873) = 119875(119885

119873=

119896 | 119884119873) for 119896 = 0 1 We also assume that the sensors work

independently and identicallySince we assume that there is at most one event hexagon

sum119873isin119877

119883119873= 1 or 0

The possible true scenarios are therefore represented bythe following |119877| + 1 different models

M0 (119883119873= 0 for all119873 isin 119877)

and for each119873 isin 119877

M119873 (119883119873= 1 and119883

1198731015840 = 0 for all1198731015840 isin 119877 119873)

Let Pr(M0) = 119875(ROI is normal) = 119901norm and for all119873 isin

119877 Pr(M119873) = Pr(event occurs at the hexagon 119867

119873) = 119901119873

In particular we may assume 119901119873rsquos to be the same for all

119873 We denote any probability under the modelM0as 119875M

0

(sdot)

and under the modelM119873as 119875M

119873

(sdot)We also make the followings assumptions

(i) For all119873 isin 119877 119875M0

(119884119873= 1) = 0 and 119875M

119873

(119884119873= 1) =

1199011

(ii) For all 1198731015840

isin 119861(119873) 119875M119873

(1198841198731015840 = 1) = 119901

2

and for all1198731015840 isin 119877 [119861(119873) cup 119873] 119875M119873

(1198841198731015840 = 1) = 0

(iii) For all119873 isin 119877 119875(119885119873= 1 | 119884

119873= 1) = 119901

119888and 119875(119885

119873=

1 | 119884119873= 0) = 119901

119908

(iv) 119885119873and 119884

1198731015840 are independent for119873 =119873

1015840

(v) The responses from different nodes are independentunder a particular model that is 119885

119873rsquos are indepen-

dent underM1198731015840 for a fixed1198731015840 isin 119877

3 Theoretical Analysis of Fault Detection

In this section we discuss some theoretical results In realsituations |119877| may be very large Given the network of thesensor nodes and some prior knowledge about the nature ofevent one may have fairly good idea about the set of feasibleregions for the event Formally instead of all possible modelsone may be able to restrict to a set containing all the feasiblemodels For example if the event is known to take place in aparticular region we can restrict our models accordingly

31 Model Selection Approach Consider

forall119873 isin 119877 119875M0

(119885119873= 1)

= 119875M0

(119885119873= 1 | 119884

119873= 0) 119875M

0

(119884119873= 0)


+ 119875M0

(119885119873= 1 | 119884

119873= 1) 119875M

0

(119884119873= 1)

= 119875 (119885119873= 1 | 119884

119873= 0) 119875M

0

(119884119873= 0)

+ 119875 (119885119873= 1 | 119884

119873= 1) 119875M

0

(119884119873= 1) = 119901

119908

(1)

Hence under the modelM0 119885119873follows Ber(119901

119908) for all119873 isin

119877 and the likelihood of the data 119885119873= 119911119873 for all 119873 isin 119877

under the modelM0 is

1198710= 119875M

0

(119885119873= 119911119873 forall119873 isin 119877)

= prod

119873isin119877

119901119911119873

119908(1 minus 119901

119908)(1minus119911119873)

= (119901119908)sum119873isin119877119911119873

times (1 minus 119901119908)sum119873isin119877(1minus119911119873)

(2)

So ln 1198710= sum119873isin119877

119911119873ln119901119908+ sum119873isin119877

(1 minus 119911119873) ln(1 minus 119901

119908)

For any119873 isin 119877 we have119875M119873

(119885119873= 1)

= 119875M119873

(119885119873= 1 | 119884

119873= 0) 119875M

119873

(119884119873= 0)

+ 119875M119873

(119885119873= 1 | 119884

119873= 1) 119875M

119873

(119884119873= 1)

= 119875 (119885119873= 1 | 119884

119873= 0) 119875M

119873

(119884119873= 0)

+ 119875 (119885119873= 1 | 119884

119873= 1) 119875M

119873

(119884119873= 1)

= 119901119908(1 minus 119901

1) + 1199011198881199011= 1199011(119901119888minus 119901119908) + 119901119908

= 1198751 say

(3)

Hence for all 119873 isin 119877 under M119873 119885119873

follows Ber(1198751)

Similarly for all1198731015840 isin 119861(119873) underM119873 1198851198731015840 follows Ber(119875

2)

where 1198752= 1199012(119901119888minus 119901119908) + 119901119908and under M

119873 1198851198731015840 follows

Ber(119901119908) for all 1198731015840 isin 119877 [119861(119873) cup 119873] Note that 119875

1gt 1198752

since 1199011gt 1199012 Hence the likelihood for the modelM

119873 given

1198851198731015840 = 1199111198731015840 1198731015840isin 119877 is

119871119873= 119875M

119873

(1198851198731015840 = 1199111198731015840 forall1198731015840isin 119877)

= 119875119911119873

1(1 minus 119875

1)(1minus119911119873)

Π1198731015840isin119861(119873)

1198751199111198731015840

2(1 minus 119875

2)(1minus1199111198731015840 )

times Π1198731015840isin119877[119861(119873)cup119873]

1199011199111198731015840

119908 (1 minus 119901119908)(1minus1199111198731015840 )

= 119875119911119873

1(1 minus 119875

1)(1minus119911119873)

119875sum1198731015840isin119861(119873)119911

1198731015840

2(1 minus 119875

2)sum1198731015840isin119861(119873)(1minus1199111198731015840 )

times 119901sum1198731015840isin119877[119861(119873)cup119873]119911

1198731015840

119908 (1 minus 119901119908)sum1198731015840isin119877[119861(119873)cup119873]

(1minus1199111198731015840 )

(4)

Let 119879119873= sum1198731015840isin119861(119873)

1198851198731015840 so thatsum

1198731015840isin119861(119873)

(1 minus 1198851198731015840) = 119896(119873) minus

119879119873with the corresponding observed values denoted by

119905119873= sum

1198731015840isin119861(119873)

1199111198731015840 sum

1198731015840isin119861(119873)

(1 minus 1199111198731015840) = 119896 (119873) minus 119905

119873 (5)

Therefore

ln 119871119873= 119911119873ln1198751+ (1 minus 119911

119873) ln (1 minus 119875

1)

+ 119905119873ln1198752+ (119896 (119873) minus 119905

119873) ln (1 minus 119875

2)

+ sum

1198731015840isin119877[119861(119873)cup119873]

119911119873ln119901119908

+ sum

1198731015840isin119877[119861(119873)cup119873]

(1 minus 119911119873) ln (119901

119908(1 minus 119901

119908))

= ln 1198710+ 119911119873ln 1198751

119901119908

+ (1 minus 119911119873) ln 1 minus 119875

1

1 minus 119901119908

+ 119905119873ln 1198752

119901119908

+ (119896 (119873) minus 119905119873) ln 1 minus 119875

2

1 minus 119901119908

= ln 1198710+ 119911119873ln

1198751(1 minus 119901

119908)

119901119908(1 minus 119875

1)+ 119905119873ln

1198752(1 minus 119901

119908)

119901119908(1 minus 119875

2)

+ ln 1 minus 1198751

1 minus 119901119908

+ 119896 (119873) ln1 minus 1198752

1 minus 119901119908

= 119886 + 119887 (119888119911119873+ 119905119873minus 119889119896 (119873)) say

(6)

where

119886 = ln 1198710+ ln 1 minus 119875

1

1 minus 119901119908

119887 = ln1198752(1 minus 119901

119908)

119901119908(1 minus 119875

2)gt 0

119888 =ln (1198751(1 minus 119901

119908) 119901119908(1 minus 119875

1))

ln (1198752(1 minus 119901

119908) 119901119908(1 minus 119875

2))

119889 =ln ((1 minus 119901

119908) (1 minus 119875

2))

ln (1198752(1 minus 119901

119908) 119901119908(1 minus 119875

2))

(7)

are independent of NIn model selection approach the model resulting in the

maximum value of the likelihood is selected Note that sincethere is no parameter being estimated this is equivalent tothe well-known Akaike Information Criterion (AIC) [30]Therefore the base station will accept the modelM

0if

= ln 1 minus 1198751

1 minus 119901119908

+ 119887 (119888119911119873+ 119905119873minus 119889119896 (119873)) lt 0

forall119873 isin 119877

(8)

Otherwise as 119887 is positive accept the model M119873for which

(119888119911119873+ 119905119873minus 119889119896(119873)) is maximum among all 119873 isin 119877 If values

of (119888119911119873+ 119905119873minus 119889119896(119873)) are equal for more than one 119873 then

we can select one of the corresponding models with equalprobability If we want to maximize the likelihood for themodelsM

119873corresponding to the interior nodes only so that

119896(119873) is fixed then we need to maximize (119888119911119873+119905119873) among all

119873 isin 119877

32 Multiple Model Selection Instead of selecting one partic-ularmodel onemaywant to selectmore than onemodel with


approximately similar log likelihood values to the maximumone We can consider the set of models

M119870

119871119870

max119873isin119877

119871119873

gt 119862 (9)

where 0 lt 119862 lt 1 is a suitable constant close to 1 This 119862is usually chosen according to the resource available Thisis similar to the idea of Occamrsquos window in the context ofBayesian model selection [27]This may be interpreted as theinterval estimation for the true model

Note that119871119873is an increasing function of 119888119911

119873+119905119873minus119889119896(119873)

as 119887 is positive We consider only the following set of models

M119870 119876119870gt 119862lowastsdotmax119873isin119877

119876119873 (10)

where119876119873= 119888119911119873+119905119873minus119889119896(119873) for all119873 isin 119877 with 0 lt 119862

lowastlt 1

In particular if we consider the interior nodes only then weconsider the set of models given by

M119870 119888119911119870+ 119905119870gt 119862lowastsdotmax119873isin119877

119888119911119873+ 119905119873 (11)

We can select multiple models using some other criteria Onesuch may be to select all the models (one or more) for whichthe maximum value of the likelihood is attained LetNmax bethe set of nodes corresponding to all these models includingldquo119873 = 0rdquo corresponding toM

0if it has the maximum value of

the likelihood Then this method selects all the models M119873

with 119873 isin Nmax By another criterion one may select themodels M

1198731015840 for 1198731015840 isin Nmax cup [cup

119873isinNmax119861(119873)] that is 1198731015840

is a node inNmax or any of the neighboring nodes of a nodeinNmax Note that 119861(119873) for119873 = 0 is the empty set One cancombine these two types of criteria and come up with manyothers

33 Bayesian Model Averaging Bayesian model averagingis an effective method to solve a decision problem whenthere are many alternative hypotheses or models whichare complicated [27] Suppose that M

1M2 M

119896are

the models considered and 119863 denotes the given data Theposterior probability for modelM

119896is given by

Pr (M119896| 119863) =

Pr (119863 | M119896)Pr (M

119896)

sumPr (119863 | M119897)Pr (M

119897) (12)

where Pr(119863 | M119896) denotes the probability of observing data

119863 under themodelM119896(which is essentially the likelihood 119871

119896

underM119896) and Pr(M

119896) is the prior probability thatM

119896is the

true model (assuming that one of the models is true)In this work the data 119863 is 119885

119873= 119911119873

119873 isin 119877 and themodels areM

0M119873 119873 isin 119877 as defined in Section 32 Hence

the posterior probability for modelM119873is

Pr (M119873| 119885119873= 119911119873 119873 isin 119877) =

119901119873119871119873

sum119897isin119877

119901119897119871119897+ 119901norm1198710

and that forM0is

119901norm1198710sum119897isin119877

119901119897119871119897+ 119901norm1198710

(13)

We select the modelM0if 119901norm1198710 is greater than 119901119873119871119873

for all 119873 isin 119877 otherwise select M119873

for which 119901119873119871119873

ismaximum among all119873 isin 119877 Hence if 119901

119873rsquos are all equal then

Bayesian approach is the same as the likelihood approach

4 Some Important Considerations andError Probabilities

In this section we consider some important issues related tothe problemof fault detection and the proposedmethodologyincluding calculation of errors (eg false detection etc) anddetection probabilities

The following probabilities give some idea about therole of neighboring nodes along with the center node indetection or false detection of event For example 119875M

0

(119879119873=

0 119885119873

= 1) gives the probability of a false detection by the119873th node and not by the neighboring nodes while119875M

119873

(119879119873=

6 119885119873

= 0) gives the probability of a false negative by the119873th node with all the neighboring nodes detecting the eventSince given a particular model 119879

119873and 119885

119873are independent

calculation of such probabilities is simple as given in thefollowing For any119873 isin 119877 and 119894 = 0 1 119896(119873)

(1) 119875M0

(119879119873= 119894 119885

119873= 0) = ( 119896(119873)

119894) 119901119894

119908(1 minus 119901

119908)119896(119873)minus119894+1

(2) 119875M0

(119879119873= 119894 119885

119873= 1) = ( 119896(119873)

119894) 119901119894+1

119908(1 minus 119901

119908)119896(119873)minus119894

(3) 119875M119873

(119879119873

= 119894 119885119873

= 0) = ( 119896(119873)119894) 119875119894

2(1 minus 119875

2)119896(119873)minus119894

(1 minus

1198751)

(4) 119875M119873

(119879119873= 119894 119885

119873= 1) = ( 119896(119873)

119894) 119875119894

2(1 minus 119875

2)119896(119873)minus119894

1198751

Note that for119873 isin 119877

119875M0

(119871119873gt 1198710)

= 119875M0

(ln 119871119873gt ln 119871

0)

= 119875M0

(119885119873ln

1198751(1 minus 119901

119908)

119901119908(1 minus 119875

1)+ 119879119873ln

1198752(1 minus 119901

119908)

119901119908(1 minus 119875

2)

+ ln 1 minus 1198751

1 minus 119901119908

+ 119896 (119873) ln1 minus 1198752

1 minus 119901119908

gt 0)

= 119875M0

(119885119873ln

1198751(1 minus 119901

119908)

119901119908(1 minus 119875

1)+ 119879119873ln

1198752(1 minus 119901

119908)

119901119908(1 minus 119875

2)

gt 119896 (119873) ln1 minus 119901119908

1 minus 1198752

+ ln1 minus 119901119908

1 minus 1198751

)

(14)

which can be numerically obtained using the joint distribu-tion of 119879

119873and 119885

119873under the model M

0 The maximum of

these probabilities over all 119873 gives a lower bound for theprobability that a node is considered to be an event node


when the ROI is normal On the other hand the sum over all119873 gives an upper bound for the same Similarly for119873 isin 119877

119875M119873

(119871119873lt 1198710)

= 119875M119873

(119885119873ln

1198751(1 minus 119901

119908)

119901119908(1 minus 119875

1)+ 119879119873ln

1198752(1 minus 119901

119908)

119901119908(1 minus 119875

2)

lt 119896 (119873) ln1 minus 119901119908

1 minus 1198752

+ ln1 minus 119901119908

1 minus 1198751

)

(15)

which can be again numerically obtained using the jointdistribution of 119879

119873and 119885


119873 This

probability gives some idea about the error that when 119873thnode is the event node and it is not detected

As noted in Section 31 we select the model M119873

forwhich 119876

119873is the maximum for119873 isin 119877 The random variable

119876119873is therefore of some interest the distribution of which

under different models is useful in calculating many errorprobabilities We first find the distribution of 119876

119873under the

model M119873 Note that 119876

119873takes values 119888119894 + 119895 minus 119889119896(119873)

corresponding to 119885119873

= 119894 and 119879119873

= 119895 for 119894 = 0 1 and 119895 =

0 1 2 119896(119873) Assume that for convenience the values of119876119873for different 119894 and 119895 are all distinct Therefore for 119894 = 0 1

and 119895 = 0 1 119896(119873)

119875M119873

(119876119873= 119888119894 + 119895 minus 119889119896 (119873))

= (119896 (119873)

119895) (1198751)119894

(1 minus 1198751)(1minus119894)

(1198752)119895

(1 minus 1198752)(119896(119873)minus119895)

119875M0

(119876119873= 119888119894 + 119895 minus 119889119896 (119873))

= (119896 (119873)

119895) (119901119908)119894+119895

(1 minus 119901119908)(1minus119894+119896(119873)minus119895)

(16)

For 1198731015840 isin 119861(119873) or 1198731015840 isin 119877 [119861(119873) cup 119873] one can find119875M1198731015840(119876119873= 119888119894 + 119895 minus 119889119896(119873)) in similar manner although the

calculation is very tedious as there aremany subcases Ideallyone is interested in probability of errors occurring at the levelof base station For example the two important errors are(1) not selecting M

0when M

0is true (false positive) and

(2) selecting M0when M

119873is true for some 119873 isin 119877 (false

negative) Theoretical calculation of these error probabilitiesis complicated We therefore use simulation technique toestimate these and similar error probabilities

5 Simulation Study

We consider a 32 times 32 hexagonal grid and we run theprogramme 10000 times The simulation is performed usingthe 119862-code and required random numbers are generatedusing the standard 119862-library

In our simulation study we consider different criteriaas discussed in Sections 31 and 32 for estimating the errorprobabilities or equivalently the success rate First considerthe probability of selectingM

0 when it is true Let 119878

1denote

the proportion of correct detection of normal situationwhen model M

0is true using the model selection method

of Section 31 That is 1198781gives an estimate of 119875M

0

(0 isin

Nmax and 0 is selected by randomization)Then 1minus1198781gives

an estimate of the false positive rateWhen M

119873is true for some 119873 isin 119877 let 119878

2denote the

proportion of correct decision for the event node using themodel selection method of Section 31 so that it estimates119875M119873

(119873 isin Nmax and is selected by randomization) Notethat for each simulation run the event hexagon is chosenrandomly so that 119878

2gives an average value over all 119873 In

this context this probability is the same for all the interiornodes Then 1 minus 119878

2gives an estimate of the corresponding

error probability of not selectingM119873 when it is true

Note that in this problem of fault detection with a singleevent node the likelihood value for a given observed dataconfiguration may be equal for more than one modelsTherefore quite often the maximum value of the likelihoodmay be attained bymore than onemodelThemodel selectionmethod of Section 31 which selects one of these modelsrandomly in such cases may often not select the correctmodel Therefore the method of Section 32 which selectsmore than one model having similar likelihood value maybe preferred and will have better chance of selecting thecorrect model We now consider some of those methods inthe following

Let us first consider the method in which all the modelscorresponding to the maximum value of the likelihood areselected Let 119878

3denote the proportion of correct selection

of the model M119873 when it is true by this method Then

1198783estimates the probability 119875M

119873

(119873 isin Nmax) which isalways more than or equal to the quantity estimated by1198782 as remarked before We also consider the method in

which all the models having maximum likelihood along withtheir neighborhood models are selected A model M

1198731015840 is a

neighborhoodmodel of themodelM119873if1198731015840 is a neighboring

node of 119873 If 1198784denotes the proportion of correct selection

of the model M119873 when it is true by this method then

1198784estimates 119875M

119873

(119873 isin Nmax cup cup1198731015840isinNmax

119861(1198731015840)) Clearly

1198784

ge 1198783

ge 1198782 Similarly if 119878

5denotes the proportion

of correct selection of the model M119873 when it is true by

selecting all those models with likelihood value being morethan 90 of the maximum likelihood (ie the method ofSection 32 with 119862 = 09) then 119878

5estimates the probability

119875M119873

(119871119873gt 09119871max) with 119871max denoting the maximum value

of the likelihoodSuppose that 119873

119894denotes the average number of selected

nodes to be searched corresponding to 119878119894 119894 = 1 2 5

Clearly 1198731= 1 minus 119878

1because we need no search when M

0

is selected When event occurs and we consider only one 119873from Nmax we need at most one search (since no searchis needed if M

0is selected) and we have 119873

2le 1 In our

simulation we find 1198732= 1 in all the cases which means

that in simulation M0has not been selected when event

occurred Note that 1198733

ge 1 since we consider all 119873rsquos inNmax for searching Again as before 1198734 gt 119873

3ge 1 ge 119873

2

Also by definition1198735ge 1 Table 1 presents the different 119878

119894rsquos

and119873119894rsquos based on simulation for different values of 119901

1 1199012 119901119888

and 119901119908with 119901

1and 119901

119888taking values 09 and 099 119901

119908taking

values 001 and 0001 and 1199012taking values 00 03 04 05

and 06 The choice of 1199011and 119901

119888reflects the corresponding

high probability whereas that of 119901119908reflects small probability


which is desirable in a good sensor Since the primary interestis to study the effect of detection by neighboring nodes weconsider 119901

2as 0 (which means that there is no effect of

neighboring nodes) and some positive values less than 1199011

Note that the probability of correct detection under M0

depends only on 119901119908This is also evident in Table 1 Intuitively

if 119901119908is high then the proportion 119878

1of correct detection in

normal situation is low In Table 1 we see that 1198781is 0 for

119901119908

= 001 varies from 035 to 037 for 119901119908

= 0001 andvaries from 090 to 091 for 119901

119908= 00001 (not shown in

Table 1) If we consider smaller value of 119901119908 then the success

probability 1198781will be higher Hence 119901

119908must be low as the

number of hexagons is high to get better results in normalsituation

We see that the estimated false negative rate that is anestimate of 119875M

119873

(M0is selected) is often 0 in our simulation

(not shown in Table 1) This is because if the event occurs at119873 then detection of the event by at least one of the nodesbelonging to 119873 cup 119861(119873) is highly probable Furthermoresince the grid size is large one of the nodes belonging to119877(119873cup119861(119873))may respondwrongly though it cannot detectthe event So underM

119873 there is a small probability to select

ROI as normal If we take 119901119908and the detection probabilities

1199011and 119901

2to be very small then we may get some positive

false negative rate but this is not a desired condition for agood sensor

From simulation we see that as 1199012increases (for positive

1199012) 119878119894values increase whereas119873

119894decrease As119901

2increases it

helps to differentiate between the likelihood values resultingin lower cardinality of the set Nmax and lower values of119873119894rsquos However since the neighboring nodes help to detect the

event the success probability increases From simulation wefind that as 119901

1increases success probabilities also increase

but the effect of 1199012is more prominent than that of 119901

1 On

the other hand success probabilities also change with 119901119908

and 119901119888 Since 119901

2= 0 means 119875

2= 119901119908 so there is little

variability in the likelihood values leading to larger size ofNmax

When 119901119908= 001 effect of 119901

2on 1198783 1198784 and 119878

5and119873

3 1198734

and 1198735seems to be significant whereas the same cannot be

said for 119901119908= 0001 There is sudden change in 119878

119894rsquos and 119873

119894rsquos

when we shift from 1199012= 0 to 119901

2= 03 for 119901

119908= 001

but not 119901119908

= 0001 So when 119901119908is small the effect of the

neighborhood seems to be lessThe values of 119878

3and 119878

4are very similar for different

values of the parameters but larger increment in1198734than119873

3

suggests that the idea of neighboring search is not effectiveBut 1198783is much higher than 119878

2 so the method of searching

all the nodes inNmax is a better idea than that of searching arandom node fromNmax

We estimate the success probability 119875M119873

(119871119873gt 119862 sdot 119871max)

by simulation for different values of the threshold 119862 rangingfrom 05 to 09 (see Table 2) Note that 119878

5corresponds to

the threshold value 119862 = 09 We consider 1199011= 099 119901

119908=

0001 119901119888= 09 and four values of 119901

2= 03 04 05 06

From Table 2 we see that the success probability increasesas the threshold value 119862 decreases and 119901

2increases

Similarly the number of search decreases with both 119862

and 1199012

6 Discussion

One prime object of this paper is to show the effect of theneighboring nodes in detection of an event In this sectionwe discuss the role of the neighboring nodes and some otherrelated issues and make remarks

61 Role of the Neighboring Nodes Since ln 119871119873= 119886+ 119887(119888119911

119873+

119905119873minus 119889119896(119873)) where 119886 119887 119888 and 119889 are as defined in Section 31

119888 denotes the weight of the central node compared to theneighboring nodes in the corresponding likelihood Notethat since119875

1gt 1198752 we have 119888 gt 1 and if 119888 is close to 1 then the

six neighboring nodes are as important as the event node Soas the value of 119888 increases the importance of the neighboringnodes decreases Also 119889 gives some idea about the role of thenumber of adjacent nodes that is 119896(119873) Recall that 119875

1and 1198752

are the probabilities of responding (ie reporting the node119873as the event hexagon) by the sensors 119878

119873and 1198781198731015840 respectively

when 119873 is the event hexagon and 1198731015840 is a neighboring node

of119873 So we numerically calculate the quantities 1198751 1198752 119888 and

119889 for some values of the parameters (see Table 3)From the theoretical results in Section 31 we see that

1198751and 119888 increase as 119901

1increases while 119875

2and 119889 do not

depend on 1199011 On the other hand while 119875

2increases with 119901

2

119888 and 119889 decrease and 1198751is independent of 119901

2 Therefore the

importance of the neighboring nodes decreases with 1199011and

increases with 1199012 as expected and observed in Table 3

62 Estimation of the Parameters In practice the parameters1199011 1199012 119901119908 and 119901

119888may be unknown We can however

estimate the parameters by some experimentationNote that under M

0 119885119873follows Ber(119901

119908) for all 119873 isin 119877

Hence 119901119908is the expected value of 119885

119873given M

0 So we

perform the experiment by keeping the ROI normal Theproportion of 119885

119873rsquos having value 1 gives an estimate of 119901

119908

Repeat this experiment several times so that the average ofthe proportions over the repeated experiments can be takenas an estimate of 119901

119908

Note that 1199011is the expected value of 119884

119873under M

119873 So

we perform the experiment by keeping an event in somenode 119873 of the ROI The proportion of 119884

119873rsquos having value 1

gives an estimate of 1199011 Repeat this experiment for several

times so that the average of the proportions over the repeatedexperiments can be taken as an estimate of 119901

1 Similar

experiments will give estimates of 1199012and 119901

119888as well

63 Incorporation of Heterogeneity and Uncertainty in Param-eters Let 120579 = (119901

1 1199012 119901119888 119901119908) denote the set of parameters

which has been assumed to be the same for all the nodesWhile in practice there is no reason why the parametersshould be same for all the nodes it is also not clear how thesewould be different across119873 This unexplained heterogeneitycan be incorporated by assuming the 120579rsquos for different 119873 tobe independent realizations from a common distribution

Let 120579119873

= (1199011119873 1199012119873 119901119888119873 119901119908119873

) denote the set of param-eters for node 119873 We assume that 120579

119873 119873 isin 119877 are iid from

some distribution say 119892(120579) Also assume that given 120579119873 119873 isin

119877 119885119873rsquos are independent Note that 119892(120579) denotes the joint


Table 1 Simulation of estimated probabilities for some values of the parameters

Other parameters Simulation of different probabilities with 119901119888= 09

1199011

1199012

119901119908

1198781

1198782

1198783

1198733

1198784

1198734

1198785

1198735

09 00 001 000 008 081 1816 081 5985 082 181709 03 001 000 047 069 544 070 1481 073 57009 04 001 000 060 079 511 078 1351 080 51209 05 001 000 070 085 464 085 1150 086 48809 06 001 000 079 090 382 091 0818 092 39009 00 0001 035 050 081 317 081 717 081 31809 03 0001 036 059 082 303 083 634 086 30609 04 0001 035 067 087 289 087 618 089 30309 05 0001 036 075 090 289 089 585 093 29609 06 0001 036 083 094 274 093 533 096 279099 00 001 000 008 089 1743 090 5620 089 1770099 03 001 000 051 073 518 073 1298 079 577099 04 001 000 062 081 509 082 1301 084 521099 05 001 000 073 088 497 088 1269 089 504099 06 001 000 081 092 366 092 735 093 357099 00 0001 035 057 089 319 089 669 089 320099 03 0001 035 062 088 299 087 600 090 302099 04 0001 036 070 090 291 091 583 093 297099 05 0001 036 079 093 282 094 567 095 283099 06 0001 036 084 095 275 095 531 097 268


1199011

1199012

119901119908

1198781

1198782

1198783

1198733

1198784

1198734

1198785

1198735

09 00 001 000 008 090 182 089 5562 090 175809 03 001 000 054 076 514 076 1308 079 56409 04 001 000 067 085 505 085 1292 087 51209 05 001 000 077 091 486 090 1210 091 50209 06 001 000 086 094 357 093 724 095 35709 00 0001 036 057 090 318 089 661 089 31909 03 0001 036 065 088 298 089 634 092 30209 04 0001 036 073 092 287 092 591 094 29109 05 0001 035 081 094 281 094 551 096 28209 06 0001 037 088 096 272 096 524 097 290099 00 001 000 009 098 169 098 5140 098 176099 03 001 000 058 083 566 083 1473 087 569099 04 001 000 069 090 543 091 1438 092 569099 05 001 000 080 094 461 094 1119 095 473099 06 001 000 087 096 326 097 626 096 335099 00 0001 035 062 098 320 098 632 098 328099 03 0001 036 069 094 290 094 588 097 300099 04 0001 036 076 095 289 096 559 098 285099 05 0001 036 083 097 270 097 548 098 280099 06 0001 036 089 098 268 098 519 099 269

distribution of the four parameters For simplicity we mayassume them to be independent so that 119892(120579) can be writtenas 119892(120579) = 119892

1(1199011)1198922(1199012)119892119888(119901119888)119892119908(119901119908) In this situation the

likelihood for the modelM0is

Π119873isin119877

int119901119911119873

119908119873(1 minus 119901

119908119873)(1minus119911119873)

119892119908(119901119908119873

) 119889119901119908119873

(17)

where the integration is over the range of 119901119908119873

Similarly thelikelihood for the modelM

119873can be written as

Π1198731015840isin119877

int119871(1198731015840)

119873(120579119873) 119892 (120579119873) 119889120579119873 (18)

where the integral is over the four-dimensional space given bythe range of 120579

119873and 119871(119873

1015840)

119873(120579119873) is the contribution of the1198731015840th


Table 2 Simulation of estimated success probabilities and number of searches for different threshold values (119862) and some values of theparameters with 119901

119888= 1199011= 09

Other parameters 119862 = 06 119862 = 07 119862 = 08 119862 = 09

1199012

119901119908

Success Search Success Search Success Search Success Search00 001 081 1821 081 1825 081 1821 081 181703 001 087 1372 078 913 075 664 073 57004 001 089 886 085 647 082 546 080 51205 001 093 688 090 569 089 505 086 48806 001 097 527 096 491 093 404 092 39000 0001 080 327 080 321 080 317 080 31803 0001 091 415 091 365 087 331 086 30604 0001 094 425 094 369 093 331 089 30305 0001 097 424 097 364 096 326 093 29606 0001 099 396 098 318 098 304 096 279

Table 3 Values of 1198751 1198752 119888 and 119889 for 119901

119888= 09

Parameters 119901119908= 01 119901

119908= 02

1199011

1199012

1198751

1198752

119888 119889 1198751

1198752

119888 119889

07

03 066 034 1865 0202 069 041 2139 029804 066 042 1526 0234 069 048 1674 033005 066 050 1302 0268 069 055 1378 036306 066 058 1135 0302 069 062 1166 0397

08

03 074 034 2114 0202 076 041 2484 029804 074 042 1730 0234 076 048 1944 033005 074 050 1476 0268 076 055 1600 036306 074 058 1287 0302 076 062 1354 0397

09

03 082 034 0241 0202 083 041 2907 029804 082 042 1981 0234 083 048 2275 033005 082 050 1690 0268 083 055 1873 036306 082 058 1474 0302 083 062 1584 0397

node to the likelihood 119871119873 given the value 120579

119873 as described in

Section 31Similar technique can also be used to incorporate param-

eter uncertainty Even though the parameters can be assumedto be same for all the nodes there may be reasonableuncertainty about the constancy of the parameter valuesAs in the Bayesian paradigm the set of parameters may beassumed to be a realization from a distribution say 119892(120579)Then the likelihoods for the modelsM

0andM

119873are

intΠ119873isin119877

119901119911119873

119908(1 minus 119901

119908)(1minus119911119873)

119892120579(119901119908) 119889119901119908

intΠ1198731015840isin119877119871(1198731015840)

119873(120579) 119892 (120579) 119889120579 respectively

(19)

The choice of119892(120579)maybe a difficult oneHowever sometimesthere may be specific information available regarding thedistribution of 120579 which can be incorporated in the model

64 When More Sensors Can Detect the Event Square Wemay consider the situation when sensing radii are larger andmore sensors can detect the event hexagon but with differentprobabilities With respect to a particular node classify the

remaining nodes with respect to the probability of detectingthe event at that node which may as well depend on thedistance from the particular node Suppose that the sensorsin the 119894th class detect the event hexagon with probability119901119894 119894 = 1 2 3 The theoretical analysis is similar to that of

Section 3 but having more probability terms

65 Concluding Remarks In this paper we consider theproblem of fault detection in wireless sensor network (WSN)We discuss how to address both the noise-related mea-surement errors (119901

1and 119901

2) and sensor fault (119901

119888and 119901

119908)

simultaneously in fault detection where the ROI is parti-tioned into regular hexagons with the event occurring atonly one hexagon We propose fault detection schemes thatexplicitly introduce the error probabilities into the optimalevent detection process We develop the schemes under theconsideration of model selection technique multiple modelselection technique and Bayesian model averaging methodThe different error probabilities are calculated by means ofsimulation Note that the same analysis can be carried outwhen ROI is partitioned into squares and sensors are placedat the centers


Nandi et al [25] consider similar problem in wirelesssensor network (WSN) in which the event can take placeat the center of one particular square (or hexagon) of thegrid covering the ROI In our paper we allow the eventsquare to be any one in the grid Our approach can alsobe used for the problem of [25] with only two models tobe considered for selection In [25] the authors developthe scheme under the consideration of Neyman-Pearsonhypothesis test where the null and alternative hypothesescorrespond to the two models In model selection approachwe select the model with higher likelihood In classicalNeyman-Pearson hypothesis test a model is selected if itslikelihood is greater than some constant times the likelihoodof the other This constant is fixed before the test dependingon the size of the test In model selection approach theconstant is 1 leaving no choice for the size of the test On theother hand we cannot apply the classical Neyman-Pearsontest withmore than twomodels to be considered for selection

The principle of hypothesis testing places a large confi-dence in the null hypothesis and does not reject it unless thereis strong evidence against itThis safeguard of null hypothesiscannot be ensured in the model selection approach ofSection 31 However the multiple model selection approachof Section 32 provides some safeguard in this regard

This principle of model selection can be extended tothe situation when there are more than one event hexagonand the objective is to detect the event hexagons We mayalso assume that the sensors can detect different types ofevents That is response of sensors may not be only binarysensors can measure distance direction speed humiditywind speed soil makeup temperature and so forth and sendthe measurement of continuous type variables to the basestation One needs a different formulation of the problem insuch case which will be taken up in future

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are sincerely thankful to the anonymous review-ers for their detailed comments that helped improve theoverall quality of this paper The authors are also thankful toSourav Sengupta Indian Statistical Institute Kolkata for hisvaluable suggestions

References

[1] I F Akyildiz W Su Y Sankarasubramaniam and E CayircildquoWireless sensor networks a surveyrdquo Computer Networks vol38 no 4 pp 393ndash422 2002

[2] G J Pottie and W J Kaiser ldquoWireless integrated networksensorsrdquo Communications of the ACM vol 43 no 5 pp 51ndash582000

[3] D P Agrawal M Lu T C Keener M Dong and V KumarldquoExploiting the use of WSNs for environmental monitoringrdquoEMMagazine pp 27ndash33 2004

[4] C Farah F Schwaner A Abedi andMWorboys ldquoA distributedhomology algorithm to detect topological events via wirelesssensor networksrdquoACMTransaction on Sensor Networks vol 10no 10 2010

[5] FMartincic and L Schwiebert ldquoIntroduction to wireless sensornetworkingrdquo in Handbook of Sensor Networks Algorithms andArchitectures I Stojmenovic Ed chapter 1 JohnWiley amp Sons2005

[6] X Bai S Kumar D Xuan Z Yun and T H Lai ldquoDeployingwireless sensors to achieve both coverage and connectivityrdquo inProceedings of the 7th ACM International Symposium on MobileAd Hoc Networking and Computing (MOBIHOC rsquo06) pp 131ndash142 Florence Italy May 2006

[7] M A Batalin and G S Sukhatme ldquoThe analysis of anefficient algorithm for robot coverage and exploration basedon sensor network deploymentrdquo in Proceedings of the IEEEInternational Conference on Robotics andAutomation pp 3478ndash3485 Barcelona Spain April 2005

[8] J Cortes SMartınez T Karatas and F Bullo ldquoCoverage controlfor mobile sensing networksrdquo IEEE Transactions on Roboticsand Automation vol 20 no 2 pp 243ndash255 2004

[9] A Filippou D A Karras and R C Papademetriou ldquoCov-erage problem for sensor networks an overview of solutionstrategiesrdquo in Proceedings of the 17th Telecommunication Forum(TELFOR rsquo09) 2009

[10] X Li A Nayak D Simplot-Ryl and I Stojmenovic ldquoSensorplacement in sensor and actuator networksrdquo inWireless Sensorand Actuator Networks Algorithms and Protocols for ScalableCoordination and Data Communication John Wiley amp Sons2010

[11] J Chen S Kher and A Somani ldquoDistributed fault detectionof wireless sensor networksrdquo in Proceedings of the Workshop onDependability Issues in Wireless Ad Hoc Networks and SensorNetworks (DIWANS rsquo06) pp 65ndash71 ACM New York NY USASeptember 2006

[12] A B Sharma L Golubchik and R Govindan ldquoSensor faultsdetectionmethods and prevalence in real-world datasetsrdquoACMTransactions on Sensor Networks vol 6 no 3 article 23 2010

[13] B Krishnamachari and S Iyengar ldquoDistributed Bayesian algo-rithms for fault-tolerant event region detection in wirelesssensor networksrdquo IEEE Transactions on Computers vol 53 no3 pp 241ndash250 2004

[14] K Chintalapudi and R Govidan ldquoLocalized edge detectionin sensor fieldsrdquo in Proceedings of the 1st IEEE InternationalWorkshop on Sensor Network Protocols and Applications pp 59ndash70 2003

[15] R Nowak and U Mitra ldquoBoundary estimation in sensornetworks theory and methodsrdquo in Proceedings of the 1st IEEEInternational Workshop Sensor Network Protocols and Applica-tions 2003

[16] J N Tsitsiklis ldquoDecentralized detectionrdquoAdvances in StatisticalSignal Processing vol 2 pp 297ndash344 1993

[17] S Li and F Qin ldquoA dynamic neural network approach forsolving nonlinear inequalities defined on a graph and itsapplication to distributed routing-free range-free localizationof WSNsrdquo Neurocomputing vol 117 pp 72ndash80 2013

[18] S Li ZWang and Y Li ldquoUsing laplacian eigenmap as heuristicinformation to solve nonlinear constraints defined on a graphand its application in distributed range-free localization ofwireless sensor networksrdquoNeural Process Letters vol 37 pp 411ndash424 2013


[19] S Li B Liu B Chen and Y Lou ldquoNeural network basedmobile phone localization using bluetooth connectivityrdquoNeuralComputing and Applications vol 23 no 3-4 pp 667ndash675 2013

[20] S Li S Chen B Liu Y Li and Y Liang ldquoDecentralized kine-matic control of a class of collaborative redundantmanipulatorsvia recurrent neural networksrdquo Neurocomputing vol 91 pp 1ndash10 2012

[21] S Li H Cui Y Li B Li and Y Lou ldquoDecentralized controlof collaborative redundant manipulators with partial commandcoverage via locally connected recurrent neural networksrdquoNeural Computing and Applications vol 23 pp 1051ndash1060 2012

[22] S Li B Liu and Y Li ldquoSelective positivemdashnegative feedbackproduces the winner-take-all competition in recurrent neuralnetworksrdquo IEEE Transactions on Neural Networks and LearningSystems vol 24 no 2 pp 301ndash309 2013

[23] S Li Y Li and Z Wang ldquoA class of finite-time dual neuralnetworks for solving quadratic programming problems and itsk-winners-take-all applicationrdquo Neural Networks vol 39 pp27ndash39 2013

[24] X Luo M Dong and Y Huang ldquoOn distributed fault-tolerantdetection in wireless sensor networksrdquo IEEE Transactions onComputers vol 55 no 1 pp 58ndash70 2006

[25] M Nandi A Nayak B Roy and S Sarkar ldquoHypothesis testingand decision theoretic approach for fault detection in wirelesssensornetworksrdquoThe International Journal of Parallel Emergentand Distributed Systems In press

[26] R Williams The Geometrical Foundation of Natural StructureA Source Book of Design Dover New York NY USA 1979

[27] J A Hoeting D Madigan A E Raftery and C T VolinskyldquoBayesian model averaging a tutorialrdquo Statistical Science vol14 no 4 pp 382ndash417 1999

[28] DMadigan and J York ldquoBayesian graphical models for discretedatardquo International Statistical Review vol 63 pp 215ndash232 1995

[29] N Bulusu J Heidemann and D Estrin ldquoGPS-less low-costoutdoor localization for very small devicesrdquo IEEE PersonalCommunications vol 7 no 5 pp 28ndash34 2000

[30] J K Ghosh M Delampady and T Samanta An Introductionto Bayesian Analysis Theory and Methods Springer New YorkNY USA 2011

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Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

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Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



The proposed algorithm is analyzed using a probabilisticapproach Sharma et al [12] have characterized the differenttypes of fault and proposed a different algorithm for faultdetection considering different types of fault Some of themethods are statistical like using histogram and so forthBoth works can only detect the faulty sensors but not theevent

One of the important sensor network applications ismonitoring inaccessible environments Sensor networks areused to determine event regions and boundaries in theenvironment with a distinguishable characteristic [13ndash15]The basic idea of distributed detection [16] is to have eachof the independent sensors make a local decision (typicallya binary one ie an event occurs or not) and then combinethese decisions at a fusion sensor (the sensor which collectsthe local information and takes the decision) or at a basestation to generate a global decision

A closely related area is neural network Several worksare there in the literature on this area Li and Qin [17]find a feasible solution to a class of nonlinear inequalitiesdefined on a graph proposing a recurrent neural networkTheconvergence of the neural network and the solution feasibilityto the defined problem are both theoretically proven Theyproposed neural network features as a parallel computingmechanism and a distributed topology isomorphic to thecorresponding graph which is suitable for distributed real-time computation The proposed neural network is appliedto range-free localization of WSNs Li et al [18] show thatfeasible solution set to the same problem is often infinityand Laplacian eigenmap is used as heuristic information togain better performance in the solution A continuous-timeprojected neural network and the corresponding discrete-time projected neural network are both given to tacklethis problem iteratively The effectiveness of the proposedneural networks is tested and compared with others via itsapplications in the range-free localization of WSNs Locationinformation is useful for mobile phones There exists adilemma between the relatively high price of GPS devicesand the dependence of location information acquisition onGPS for most phones in current stage Li et al [19] formulatethe problem as an optimization problem defined on theBluetooth networkThe solution to this optimization problemis not unique Heuristic information is employed to improvethe performance of the result in the feasible set They usedrecurrent neural networks to solve the problem distributivelyin real time The convergence of the neural network andthe solution feasibility to the defined problem are boththeoretically proven The hardware implementation of theproposed neural network is also explored in this paper

Distributed algorithms are also used for a networkdynamic system Li et al [20 21] studied the decentral-ized control and kinematic control of multiple redundantmanipulators for the cooperative task execution problemThe problem is formulated as a constrained quadratic pro-gramming problem and then a recurrent neural networkwith independent modules is proposed to solve the problemin a distributed manner They proposed a novel strategycapable of solving the problem even though there exist somemanipulators unable to access the command signal directly

Another related area is the winner-take-all (WTA) com-petition which is widely observed in both inanimate andbiological media and society Many mathematical modelsare proposed to describe the phenomena discovered indifferent fields These models are capable of demonstratingthe WTA competition Li et al [22 23] make steps in thatdirection and present a simple model which produces theWTA competition by taking advantage of selective positive-negative feedback through the interaction of neurons via 119901-norm They also present a class of recurrent neural networksto solve quadratic programming problems Different frommost existing recurrent neural networks for solving quadraticprogramming problems the proposed neural networkmodelconverges in finite time and the activation function is notrequired to be a hard-limiting function for finite convergencetime The stability finite-time convergence property andthe optimality of the proposed neural network for solvingthe original quadratic programming problem are proven intheory Extensive simulations are performed to evaluate theperformance of the neural networkwith different parametersIn addition the proposed neural network is applied to solvingthe k-winner-take-all (k-WTA) problem

11 Our Motivation In this paper we are interested in oneparticular query determining event in the environment (ieROI) with a distinguishable characteristic We assume theROI to be partitioned into suitable number of congruentregular hexagonal cells (ie we can think of ROI as a regularhexagonal grid) This physical structure of ROI is not arequirement for the theoretical analysis and we can do thesimilar analysis with another structure also Suppose thatsensors are placed a priori at the center (which are knownas nodes) of every hexagon of the grid We assume that thesensors are connected to its adjacent sensor nodes in the sensethat a hexagon will be strongly covered by its center node andweakly covered by the adjacent nodes If event occurs in thehexagon where a particular sensor lies then that particularsensor can detect the event with a greater probability whereasif event occurs in any adjacent hexagon then the particularsensor can detect the event with a lesser probability Henceonly one node (center node of the event hexagon) candetect the event hexagon with greater probability say 119901

1 and

adjacent nodes (six for interior nodes and less for boundarynodes) can detect the event hexagon with lesser probabilitysay 1199012 with 119901

1gt 1199012 We assume that no other sensor can

detect the event hexagon In this paper unlike the previousworks we assume that if the event occurs then it occursat only one hexagon of the grid which will be known asevent hexagon and there is no fusion sensor All sensorscan communicate with the base station and the base stationtakes the decision about the query As an example considera network of devices that are capable of sensing mines orbombs if we assume that a fewmines or bombsmay be placedon a particular area of ROI Information from these devicescan be sent to a nearby police station or a central facilityThen an important query in this situation could be whethera particular hexagon is the event hexagon or not (ie minesor bombs are placed or not)










1












Adjacent node

Adjacent node

Adjacent node

Adjacent node

Adjacent node

Adjacent node

Center node












119873= 1


119873= 0



119873


119873= 0 if 119878

119873does not


119873or 1198671198731015840 for 1198731015840 isin 119861(119873) and 119885

119873= 1 if





119873= 119896 | 119884

119873 119883119873) = 119875(119885

119873=



sum119873isin119877

119883119873= 1 or 0


M0 (119883119873= 0 for all119873 isin 119877)


M119873 (119883119873= 1 and119883

1198731015840 = 0 for all1198731015840 isin 119877 119873)



119873) = 119901119873



0

(sdot)


119873


(i) For all119873 isin 119877 119875M0

(119884119873= 1) = 0 and 119875M

119873

(119884119873= 1) =

1199011

(ii) For all 1198731015840

isin 119861(119873) 119875M119873

(1198841198731015840 = 1) = 119901

2


(1198841198731015840 = 1) = 0

(iii) For all119873 isin 119877 119875(119885119873= 1 | 119884

119873= 1) = 119901

119888and 119875(119885

119873=

1 | 119884119873= 0) = 119901

119908

(iv) 119885119873and 119884


1015840







forall119873 isin 119877 119875M0

(119885119873= 1)

= 119875M0

(119885119873= 1 | 119884

119873= 0) 119875M

0

(119884119873= 0)


+ 119875M0

(119885119873= 1 | 119884

119873= 1) 119875M

0

(119884119873= 1)

= 119875 (119885119873= 1 | 119884

119873= 0) 119875M

0

(119884119873= 0)

+ 119875 (119885119873= 1 | 119884

119873= 1) 119875M

0

(119884119873= 1) = 119901

119908

(1)


119908) for all119873 isin



1198710= 119875M

0

(119885119873= 119911119873 forall119873 isin 119877)

= prod

119873isin119877

119901119911119873

119908(1 minus 119901

119908)(1minus119911119873)

= (119901119908)sum119873isin119877119911119873


(2)

So ln 1198710= sum119873isin119877

119911119873ln119901119908+ sum119873isin119877

(1 minus 119911119873) ln(1 minus 119901

119908)


(119885119873= 1)

= 119875M119873

(119885119873= 1 | 119884

119873= 0) 119875M

119873

(119884119873= 0)

+ 119875M119873

(119885119873= 1 | 119884

119873= 1) 119875M

119873

(119884119873= 1)

= 119875 (119885119873= 1 | 119884

119873= 0) 119875M

119873

(119884119873= 0)

+ 119875 (119885119873= 1 | 119884

119873= 1) 119875M

119873

(119884119873= 1)

= 119901119908(1 minus 119901

1) + 1199011198881199011= 1199011(119901119888minus 119901119908) + 119901119908

= 1198751 say

(3)




2)


119873 1198851198731015840 follows


1gt 1198752


119873 given

1198851198731015840 = 1199111198731015840 1198731015840isin 119877 is

119871119873= 119875M

119873

(1198851198731015840 = 1199111198731015840 forall1198731015840isin 119877)

= 119875119911119873

1(1 minus 119875

1)(1minus119911119873)

Π1198731015840isin119861(119873)

1198751199111198731015840

2(1 minus 119875

2)(1minus1199111198731015840 )

times Π1198731015840isin119877[119861(119873)cup119873]

1199011199111198731015840

119908 (1 minus 119901119908)(1minus1199111198731015840 )

= 119875119911119873

1(1 minus 119875

1)(1minus119911119873)

119875sum1198731015840isin119861(119873)119911

1198731015840

2(1 minus 119875

2)sum1198731015840isin119861(119873)(1minus1199111198731015840 )


1198731015840


(1minus1199111198731015840 )

(4)

Let 119879119873= sum1198731015840isin119861(119873)

1198851198731015840 so thatsum

1198731015840isin119861(119873)

(1 minus 1198851198731015840) = 119896(119873) minus


119905119873= sum

1198731015840isin119861(119873)

1199111198731015840 sum

1198731015840isin119861(119873)

(1 minus 1199111198731015840) = 119896 (119873) minus 119905

119873 (5)

Therefore

ln 119871119873= 119911119873ln1198751+ (1 minus 119911

119873) ln (1 minus 119875

1)

+ 119905119873ln1198752+ (119896 (119873) minus 119905

119873) ln (1 minus 119875

2)

+ sum

1198731015840isin119877[119861(119873)cup119873]

119911119873ln119901119908

+ sum

1198731015840isin119877[119861(119873)cup119873]

(1 minus 119911119873) ln (119901

119908(1 minus 119901

119908))

= ln 1198710+ 119911119873ln 1198751

119901119908

+ (1 minus 119911119873) ln 1 minus 119875

1

1 minus 119901119908

+ 119905119873ln 1198752

119901119908

+ (119896 (119873) minus 119905119873) ln 1 minus 119875

2

1 minus 119901119908

= ln 1198710+ 119911119873ln

1198751(1 minus 119901

119908)

119901119908(1 minus 119875

1)+ 119905119873ln

1198752(1 minus 119901

119908)

119901119908(1 minus 119875

2)

+ ln 1 minus 1198751

1 minus 119901119908

+ 119896 (119873) ln1 minus 1198752

1 minus 119901119908

= 119886 + 119887 (119888119911119873+ 119905119873minus 119889119896 (119873)) say

(6)

where

119886 = ln 1198710+ ln 1 minus 119875

1

1 minus 119901119908

119887 = ln1198752(1 minus 119901

119908)

119901119908(1 minus 119875

2)gt 0

119888 =ln (1198751(1 minus 119901

119908) 119901119908(1 minus 119875

1))

ln (1198752(1 minus 119901

119908) 119901119908(1 minus 119875

2))

119889 =ln ((1 minus 119901

119908) (1 minus 119875

2))

ln (1198752(1 minus 119901

119908) 119901119908(1 minus 119875

2))

(7)



0if

= ln 1 minus 1198751

1 minus 119901119908

+ 119887 (119888119911119873+ 119905119873minus 119889119896 (119873)) lt 0


(8)







119873 isin 119877




M119870

119871119870

max119873isin119877

119871119873

gt 119862 (9)



119873+119905119873minus119889119896(119873)



119876119873 (10)


lowastlt 1



119888119911119873+ 119905119873 (11)






119873isinNmax119861(119873)] that is 1198731015840



1M2 M

119896are


119896is given by

Pr (M119896| 119863) =

Pr (119863 | M119896)Pr (M

119896)

sumPr (119863 | M119897)Pr (M

119897) (12)



119896



119896is the


119873= 119911119873




Pr (M119873| 119885119873= 119911119873 119873 isin 119877) =

119901119873119871119873

sum119897isin119877

119901119897119871119897+ 119901norm1198710

and that forM0is

119901norm1198710sum119897isin119877

119901119897119871119897+ 119901norm1198710

(13)



for which 119901119873119871119873







0

(119879119873=

0 119885119873


119873

(119879119873=

6 119885119873


119873and 119885



(1) 119875M0

(119879119873= 119894 119885

119873= 0) = ( 119896(119873)

119894) 119901119894

119908(1 minus 119901

119908)119896(119873)minus119894+1

(2) 119875M0

(119879119873= 119894 119885

119873= 1) = ( 119896(119873)

119894) 119901119894+1

119908(1 minus 119901

119908)119896(119873)minus119894

(3) 119875M119873

(119879119873

= 119894 119885119873

= 0) = ( 119896(119873)119894) 119875119894

2(1 minus 119875

2)119896(119873)minus119894

(1 minus

1198751)

(4) 119875M119873

(119879119873= 119894 119885

119873= 1) = ( 119896(119873)

119894) 119875119894

2(1 minus 119875

2)119896(119873)minus119894

1198751


119875M0

(119871119873gt 1198710)

= 119875M0

(ln 119871119873gt ln 119871

0)

= 119875M0

(119885119873ln

1198751(1 minus 119901

119908)

119901119908(1 minus 119875

1)+ 119879119873ln

1198752(1 minus 119901

119908)

119901119908(1 minus 119875

2)

+ ln 1 minus 1198751

1 minus 119901119908

+ 119896 (119873) ln1 minus 1198752

1 minus 119901119908

gt 0)

= 119875M0

(119885119873ln

1198751(1 minus 119901

119908)

119901119908(1 minus 119875

1)+ 119879119873ln

1198752(1 minus 119901

119908)

119901119908(1 minus 119875

2)

gt 119896 (119873) ln1 minus 119901119908

1 minus 1198752

+ ln1 minus 119901119908

1 minus 1198751

)

(14)


119873and 119885


0 The maximum of




119875M119873

(119871119873lt 1198710)

= 119875M119873

(119885119873ln

1198751(1 minus 119901

119908)

119901119908(1 minus 119875

1)+ 119879119873ln

1198752(1 minus 119901

119908)

119901119908(1 minus 119875

2)

lt 119896 (119873) ln1 minus 119901119908

1 minus 1198752

+ ln1 minus 119901119908

1 minus 1198751

)

(15)


119873and 119885


119873 This



forwhich 119876




119873under the




= 119894 and 119879119873

= 119895 for 119894 = 0 1 and 119895 =


and 119895 = 0 1 119896(119873)

119875M119873

(119876119873= 119888119894 + 119895 minus 119889119896 (119873))

= (119896 (119873)

119895) (1198751)119894

(1 minus 1198751)(1minus119894)

(1198752)119895

(1 minus 1198752)(119896(119873)minus119895)

119875M0

(119876119873= 119888119894 + 119895 minus 119889119896 (119873))

= (119896 (119873)

119895) (119901119908)119894+119895


(16)



0when M





5 Simulation Study




1denote




0

(0 isin




2denote the












119873



1198731015840 is a





119873


119861(1198731015840)) Clearly

1198784

ge 1198783






119875M119873





Clearly 1198731= 1 minus 119878


0



2le 1 In our





3ge 1 ge 119873

2


119894rsquos


1 1199012 119901119888

and 119901119908with 119901

1and 119901


119908taking














119901119908



119908= 00001 (not shown in






119873





1199011and 119901






2increases it





1 On


and 119901119888 Since 119901

2= 0 means 119875



When 119901119908= 001 effect of 119901

2on 1198783 1198784 and 119878

5and119873

3 1198734



119894rsquos and 119873

119894rsquos


2= 03 for 119901

119908= 001

but not 119901119908



3and 119878



3





(119871119873gt 119862 sdot 119871max)


5corresponds to


119908=


2= 03 04 05 06


2increases


and 1199012

6 Discussion



119873+





1and 1198752






1198751and 119888 increase as 119901


2and 119889 do not



2


2 Therefore the






0 119885119873follows Ber(119901

119908) for all 119873 isin 119877


119873given M

0 So we



119908


119908


119873under M

119873 So





1 Similar


119888as well




Let 120579119873

= (1199011119873 1199012119873 119901119888119873 119901119908119873


119873 119873 isin 119877 are iid from






1199011

1199012

119901119908

1198781

1198782

1198783

1198733

1198784

1198734

1198785

1198735

09 00 001 000 008 081 1816 081 5985 082 181709 03 001 000 047 069 544 070 1481 073 57009 04 001 000 060 079 511 078 1351 080 51209 05 001 000 070 085 464 085 1150 086 48809 06 001 000 079 090 382 091 0818 092 39009 00 0001 035 050 081 317 081 717 081 31809 03 0001 036 059 082 303 083 634 086 30609 04 0001 035 067 087 289 087 618 089 30309 05 0001 036 075 090 289 089 585 093 29609 06 0001 036 083 094 274 093 533 096 279099 00 001 000 008 089 1743 090 5620 089 1770099 03 001 000 051 073 518 073 1298 079 577099 04 001 000 062 081 509 082 1301 084 521099 05 001 000 073 088 497 088 1269 089 504099 06 001 000 081 092 366 092 735 093 357099 00 0001 035 057 089 319 089 669 089 320099 03 0001 035 062 088 299 087 600 090 302099 04 0001 036 070 090 291 091 583 093 297099 05 0001 036 079 093 282 094 567 095 283099 06 0001 036 084 095 275 095 531 097 268


1199011

1199012

119901119908

1198781

1198782

1198783

1198733

1198784

1198734

1198785

1198735

09 00 001 000 008 090 182 089 5562 090 175809 03 001 000 054 076 514 076 1308 079 56409 04 001 000 067 085 505 085 1292 087 51209 05 001 000 077 091 486 090 1210 091 50209 06 001 000 086 094 357 093 724 095 35709 00 0001 036 057 090 318 089 661 089 31909 03 0001 036 065 088 298 089 634 092 30209 04 0001 036 073 092 287 092 591 094 29109 05 0001 035 081 094 281 094 551 096 28209 06 0001 037 088 096 272 096 524 097 290099 00 001 000 009 098 169 098 5140 098 176099 03 001 000 058 083 566 083 1473 087 569099 04 001 000 069 090 543 091 1438 092 569099 05 001 000 080 094 461 094 1119 095 473099 06 001 000 087 096 326 097 626 096 335099 00 0001 035 062 098 320 098 632 098 328099 03 0001 036 069 094 290 094 588 097 300099 04 0001 036 076 095 289 096 559 098 285099 05 0001 036 083 097 270 097 548 098 280099 06 0001 036 089 098 268 098 519 099 269




Π119873isin119877

int119901119911119873

119908119873(1 minus 119901

119908119873)(1minus119911119873)

119892119908(119901119908119873

) 119889119901119908119873

(17)




Π1198731015840isin119877

int119871(1198731015840)

119873(120579119873) 119892 (120579119873) 119889120579119873 (18)


119873and 119871(119873

1015840)




119888= 1199011= 09


1199012

119901119908



119888= 09

Parameters 119901119908= 01 119901

119908= 02

1199011

1199012

1198751

1198752

119888 119889 1198751

1198752

119888 119889

07

03 066 034 1865 0202 069 041 2139 029804 066 042 1526 0234 069 048 1674 033005 066 050 1302 0268 069 055 1378 036306 066 058 1135 0302 069 062 1166 0397

08

03 074 034 2114 0202 076 041 2484 029804 074 042 1730 0234 076 048 1944 033005 074 050 1476 0268 076 055 1600 036306 074 058 1287 0302 076 062 1354 0397

09

03 082 034 0241 0202 083 041 2907 029804 082 042 1981 0234 083 048 2275 033005 082 050 1690 0268 083 055 1873 036306 082 058 1474 0302 083 062 1584 0397





0andM

119873are


119901119911119873

119908(1 minus 119901

119908)(1minus119911119873)

119892120579(119901119908) 119889119901119908

intΠ1198731015840isin119877119871(1198731015840)

119873(120579) 119892 (120579) 119889120579 respectively

(19)






1and 119901


119888and 119901

119908)








Acknowledgments


References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation


















1












Adjacent node

Adjacent node

Adjacent node

Adjacent node

Adjacent node

Adjacent node

Center node












119873= 1


119873= 0



119873


119873= 0 if 119878

119873does not


119873or 1198671198731015840 for 1198731015840 isin 119861(119873) and 119885

119873= 1 if





119873= 119896 | 119884

119873 119883119873) = 119875(119885

119873=



sum119873isin119877

119883119873= 1 or 0


M0 (119883119873= 0 for all119873 isin 119877)


M119873 (119883119873= 1 and119883

1198731015840 = 0 for all1198731015840 isin 119877 119873)



119873) = 119901119873



0

(sdot)


119873


(i) For all119873 isin 119877 119875M0

(119884119873= 1) = 0 and 119875M

119873

(119884119873= 1) =

1199011

(ii) For all 1198731015840

isin 119861(119873) 119875M119873

(1198841198731015840 = 1) = 119901

2


(1198841198731015840 = 1) = 0

(iii) For all119873 isin 119877 119875(119885119873= 1 | 119884

119873= 1) = 119901

119888and 119875(119885

119873=

1 | 119884119873= 0) = 119901

119908

(iv) 119885119873and 119884


1015840







forall119873 isin 119877 119875M0

(119885119873= 1)

= 119875M0

(119885119873= 1 | 119884

119873= 0) 119875M

0

(119884119873= 0)


+ 119875M0

(119885119873= 1 | 119884

119873= 1) 119875M

0

(119884119873= 1)

= 119875 (119885119873= 1 | 119884

119873= 0) 119875M

0

(119884119873= 0)

+ 119875 (119885119873= 1 | 119884

119873= 1) 119875M

0

(119884119873= 1) = 119901

119908

(1)


119908) for all119873 isin



1198710= 119875M

0

(119885119873= 119911119873 forall119873 isin 119877)

= prod

119873isin119877

119901119911119873

119908(1 minus 119901

119908)(1minus119911119873)

= (119901119908)sum119873isin119877119911119873


(2)

So ln 1198710= sum119873isin119877

119911119873ln119901119908+ sum119873isin119877

(1 minus 119911119873) ln(1 minus 119901

119908)


(119885119873= 1)

= 119875M119873

(119885119873= 1 | 119884

119873= 0) 119875M

119873

(119884119873= 0)

+ 119875M119873

(119885119873= 1 | 119884

119873= 1) 119875M

119873

(119884119873= 1)

= 119875 (119885119873= 1 | 119884

119873= 0) 119875M

119873

(119884119873= 0)

+ 119875 (119885119873= 1 | 119884

119873= 1) 119875M

119873

(119884119873= 1)

= 119901119908(1 minus 119901

1) + 1199011198881199011= 1199011(119901119888minus 119901119908) + 119901119908

= 1198751 say

(3)




2)


119873 1198851198731015840 follows


1gt 1198752


119873 given

1198851198731015840 = 1199111198731015840 1198731015840isin 119877 is

119871119873= 119875M

119873

(1198851198731015840 = 1199111198731015840 forall1198731015840isin 119877)

= 119875119911119873

1(1 minus 119875

1)(1minus119911119873)

Π1198731015840isin119861(119873)

1198751199111198731015840

2(1 minus 119875

2)(1minus1199111198731015840 )

times Π1198731015840isin119877[119861(119873)cup119873]

1199011199111198731015840

119908 (1 minus 119901119908)(1minus1199111198731015840 )

= 119875119911119873

1(1 minus 119875

1)(1minus119911119873)

119875sum1198731015840isin119861(119873)119911

1198731015840

2(1 minus 119875

2)sum1198731015840isin119861(119873)(1minus1199111198731015840 )


1198731015840


(1minus1199111198731015840 )

(4)

Let 119879119873= sum1198731015840isin119861(119873)

1198851198731015840 so thatsum

1198731015840isin119861(119873)

(1 minus 1198851198731015840) = 119896(119873) minus


119905119873= sum

1198731015840isin119861(119873)

1199111198731015840 sum

1198731015840isin119861(119873)

(1 minus 1199111198731015840) = 119896 (119873) minus 119905

119873 (5)

Therefore

ln 119871119873= 119911119873ln1198751+ (1 minus 119911

119873) ln (1 minus 119875

1)

+ 119905119873ln1198752+ (119896 (119873) minus 119905

119873) ln (1 minus 119875

2)

+ sum

1198731015840isin119877[119861(119873)cup119873]

119911119873ln119901119908

+ sum

1198731015840isin119877[119861(119873)cup119873]

(1 minus 119911119873) ln (119901

119908(1 minus 119901

119908))

= ln 1198710+ 119911119873ln 1198751

119901119908

+ (1 minus 119911119873) ln 1 minus 119875

1

1 minus 119901119908

+ 119905119873ln 1198752

119901119908

+ (119896 (119873) minus 119905119873) ln 1 minus 119875

2

1 minus 119901119908

= ln 1198710+ 119911119873ln

1198751(1 minus 119901

119908)

119901119908(1 minus 119875

1)+ 119905119873ln

1198752(1 minus 119901

119908)

119901119908(1 minus 119875

2)

+ ln 1 minus 1198751

1 minus 119901119908

+ 119896 (119873) ln1 minus 1198752

1 minus 119901119908

= 119886 + 119887 (119888119911119873+ 119905119873minus 119889119896 (119873)) say

(6)

where

119886 = ln 1198710+ ln 1 minus 119875

1

1 minus 119901119908

119887 = ln1198752(1 minus 119901

119908)

119901119908(1 minus 119875

2)gt 0

119888 =ln (1198751(1 minus 119901

119908) 119901119908(1 minus 119875

1))

ln (1198752(1 minus 119901

119908) 119901119908(1 minus 119875

2))

119889 =ln ((1 minus 119901

119908) (1 minus 119875

2))

ln (1198752(1 minus 119901

119908) 119901119908(1 minus 119875

2))

(7)



0if

= ln 1 minus 1198751

1 minus 119901119908

+ 119887 (119888119911119873+ 119905119873minus 119889119896 (119873)) lt 0


(8)







119873 isin 119877




M119870

119871119870

max119873isin119877

119871119873

gt 119862 (9)



119873+119905119873minus119889119896(119873)



119876119873 (10)


lowastlt 1



119888119911119873+ 119905119873 (11)






119873isinNmax119861(119873)] that is 1198731015840



1M2 M

119896are


119896is given by

Pr (M119896| 119863) =

Pr (119863 | M119896)Pr (M

119896)

sumPr (119863 | M119897)Pr (M

119897) (12)



119896



119896is the


119873= 119911119873




Pr (M119873| 119885119873= 119911119873 119873 isin 119877) =

119901119873119871119873

sum119897isin119877

119901119897119871119897+ 119901norm1198710

and that forM0is

119901norm1198710sum119897isin119877

119901119897119871119897+ 119901norm1198710

(13)



for which 119901119873119871119873







0

(119879119873=

0 119885119873


119873

(119879119873=

6 119885119873


119873and 119885



(1) 119875M0

(119879119873= 119894 119885

119873= 0) = ( 119896(119873)

119894) 119901119894

119908(1 minus 119901

119908)119896(119873)minus119894+1

(2) 119875M0

(119879119873= 119894 119885

119873= 1) = ( 119896(119873)

119894) 119901119894+1

119908(1 minus 119901

119908)119896(119873)minus119894

(3) 119875M119873

(119879119873

= 119894 119885119873

= 0) = ( 119896(119873)119894) 119875119894

2(1 minus 119875

2)119896(119873)minus119894

(1 minus

1198751)

(4) 119875M119873

(119879119873= 119894 119885

119873= 1) = ( 119896(119873)

119894) 119875119894

2(1 minus 119875

2)119896(119873)minus119894

1198751


119875M0

(119871119873gt 1198710)

= 119875M0

(ln 119871119873gt ln 119871

0)

= 119875M0

(119885119873ln

1198751(1 minus 119901

119908)

119901119908(1 minus 119875

1)+ 119879119873ln

1198752(1 minus 119901

119908)

119901119908(1 minus 119875

2)

+ ln 1 minus 1198751

1 minus 119901119908

+ 119896 (119873) ln1 minus 1198752

1 minus 119901119908

gt 0)

= 119875M0

(119885119873ln

1198751(1 minus 119901

119908)

119901119908(1 minus 119875

1)+ 119879119873ln

1198752(1 minus 119901

119908)

119901119908(1 minus 119875

2)

gt 119896 (119873) ln1 minus 119901119908

1 minus 1198752

+ ln1 minus 119901119908

1 minus 1198751

)

(14)


119873and 119885


0 The maximum of




119875M119873

(119871119873lt 1198710)

= 119875M119873

(119885119873ln

1198751(1 minus 119901

119908)

119901119908(1 minus 119875

1)+ 119879119873ln

1198752(1 minus 119901

119908)

119901119908(1 minus 119875

2)

lt 119896 (119873) ln1 minus 119901119908

1 minus 1198752

+ ln1 minus 119901119908

1 minus 1198751

)

(15)


119873and 119885


119873 This



forwhich 119876




119873under the




= 119894 and 119879119873

= 119895 for 119894 = 0 1 and 119895 =


and 119895 = 0 1 119896(119873)

119875M119873

(119876119873= 119888119894 + 119895 minus 119889119896 (119873))

= (119896 (119873)

119895) (1198751)119894

(1 minus 1198751)(1minus119894)

(1198752)119895

(1 minus 1198752)(119896(119873)minus119895)

119875M0

(119876119873= 119888119894 + 119895 minus 119889119896 (119873))

= (119896 (119873)

119895) (119901119908)119894+119895


(16)



0when M





5 Simulation Study




1denote




0

(0 isin




2denote the












119873



1198731015840 is a





119873


119861(1198731015840)) Clearly

1198784

ge 1198783






119875M119873





Clearly 1198731= 1 minus 119878


0



2le 1 In our





3ge 1 ge 119873

2


119894rsquos


1 1199012 119901119888

and 119901119908with 119901

1and 119901


119908taking














119901119908



119908= 00001 (not shown in






119873





1199011and 119901






2increases it





1 On


and 119901119888 Since 119901

2= 0 means 119875



When 119901119908= 001 effect of 119901

2on 1198783 1198784 and 119878

5and119873

3 1198734



119894rsquos and 119873

119894rsquos


2= 03 for 119901

119908= 001

but not 119901119908



3and 119878



3





(119871119873gt 119862 sdot 119871max)


5corresponds to


119908=


2= 03 04 05 06


2increases


and 1199012

6 Discussion



119873+





1and 1198752






1198751and 119888 increase as 119901


2and 119889 do not



2


2 Therefore the






0 119885119873follows Ber(119901

119908) for all 119873 isin 119877


119873given M

0 So we



119908


119908


119873under M

119873 So





1 Similar


119888as well




Let 120579119873

= (1199011119873 1199012119873 119901119888119873 119901119908119873


119873 119873 isin 119877 are iid from






1199011

1199012

119901119908

1198781

1198782

1198783

1198733

1198784

1198734

1198785

1198735

09 00 001 000 008 081 1816 081 5985 082 181709 03 001 000 047 069 544 070 1481 073 57009 04 001 000 060 079 511 078 1351 080 51209 05 001 000 070 085 464 085 1150 086 48809 06 001 000 079 090 382 091 0818 092 39009 00 0001 035 050 081 317 081 717 081 31809 03 0001 036 059 082 303 083 634 086 30609 04 0001 035 067 087 289 087 618 089 30309 05 0001 036 075 090 289 089 585 093 29609 06 0001 036 083 094 274 093 533 096 279099 00 001 000 008 089 1743 090 5620 089 1770099 03 001 000 051 073 518 073 1298 079 577099 04 001 000 062 081 509 082 1301 084 521099 05 001 000 073 088 497 088 1269 089 504099 06 001 000 081 092 366 092 735 093 357099 00 0001 035 057 089 319 089 669 089 320099 03 0001 035 062 088 299 087 600 090 302099 04 0001 036 070 090 291 091 583 093 297099 05 0001 036 079 093 282 094 567 095 283099 06 0001 036 084 095 275 095 531 097 268


1199011

1199012

119901119908

1198781

1198782

1198783

1198733

1198784

1198734

1198785

1198735

09 00 001 000 008 090 182 089 5562 090 175809 03 001 000 054 076 514 076 1308 079 56409 04 001 000 067 085 505 085 1292 087 51209 05 001 000 077 091 486 090 1210 091 50209 06 001 000 086 094 357 093 724 095 35709 00 0001 036 057 090 318 089 661 089 31909 03 0001 036 065 088 298 089 634 092 30209 04 0001 036 073 092 287 092 591 094 29109 05 0001 035 081 094 281 094 551 096 28209 06 0001 037 088 096 272 096 524 097 290099 00 001 000 009 098 169 098 5140 098 176099 03 001 000 058 083 566 083 1473 087 569099 04 001 000 069 090 543 091 1438 092 569099 05 001 000 080 094 461 094 1119 095 473099 06 001 000 087 096 326 097 626 096 335099 00 0001 035 062 098 320 098 632 098 328099 03 0001 036 069 094 290 094 588 097 300099 04 0001 036 076 095 289 096 559 098 285099 05 0001 036 083 097 270 097 548 098 280099 06 0001 036 089 098 268 098 519 099 269




Π119873isin119877

int119901119911119873

119908119873(1 minus 119901

119908119873)(1minus119911119873)

119892119908(119901119908119873

) 119889119901119908119873

(17)




Π1198731015840isin119877

int119871(1198731015840)

119873(120579119873) 119892 (120579119873) 119889120579119873 (18)


119873and 119871(119873

1015840)




119888= 1199011= 09


1199012

119901119908



119888= 09

Parameters 119901119908= 01 119901

119908= 02

1199011

1199012

1198751

1198752

119888 119889 1198751

1198752

119888 119889

07

03 066 034 1865 0202 069 041 2139 029804 066 042 1526 0234 069 048 1674 033005 066 050 1302 0268 069 055 1378 036306 066 058 1135 0302 069 062 1166 0397

08

03 074 034 2114 0202 076 041 2484 029804 074 042 1730 0234 076 048 1944 033005 074 050 1476 0268 076 055 1600 036306 074 058 1287 0302 076 062 1354 0397

09

03 082 034 0241 0202 083 041 2907 029804 082 042 1981 0234 083 048 2275 033005 082 050 1690 0268 083 055 1873 036306 082 058 1474 0302 083 062 1584 0397





0andM

119873are


119901119911119873

119908(1 minus 119901

119908)(1minus119911119873)

119892120579(119901119908) 119889119901119908

intΠ1198731015840isin119877119871(1198731015840)

119873(120579) 119892 (120579) 119889120579 respectively

(19)






1and 119901


119888and 119901

119908)








Acknowledgments


References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Adjacent node

Adjacent node

Adjacent node

Adjacent node

Adjacent node

Adjacent node

Center node












119873= 1


119873= 0



119873


119873= 0 if 119878

119873does not


119873or 1198671198731015840 for 1198731015840 isin 119861(119873) and 119885

119873= 1 if





119873= 119896 | 119884

119873 119883119873) = 119875(119885

119873=



sum119873isin119877

119883119873= 1 or 0


M0 (119883119873= 0 for all119873 isin 119877)


M119873 (119883119873= 1 and119883

1198731015840 = 0 for all1198731015840 isin 119877 119873)



119873) = 119901119873



0

(sdot)


119873


(i) For all119873 isin 119877 119875M0

(119884119873= 1) = 0 and 119875M

119873

(119884119873= 1) =

1199011

(ii) For all 1198731015840

isin 119861(119873) 119875M119873

(1198841198731015840 = 1) = 119901

2


(1198841198731015840 = 1) = 0

(iii) For all119873 isin 119877 119875(119885119873= 1 | 119884

119873= 1) = 119901

119888and 119875(119885

119873=

1 | 119884119873= 0) = 119901

119908

(iv) 119885119873and 119884


1015840







forall119873 isin 119877 119875M0

(119885119873= 1)

= 119875M0

(119885119873= 1 | 119884

119873= 0) 119875M

0

(119884119873= 0)


+ 119875M0

(119885119873= 1 | 119884

119873= 1) 119875M

0

(119884119873= 1)

= 119875 (119885119873= 1 | 119884

119873= 0) 119875M

0

(119884119873= 0)

+ 119875 (119885119873= 1 | 119884

119873= 1) 119875M

0

(119884119873= 1) = 119901

119908

(1)


119908) for all119873 isin



1198710= 119875M

0

(119885119873= 119911119873 forall119873 isin 119877)

= prod

119873isin119877

119901119911119873

119908(1 minus 119901

119908)(1minus119911119873)

= (119901119908)sum119873isin119877119911119873


(2)

So ln 1198710= sum119873isin119877

119911119873ln119901119908+ sum119873isin119877

(1 minus 119911119873) ln(1 minus 119901

119908)


(119885119873= 1)

= 119875M119873

(119885119873= 1 | 119884

119873= 0) 119875M

119873

(119884119873= 0)

+ 119875M119873

(119885119873= 1 | 119884

119873= 1) 119875M

119873

(119884119873= 1)

= 119875 (119885119873= 1 | 119884

119873= 0) 119875M

119873

(119884119873= 0)

+ 119875 (119885119873= 1 | 119884

119873= 1) 119875M

119873

(119884119873= 1)

= 119901119908(1 minus 119901

1) + 1199011198881199011= 1199011(119901119888minus 119901119908) + 119901119908

= 1198751 say

(3)




2)


119873 1198851198731015840 follows


1gt 1198752


119873 given

1198851198731015840 = 1199111198731015840 1198731015840isin 119877 is

119871119873= 119875M

119873

(1198851198731015840 = 1199111198731015840 forall1198731015840isin 119877)

= 119875119911119873

1(1 minus 119875

1)(1minus119911119873)

Π1198731015840isin119861(119873)

1198751199111198731015840

2(1 minus 119875

2)(1minus1199111198731015840 )

times Π1198731015840isin119877[119861(119873)cup119873]

1199011199111198731015840

119908 (1 minus 119901119908)(1minus1199111198731015840 )

= 119875119911119873

1(1 minus 119875

1)(1minus119911119873)

119875sum1198731015840isin119861(119873)119911

1198731015840

2(1 minus 119875

2)sum1198731015840isin119861(119873)(1minus1199111198731015840 )


1198731015840


(1minus1199111198731015840 )

(4)

Let 119879119873= sum1198731015840isin119861(119873)

1198851198731015840 so thatsum

1198731015840isin119861(119873)

(1 minus 1198851198731015840) = 119896(119873) minus


119905119873= sum

1198731015840isin119861(119873)

1199111198731015840 sum

1198731015840isin119861(119873)

(1 minus 1199111198731015840) = 119896 (119873) minus 119905

119873 (5)

Therefore

ln 119871119873= 119911119873ln1198751+ (1 minus 119911

119873) ln (1 minus 119875

1)

+ 119905119873ln1198752+ (119896 (119873) minus 119905

119873) ln (1 minus 119875

2)

+ sum

1198731015840isin119877[119861(119873)cup119873]

119911119873ln119901119908

+ sum

1198731015840isin119877[119861(119873)cup119873]

(1 minus 119911119873) ln (119901

119908(1 minus 119901

119908))

= ln 1198710+ 119911119873ln 1198751

119901119908

+ (1 minus 119911119873) ln 1 minus 119875

1

1 minus 119901119908

+ 119905119873ln 1198752

119901119908

+ (119896 (119873) minus 119905119873) ln 1 minus 119875

2

1 minus 119901119908

= ln 1198710+ 119911119873ln

1198751(1 minus 119901

119908)

119901119908(1 minus 119875

1)+ 119905119873ln

1198752(1 minus 119901

119908)

119901119908(1 minus 119875

2)

+ ln 1 minus 1198751

1 minus 119901119908

+ 119896 (119873) ln1 minus 1198752

1 minus 119901119908

= 119886 + 119887 (119888119911119873+ 119905119873minus 119889119896 (119873)) say

(6)

where

119886 = ln 1198710+ ln 1 minus 119875

1

1 minus 119901119908

119887 = ln1198752(1 minus 119901

119908)

119901119908(1 minus 119875

2)gt 0

119888 =ln (1198751(1 minus 119901

119908) 119901119908(1 minus 119875

1))

ln (1198752(1 minus 119901

119908) 119901119908(1 minus 119875

2))

119889 =ln ((1 minus 119901

119908) (1 minus 119875

2))

ln (1198752(1 minus 119901

119908) 119901119908(1 minus 119875

2))

(7)



0if

= ln 1 minus 1198751

1 minus 119901119908

+ 119887 (119888119911119873+ 119905119873minus 119889119896 (119873)) lt 0


(8)







119873 isin 119877




M119870

119871119870

max119873isin119877

119871119873

gt 119862 (9)



119873+119905119873minus119889119896(119873)



119876119873 (10)


lowastlt 1



119888119911119873+ 119905119873 (11)






119873isinNmax119861(119873)] that is 1198731015840



1M2 M

119896are


119896is given by

Pr (M119896| 119863) =

Pr (119863 | M119896)Pr (M

119896)

sumPr (119863 | M119897)Pr (M

119897) (12)



119896



119896is the


119873= 119911119873




Pr (M119873| 119885119873= 119911119873 119873 isin 119877) =

119901119873119871119873

sum119897isin119877

119901119897119871119897+ 119901norm1198710

and that forM0is

119901norm1198710sum119897isin119877

119901119897119871119897+ 119901norm1198710

(13)



for which 119901119873119871119873







0

(119879119873=

0 119885119873


119873

(119879119873=

6 119885119873


119873and 119885



(1) 119875M0

(119879119873= 119894 119885

119873= 0) = ( 119896(119873)

119894) 119901119894

119908(1 minus 119901

119908)119896(119873)minus119894+1

(2) 119875M0

(119879119873= 119894 119885

119873= 1) = ( 119896(119873)

119894) 119901119894+1

119908(1 minus 119901

119908)119896(119873)minus119894

(3) 119875M119873

(119879119873

= 119894 119885119873

= 0) = ( 119896(119873)119894) 119875119894

2(1 minus 119875

2)119896(119873)minus119894

(1 minus

1198751)

(4) 119875M119873

(119879119873= 119894 119885

119873= 1) = ( 119896(119873)

119894) 119875119894

2(1 minus 119875

2)119896(119873)minus119894

1198751


119875M0

(119871119873gt 1198710)

= 119875M0

(ln 119871119873gt ln 119871

0)

= 119875M0

(119885119873ln

1198751(1 minus 119901

119908)

119901119908(1 minus 119875

1)+ 119879119873ln

1198752(1 minus 119901

119908)

119901119908(1 minus 119875

2)

+ ln 1 minus 1198751

1 minus 119901119908

+ 119896 (119873) ln1 minus 1198752

1 minus 119901119908

gt 0)

= 119875M0

(119885119873ln

1198751(1 minus 119901

119908)

119901119908(1 minus 119875

1)+ 119879119873ln

1198752(1 minus 119901

119908)

119901119908(1 minus 119875

2)

gt 119896 (119873) ln1 minus 119901119908

1 minus 1198752

+ ln1 minus 119901119908

1 minus 1198751

)

(14)


119873and 119885


0 The maximum of




119875M119873

(119871119873lt 1198710)

= 119875M119873

(119885119873ln

1198751(1 minus 119901

119908)

119901119908(1 minus 119875

1)+ 119879119873ln

1198752(1 minus 119901

119908)

119901119908(1 minus 119875

2)

lt 119896 (119873) ln1 minus 119901119908

1 minus 1198752

+ ln1 minus 119901119908

1 minus 1198751

)

(15)


119873and 119885


119873 This



forwhich 119876




119873under the




= 119894 and 119879119873

= 119895 for 119894 = 0 1 and 119895 =


and 119895 = 0 1 119896(119873)

119875M119873

(119876119873= 119888119894 + 119895 minus 119889119896 (119873))

= (119896 (119873)

119895) (1198751)119894

(1 minus 1198751)(1minus119894)

(1198752)119895

(1 minus 1198752)(119896(119873)minus119895)

119875M0

(119876119873= 119888119894 + 119895 minus 119889119896 (119873))

= (119896 (119873)

119895) (119901119908)119894+119895


(16)



0when M





5 Simulation Study




1denote




0

(0 isin




2denote the












119873



1198731015840 is a





119873


119861(1198731015840)) Clearly

1198784

ge 1198783






119875M119873





Clearly 1198731= 1 minus 119878


0



2le 1 In our





3ge 1 ge 119873

2


119894rsquos


1 1199012 119901119888

and 119901119908with 119901

1and 119901


119908taking














119901119908



119908= 00001 (not shown in






119873





1199011and 119901






2increases it





1 On


and 119901119888 Since 119901

2= 0 means 119875



When 119901119908= 001 effect of 119901

2on 1198783 1198784 and 119878

5and119873

3 1198734



119894rsquos and 119873

119894rsquos


2= 03 for 119901

119908= 001

but not 119901119908



3and 119878



3





(119871119873gt 119862 sdot 119871max)


5corresponds to


119908=


2= 03 04 05 06


2increases


and 1199012

6 Discussion



119873+





1and 1198752






1198751and 119888 increase as 119901


2and 119889 do not



2


2 Therefore the






0 119885119873follows Ber(119901

119908) for all 119873 isin 119877


119873given M

0 So we



119908


119908


119873under M

119873 So





1 Similar


119888as well




Let 120579119873

= (1199011119873 1199012119873 119901119888119873 119901119908119873


119873 119873 isin 119877 are iid from






1199011

1199012

119901119908

1198781

1198782

1198783

1198733

1198784

1198734

1198785

1198735

09 00 001 000 008 081 1816 081 5985 082 181709 03 001 000 047 069 544 070 1481 073 57009 04 001 000 060 079 511 078 1351 080 51209 05 001 000 070 085 464 085 1150 086 48809 06 001 000 079 090 382 091 0818 092 39009 00 0001 035 050 081 317 081 717 081 31809 03 0001 036 059 082 303 083 634 086 30609 04 0001 035 067 087 289 087 618 089 30309 05 0001 036 075 090 289 089 585 093 29609 06 0001 036 083 094 274 093 533 096 279099 00 001 000 008 089 1743 090 5620 089 1770099 03 001 000 051 073 518 073 1298 079 577099 04 001 000 062 081 509 082 1301 084 521099 05 001 000 073 088 497 088 1269 089 504099 06 001 000 081 092 366 092 735 093 357099 00 0001 035 057 089 319 089 669 089 320099 03 0001 035 062 088 299 087 600 090 302099 04 0001 036 070 090 291 091 583 093 297099 05 0001 036 079 093 282 094 567 095 283099 06 0001 036 084 095 275 095 531 097 268


1199011

1199012

119901119908

1198781

1198782

1198783

1198733

1198784

1198734

1198785

1198735

09 00 001 000 008 090 182 089 5562 090 175809 03 001 000 054 076 514 076 1308 079 56409 04 001 000 067 085 505 085 1292 087 51209 05 001 000 077 091 486 090 1210 091 50209 06 001 000 086 094 357 093 724 095 35709 00 0001 036 057 090 318 089 661 089 31909 03 0001 036 065 088 298 089 634 092 30209 04 0001 036 073 092 287 092 591 094 29109 05 0001 035 081 094 281 094 551 096 28209 06 0001 037 088 096 272 096 524 097 290099 00 001 000 009 098 169 098 5140 098 176099 03 001 000 058 083 566 083 1473 087 569099 04 001 000 069 090 543 091 1438 092 569099 05 001 000 080 094 461 094 1119 095 473099 06 001 000 087 096 326 097 626 096 335099 00 0001 035 062 098 320 098 632 098 328099 03 0001 036 069 094 290 094 588 097 300099 04 0001 036 076 095 289 096 559 098 285099 05 0001 036 083 097 270 097 548 098 280099 06 0001 036 089 098 268 098 519 099 269




Π119873isin119877

int119901119911119873

119908119873(1 minus 119901

119908119873)(1minus119911119873)

119892119908(119901119908119873

) 119889119901119908119873

(17)




Π1198731015840isin119877

int119871(1198731015840)

119873(120579119873) 119892 (120579119873) 119889120579119873 (18)


119873and 119871(119873

1015840)




119888= 1199011= 09


1199012

119901119908



119888= 09

Parameters 119901119908= 01 119901

119908= 02

1199011

1199012

1198751

1198752

119888 119889 1198751

1198752

119888 119889

07

03 066 034 1865 0202 069 041 2139 029804 066 042 1526 0234 069 048 1674 033005 066 050 1302 0268 069 055 1378 036306 066 058 1135 0302 069 062 1166 0397

08

03 074 034 2114 0202 076 041 2484 029804 074 042 1730 0234 076 048 1944 033005 074 050 1476 0268 076 055 1600 036306 074 058 1287 0302 076 062 1354 0397

09

03 082 034 0241 0202 083 041 2907 029804 082 042 1981 0234 083 048 2275 033005 082 050 1690 0268 083 055 1873 036306 082 058 1474 0302 083 062 1584 0397





0andM

119873are


119901119911119873

119908(1 minus 119901

119908)(1minus119911119873)

119892120579(119901119908) 119889119901119908

intΠ1198731015840isin119877119871(1198731015840)

119873(120579) 119892 (120579) 119889120579 respectively

(19)






1and 119901


119888and 119901

119908)








Acknowledgments


References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










+ 119875M0

(119885119873= 1 | 119884

119873= 1) 119875M

0

(119884119873= 1)

= 119875 (119885119873= 1 | 119884

119873= 0) 119875M

0

(119884119873= 0)

+ 119875 (119885119873= 1 | 119884

119873= 1) 119875M

0

(119884119873= 1) = 119901

119908

(1)


119908) for all119873 isin



1198710= 119875M

0

(119885119873= 119911119873 forall119873 isin 119877)

= prod

119873isin119877

119901119911119873

119908(1 minus 119901

119908)(1minus119911119873)

= (119901119908)sum119873isin119877119911119873


(2)

So ln 1198710= sum119873isin119877

119911119873ln119901119908+ sum119873isin119877

(1 minus 119911119873) ln(1 minus 119901

119908)


(119885119873= 1)

= 119875M119873

(119885119873= 1 | 119884

119873= 0) 119875M

119873

(119884119873= 0)

+ 119875M119873

(119885119873= 1 | 119884

119873= 1) 119875M

119873

(119884119873= 1)

= 119875 (119885119873= 1 | 119884

119873= 0) 119875M

119873

(119884119873= 0)

+ 119875 (119885119873= 1 | 119884

119873= 1) 119875M

119873

(119884119873= 1)

= 119901119908(1 minus 119901

1) + 1199011198881199011= 1199011(119901119888minus 119901119908) + 119901119908

= 1198751 say

(3)




2)


119873 1198851198731015840 follows


1gt 1198752


119873 given

1198851198731015840 = 1199111198731015840 1198731015840isin 119877 is

119871119873= 119875M

119873

(1198851198731015840 = 1199111198731015840 forall1198731015840isin 119877)

= 119875119911119873

1(1 minus 119875

1)(1minus119911119873)

Π1198731015840isin119861(119873)

1198751199111198731015840

2(1 minus 119875

2)(1minus1199111198731015840 )

times Π1198731015840isin119877[119861(119873)cup119873]

1199011199111198731015840

119908 (1 minus 119901119908)(1minus1199111198731015840 )

= 119875119911119873

1(1 minus 119875

1)(1minus119911119873)

119875sum1198731015840isin119861(119873)119911

1198731015840

2(1 minus 119875

2)sum1198731015840isin119861(119873)(1minus1199111198731015840 )


1198731015840


(1minus1199111198731015840 )

(4)

Let 119879119873= sum1198731015840isin119861(119873)

1198851198731015840 so thatsum

1198731015840isin119861(119873)

(1 minus 1198851198731015840) = 119896(119873) minus


119905119873= sum

1198731015840isin119861(119873)

1199111198731015840 sum

1198731015840isin119861(119873)

(1 minus 1199111198731015840) = 119896 (119873) minus 119905

119873 (5)

Therefore

ln 119871119873= 119911119873ln1198751+ (1 minus 119911

119873) ln (1 minus 119875

1)

+ 119905119873ln1198752+ (119896 (119873) minus 119905

119873) ln (1 minus 119875

2)

+ sum

1198731015840isin119877[119861(119873)cup119873]

119911119873ln119901119908

+ sum

1198731015840isin119877[119861(119873)cup119873]

(1 minus 119911119873) ln (119901

119908(1 minus 119901

119908))

= ln 1198710+ 119911119873ln 1198751

119901119908

+ (1 minus 119911119873) ln 1 minus 119875

1

1 minus 119901119908

+ 119905119873ln 1198752

119901119908

+ (119896 (119873) minus 119905119873) ln 1 minus 119875

2

1 minus 119901119908

= ln 1198710+ 119911119873ln

1198751(1 minus 119901

119908)

119901119908(1 minus 119875

1)+ 119905119873ln

1198752(1 minus 119901

119908)

119901119908(1 minus 119875

2)

+ ln 1 minus 1198751

1 minus 119901119908

+ 119896 (119873) ln1 minus 1198752

1 minus 119901119908

= 119886 + 119887 (119888119911119873+ 119905119873minus 119889119896 (119873)) say

(6)

where

119886 = ln 1198710+ ln 1 minus 119875

1

1 minus 119901119908

119887 = ln1198752(1 minus 119901

119908)

119901119908(1 minus 119875

2)gt 0

119888 =ln (1198751(1 minus 119901

119908) 119901119908(1 minus 119875

1))

ln (1198752(1 minus 119901

119908) 119901119908(1 minus 119875

2))

119889 =ln ((1 minus 119901

119908) (1 minus 119875

2))

ln (1198752(1 minus 119901

119908) 119901119908(1 minus 119875

2))

(7)



0if

= ln 1 minus 1198751

1 minus 119901119908

+ 119887 (119888119911119873+ 119905119873minus 119889119896 (119873)) lt 0


(8)







119873 isin 119877




M119870

119871119870

max119873isin119877

119871119873

gt 119862 (9)



119873+119905119873minus119889119896(119873)



119876119873 (10)


lowastlt 1



119888119911119873+ 119905119873 (11)






119873isinNmax119861(119873)] that is 1198731015840



1M2 M

119896are


119896is given by

Pr (M119896| 119863) =

Pr (119863 | M119896)Pr (M

119896)

sumPr (119863 | M119897)Pr (M

119897) (12)



119896



119896is the


119873= 119911119873




Pr (M119873| 119885119873= 119911119873 119873 isin 119877) =

119901119873119871119873

sum119897isin119877

119901119897119871119897+ 119901norm1198710

and that forM0is

119901norm1198710sum119897isin119877

119901119897119871119897+ 119901norm1198710

(13)



for which 119901119873119871119873







0

(119879119873=

0 119885119873


119873

(119879119873=

6 119885119873


119873and 119885



(1) 119875M0

(119879119873= 119894 119885

119873= 0) = ( 119896(119873)

119894) 119901119894

119908(1 minus 119901

119908)119896(119873)minus119894+1

(2) 119875M0

(119879119873= 119894 119885

119873= 1) = ( 119896(119873)

119894) 119901119894+1

119908(1 minus 119901

119908)119896(119873)minus119894

(3) 119875M119873

(119879119873

= 119894 119885119873

= 0) = ( 119896(119873)119894) 119875119894

2(1 minus 119875

2)119896(119873)minus119894

(1 minus

1198751)

(4) 119875M119873

(119879119873= 119894 119885

119873= 1) = ( 119896(119873)

119894) 119875119894

2(1 minus 119875

2)119896(119873)minus119894

1198751


119875M0

(119871119873gt 1198710)

= 119875M0

(ln 119871119873gt ln 119871

0)

= 119875M0

(119885119873ln

1198751(1 minus 119901

119908)

119901119908(1 minus 119875

1)+ 119879119873ln

1198752(1 minus 119901

119908)

119901119908(1 minus 119875

2)

+ ln 1 minus 1198751

1 minus 119901119908

+ 119896 (119873) ln1 minus 1198752

1 minus 119901119908

gt 0)

= 119875M0

(119885119873ln

1198751(1 minus 119901

119908)

119901119908(1 minus 119875

1)+ 119879119873ln

1198752(1 minus 119901

119908)

119901119908(1 minus 119875

2)

gt 119896 (119873) ln1 minus 119901119908

1 minus 1198752

+ ln1 minus 119901119908

1 minus 1198751

)

(14)


119873and 119885


0 The maximum of




119875M119873

(119871119873lt 1198710)

= 119875M119873

(119885119873ln

1198751(1 minus 119901

119908)

119901119908(1 minus 119875

1)+ 119879119873ln

1198752(1 minus 119901

119908)

119901119908(1 minus 119875

2)

lt 119896 (119873) ln1 minus 119901119908

1 minus 1198752

+ ln1 minus 119901119908

1 minus 1198751

)

(15)


119873and 119885


119873 This



forwhich 119876




119873under the




= 119894 and 119879119873

= 119895 for 119894 = 0 1 and 119895 =


and 119895 = 0 1 119896(119873)

119875M119873

(119876119873= 119888119894 + 119895 minus 119889119896 (119873))

= (119896 (119873)

119895) (1198751)119894

(1 minus 1198751)(1minus119894)

(1198752)119895

(1 minus 1198752)(119896(119873)minus119895)

119875M0

(119876119873= 119888119894 + 119895 minus 119889119896 (119873))

= (119896 (119873)

119895) (119901119908)119894+119895


(16)



0when M





5 Simulation Study




1denote




0

(0 isin




2denote the












119873



1198731015840 is a





119873


119861(1198731015840)) Clearly

1198784

ge 1198783






119875M119873





Clearly 1198731= 1 minus 119878


0



2le 1 In our





3ge 1 ge 119873

2


119894rsquos


1 1199012 119901119888

and 119901119908with 119901

1and 119901


119908taking














119901119908



119908= 00001 (not shown in






119873





1199011and 119901






2increases it





1 On


and 119901119888 Since 119901

2= 0 means 119875



When 119901119908= 001 effect of 119901

2on 1198783 1198784 and 119878

5and119873

3 1198734



119894rsquos and 119873

119894rsquos


2= 03 for 119901

119908= 001

but not 119901119908



3and 119878



3





(119871119873gt 119862 sdot 119871max)


5corresponds to


119908=


2= 03 04 05 06


2increases


and 1199012

6 Discussion



119873+





1and 1198752






1198751and 119888 increase as 119901


2and 119889 do not



2


2 Therefore the






0 119885119873follows Ber(119901

119908) for all 119873 isin 119877


119873given M

0 So we



119908


119908


119873under M

119873 So





1 Similar


119888as well




Let 120579119873

= (1199011119873 1199012119873 119901119888119873 119901119908119873


119873 119873 isin 119877 are iid from






1199011

1199012

119901119908

1198781

1198782

1198783

1198733

1198784

1198734

1198785

1198735

09 00 001 000 008 081 1816 081 5985 082 181709 03 001 000 047 069 544 070 1481 073 57009 04 001 000 060 079 511 078 1351 080 51209 05 001 000 070 085 464 085 1150 086 48809 06 001 000 079 090 382 091 0818 092 39009 00 0001 035 050 081 317 081 717 081 31809 03 0001 036 059 082 303 083 634 086 30609 04 0001 035 067 087 289 087 618 089 30309 05 0001 036 075 090 289 089 585 093 29609 06 0001 036 083 094 274 093 533 096 279099 00 001 000 008 089 1743 090 5620 089 1770099 03 001 000 051 073 518 073 1298 079 577099 04 001 000 062 081 509 082 1301 084 521099 05 001 000 073 088 497 088 1269 089 504099 06 001 000 081 092 366 092 735 093 357099 00 0001 035 057 089 319 089 669 089 320099 03 0001 035 062 088 299 087 600 090 302099 04 0001 036 070 090 291 091 583 093 297099 05 0001 036 079 093 282 094 567 095 283099 06 0001 036 084 095 275 095 531 097 268


1199011

1199012

119901119908

1198781

1198782

1198783

1198733

1198784

1198734

1198785

1198735

09 00 001 000 008 090 182 089 5562 090 175809 03 001 000 054 076 514 076 1308 079 56409 04 001 000 067 085 505 085 1292 087 51209 05 001 000 077 091 486 090 1210 091 50209 06 001 000 086 094 357 093 724 095 35709 00 0001 036 057 090 318 089 661 089 31909 03 0001 036 065 088 298 089 634 092 30209 04 0001 036 073 092 287 092 591 094 29109 05 0001 035 081 094 281 094 551 096 28209 06 0001 037 088 096 272 096 524 097 290099 00 001 000 009 098 169 098 5140 098 176099 03 001 000 058 083 566 083 1473 087 569099 04 001 000 069 090 543 091 1438 092 569099 05 001 000 080 094 461 094 1119 095 473099 06 001 000 087 096 326 097 626 096 335099 00 0001 035 062 098 320 098 632 098 328099 03 0001 036 069 094 290 094 588 097 300099 04 0001 036 076 095 289 096 559 098 285099 05 0001 036 083 097 270 097 548 098 280099 06 0001 036 089 098 268 098 519 099 269




Π119873isin119877

int119901119911119873

119908119873(1 minus 119901

119908119873)(1minus119911119873)

119892119908(119901119908119873

) 119889119901119908119873

(17)




Π1198731015840isin119877

int119871(1198731015840)

119873(120579119873) 119892 (120579119873) 119889120579119873 (18)


119873and 119871(119873

1015840)




119888= 1199011= 09


1199012

119901119908



119888= 09

Parameters 119901119908= 01 119901

119908= 02

1199011

1199012

1198751

1198752

119888 119889 1198751

1198752

119888 119889

07

03 066 034 1865 0202 069 041 2139 029804 066 042 1526 0234 069 048 1674 033005 066 050 1302 0268 069 055 1378 036306 066 058 1135 0302 069 062 1166 0397

08

03 074 034 2114 0202 076 041 2484 029804 074 042 1730 0234 076 048 1944 033005 074 050 1476 0268 076 055 1600 036306 074 058 1287 0302 076 062 1354 0397

09

03 082 034 0241 0202 083 041 2907 029804 082 042 1981 0234 083 048 2275 033005 082 050 1690 0268 083 055 1873 036306 082 058 1474 0302 083 062 1584 0397





0andM

119873are


119901119911119873

119908(1 minus 119901

119908)(1minus119911119873)

119892120579(119901119908) 119889119901119908

intΠ1198731015840isin119877119871(1198731015840)

119873(120579) 119892 (120579) 119889120579 respectively

(19)






1and 119901


119888and 119901

119908)








Acknowledgments


References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











M119870

119871119870

max119873isin119877

119871119873

gt 119862 (9)



119873+119905119873minus119889119896(119873)



119876119873 (10)


lowastlt 1



119888119911119873+ 119905119873 (11)






119873isinNmax119861(119873)] that is 1198731015840



1M2 M

119896are


119896is given by

Pr (M119896| 119863) =

Pr (119863 | M119896)Pr (M

119896)

sumPr (119863 | M119897)Pr (M

119897) (12)



119896



119896is the


119873= 119911119873




Pr (M119873| 119885119873= 119911119873 119873 isin 119877) =

119901119873119871119873

sum119897isin119877

119901119897119871119897+ 119901norm1198710

and that forM0is

119901norm1198710sum119897isin119877

119901119897119871119897+ 119901norm1198710

(13)



for which 119901119873119871119873







0

(119879119873=

0 119885119873


119873

(119879119873=

6 119885119873


119873and 119885



(1) 119875M0

(119879119873= 119894 119885

119873= 0) = ( 119896(119873)

119894) 119901119894

119908(1 minus 119901

119908)119896(119873)minus119894+1

(2) 119875M0

(119879119873= 119894 119885

119873= 1) = ( 119896(119873)

119894) 119901119894+1

119908(1 minus 119901

119908)119896(119873)minus119894

(3) 119875M119873

(119879119873

= 119894 119885119873

= 0) = ( 119896(119873)119894) 119875119894

2(1 minus 119875

2)119896(119873)minus119894

(1 minus

1198751)

(4) 119875M119873

(119879119873= 119894 119885

119873= 1) = ( 119896(119873)

119894) 119875119894

2(1 minus 119875

2)119896(119873)minus119894

1198751


119875M0

(119871119873gt 1198710)

= 119875M0

(ln 119871119873gt ln 119871

0)

= 119875M0

(119885119873ln

1198751(1 minus 119901

119908)

119901119908(1 minus 119875

1)+ 119879119873ln

1198752(1 minus 119901

119908)

119901119908(1 minus 119875

2)

+ ln 1 minus 1198751

1 minus 119901119908

+ 119896 (119873) ln1 minus 1198752

1 minus 119901119908

gt 0)

= 119875M0

(119885119873ln

1198751(1 minus 119901

119908)

119901119908(1 minus 119875

1)+ 119879119873ln

1198752(1 minus 119901

119908)

119901119908(1 minus 119875

2)

gt 119896 (119873) ln1 minus 119901119908

1 minus 1198752

+ ln1 minus 119901119908

1 minus 1198751

)

(14)


119873and 119885


0 The maximum of




119875M119873

(119871119873lt 1198710)

= 119875M119873

(119885119873ln

1198751(1 minus 119901

119908)

119901119908(1 minus 119875

1)+ 119879119873ln

1198752(1 minus 119901

119908)

119901119908(1 minus 119875

2)

lt 119896 (119873) ln1 minus 119901119908

1 minus 1198752

+ ln1 minus 119901119908

1 minus 1198751

)

(15)


119873and 119885


119873 This



forwhich 119876




119873under the




= 119894 and 119879119873

= 119895 for 119894 = 0 1 and 119895 =


and 119895 = 0 1 119896(119873)

119875M119873

(119876119873= 119888119894 + 119895 minus 119889119896 (119873))

= (119896 (119873)

119895) (1198751)119894

(1 minus 1198751)(1minus119894)

(1198752)119895

(1 minus 1198752)(119896(119873)minus119895)

119875M0

(119876119873= 119888119894 + 119895 minus 119889119896 (119873))

= (119896 (119873)

119895) (119901119908)119894+119895


(16)



0when M





5 Simulation Study




1denote




0

(0 isin




2denote the












119873



1198731015840 is a





119873


119861(1198731015840)) Clearly

1198784

ge 1198783






119875M119873





Clearly 1198731= 1 minus 119878


0



2le 1 In our





3ge 1 ge 119873

2


119894rsquos


1 1199012 119901119888

and 119901119908with 119901

1and 119901


119908taking














119901119908



119908= 00001 (not shown in






119873





1199011and 119901






2increases it





1 On


and 119901119888 Since 119901

2= 0 means 119875



When 119901119908= 001 effect of 119901

2on 1198783 1198784 and 119878

5and119873

3 1198734



119894rsquos and 119873

119894rsquos


2= 03 for 119901

119908= 001

but not 119901119908



3and 119878



3





(119871119873gt 119862 sdot 119871max)


5corresponds to


119908=


2= 03 04 05 06


2increases


and 1199012

6 Discussion



119873+





1and 1198752






1198751and 119888 increase as 119901


2and 119889 do not



2


2 Therefore the






0 119885119873follows Ber(119901

119908) for all 119873 isin 119877


119873given M

0 So we



119908


119908


119873under M

119873 So





1 Similar


119888as well




Let 120579119873

= (1199011119873 1199012119873 119901119888119873 119901119908119873


119873 119873 isin 119877 are iid from






1199011

1199012

119901119908

1198781

1198782

1198783

1198733

1198784

1198734

1198785

1198735

09 00 001 000 008 081 1816 081 5985 082 181709 03 001 000 047 069 544 070 1481 073 57009 04 001 000 060 079 511 078 1351 080 51209 05 001 000 070 085 464 085 1150 086 48809 06 001 000 079 090 382 091 0818 092 39009 00 0001 035 050 081 317 081 717 081 31809 03 0001 036 059 082 303 083 634 086 30609 04 0001 035 067 087 289 087 618 089 30309 05 0001 036 075 090 289 089 585 093 29609 06 0001 036 083 094 274 093 533 096 279099 00 001 000 008 089 1743 090 5620 089 1770099 03 001 000 051 073 518 073 1298 079 577099 04 001 000 062 081 509 082 1301 084 521099 05 001 000 073 088 497 088 1269 089 504099 06 001 000 081 092 366 092 735 093 357099 00 0001 035 057 089 319 089 669 089 320099 03 0001 035 062 088 299 087 600 090 302099 04 0001 036 070 090 291 091 583 093 297099 05 0001 036 079 093 282 094 567 095 283099 06 0001 036 084 095 275 095 531 097 268


1199011

1199012

119901119908

1198781

1198782

1198783

1198733

1198784

1198734

1198785

1198735

09 00 001 000 008 090 182 089 5562 090 175809 03 001 000 054 076 514 076 1308 079 56409 04 001 000 067 085 505 085 1292 087 51209 05 001 000 077 091 486 090 1210 091 50209 06 001 000 086 094 357 093 724 095 35709 00 0001 036 057 090 318 089 661 089 31909 03 0001 036 065 088 298 089 634 092 30209 04 0001 036 073 092 287 092 591 094 29109 05 0001 035 081 094 281 094 551 096 28209 06 0001 037 088 096 272 096 524 097 290099 00 001 000 009 098 169 098 5140 098 176099 03 001 000 058 083 566 083 1473 087 569099 04 001 000 069 090 543 091 1438 092 569099 05 001 000 080 094 461 094 1119 095 473099 06 001 000 087 096 326 097 626 096 335099 00 0001 035 062 098 320 098 632 098 328099 03 0001 036 069 094 290 094 588 097 300099 04 0001 036 076 095 289 096 559 098 285099 05 0001 036 083 097 270 097 548 098 280099 06 0001 036 089 098 268 098 519 099 269




Π119873isin119877

int119901119911119873

119908119873(1 minus 119901

119908119873)(1minus119911119873)

119892119908(119901119908119873

) 119889119901119908119873

(17)




Π1198731015840isin119877

int119871(1198731015840)

119873(120579119873) 119892 (120579119873) 119889120579119873 (18)


119873and 119871(119873

1015840)




119888= 1199011= 09


1199012

119901119908



119888= 09

Parameters 119901119908= 01 119901

119908= 02

1199011

1199012

1198751

1198752

119888 119889 1198751

1198752

119888 119889

07

03 066 034 1865 0202 069 041 2139 029804 066 042 1526 0234 069 048 1674 033005 066 050 1302 0268 069 055 1378 036306 066 058 1135 0302 069 062 1166 0397

08

03 074 034 2114 0202 076 041 2484 029804 074 042 1730 0234 076 048 1944 033005 074 050 1476 0268 076 055 1600 036306 074 058 1287 0302 076 062 1354 0397

09

03 082 034 0241 0202 083 041 2907 029804 082 042 1981 0234 083 048 2275 033005 082 050 1690 0268 083 055 1873 036306 082 058 1474 0302 083 062 1584 0397





0andM

119873are


119901119911119873

119908(1 minus 119901

119908)(1minus119911119873)

119892120579(119901119908) 119889119901119908

intΠ1198731015840isin119877119871(1198731015840)

119873(120579) 119892 (120579) 119889120579 respectively

(19)






1and 119901


119888and 119901

119908)








Acknowledgments


References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119875M119873

(119871119873lt 1198710)

= 119875M119873

(119885119873ln

1198751(1 minus 119901

119908)

119901119908(1 minus 119875

1)+ 119879119873ln

1198752(1 minus 119901

119908)

119901119908(1 minus 119875

2)

lt 119896 (119873) ln1 minus 119901119908

1 minus 1198752

+ ln1 minus 119901119908

1 minus 1198751

)

(15)


119873and 119885


119873 This



forwhich 119876




119873under the




= 119894 and 119879119873

= 119895 for 119894 = 0 1 and 119895 =


and 119895 = 0 1 119896(119873)

119875M119873

(119876119873= 119888119894 + 119895 minus 119889119896 (119873))

= (119896 (119873)

119895) (1198751)119894

(1 minus 1198751)(1minus119894)

(1198752)119895

(1 minus 1198752)(119896(119873)minus119895)

119875M0

(119876119873= 119888119894 + 119895 minus 119889119896 (119873))

= (119896 (119873)

119895) (119901119908)119894+119895


(16)



0when M





5 Simulation Study




1denote




0

(0 isin




2denote the












119873



1198731015840 is a





119873


119861(1198731015840)) Clearly

1198784

ge 1198783






119875M119873





Clearly 1198731= 1 minus 119878


0



2le 1 In our





3ge 1 ge 119873

2


119894rsquos


1 1199012 119901119888

and 119901119908with 119901

1and 119901


119908taking














119901119908



119908= 00001 (not shown in






119873





1199011and 119901






2increases it





1 On


and 119901119888 Since 119901

2= 0 means 119875



When 119901119908= 001 effect of 119901

2on 1198783 1198784 and 119878

5and119873

3 1198734



119894rsquos and 119873

119894rsquos


2= 03 for 119901

119908= 001

but not 119901119908



3and 119878



3





(119871119873gt 119862 sdot 119871max)


5corresponds to


119908=


2= 03 04 05 06


2increases


and 1199012

6 Discussion



119873+





1and 1198752






1198751and 119888 increase as 119901


2and 119889 do not



2


2 Therefore the






0 119885119873follows Ber(119901

119908) for all 119873 isin 119877


119873given M

0 So we



119908


119908


119873under M

119873 So





1 Similar


119888as well




Let 120579119873

= (1199011119873 1199012119873 119901119888119873 119901119908119873


119873 119873 isin 119877 are iid from






1199011

1199012

119901119908

1198781

1198782

1198783

1198733

1198784

1198734

1198785

1198735

09 00 001 000 008 081 1816 081 5985 082 181709 03 001 000 047 069 544 070 1481 073 57009 04 001 000 060 079 511 078 1351 080 51209 05 001 000 070 085 464 085 1150 086 48809 06 001 000 079 090 382 091 0818 092 39009 00 0001 035 050 081 317 081 717 081 31809 03 0001 036 059 082 303 083 634 086 30609 04 0001 035 067 087 289 087 618 089 30309 05 0001 036 075 090 289 089 585 093 29609 06 0001 036 083 094 274 093 533 096 279099 00 001 000 008 089 1743 090 5620 089 1770099 03 001 000 051 073 518 073 1298 079 577099 04 001 000 062 081 509 082 1301 084 521099 05 001 000 073 088 497 088 1269 089 504099 06 001 000 081 092 366 092 735 093 357099 00 0001 035 057 089 319 089 669 089 320099 03 0001 035 062 088 299 087 600 090 302099 04 0001 036 070 090 291 091 583 093 297099 05 0001 036 079 093 282 094 567 095 283099 06 0001 036 084 095 275 095 531 097 268


1199011

1199012

119901119908

1198781

1198782

1198783

1198733

1198784

1198734

1198785

1198735

09 00 001 000 008 090 182 089 5562 090 175809 03 001 000 054 076 514 076 1308 079 56409 04 001 000 067 085 505 085 1292 087 51209 05 001 000 077 091 486 090 1210 091 50209 06 001 000 086 094 357 093 724 095 35709 00 0001 036 057 090 318 089 661 089 31909 03 0001 036 065 088 298 089 634 092 30209 04 0001 036 073 092 287 092 591 094 29109 05 0001 035 081 094 281 094 551 096 28209 06 0001 037 088 096 272 096 524 097 290099 00 001 000 009 098 169 098 5140 098 176099 03 001 000 058 083 566 083 1473 087 569099 04 001 000 069 090 543 091 1438 092 569099 05 001 000 080 094 461 094 1119 095 473099 06 001 000 087 096 326 097 626 096 335099 00 0001 035 062 098 320 098 632 098 328099 03 0001 036 069 094 290 094 588 097 300099 04 0001 036 076 095 289 096 559 098 285099 05 0001 036 083 097 270 097 548 098 280099 06 0001 036 089 098 268 098 519 099 269




Π119873isin119877

int119901119911119873

119908119873(1 minus 119901

119908119873)(1minus119911119873)

119892119908(119901119908119873

) 119889119901119908119873

(17)




Π1198731015840isin119877

int119871(1198731015840)

119873(120579119873) 119892 (120579119873) 119889120579119873 (18)


119873and 119871(119873

1015840)




119888= 1199011= 09


1199012

119901119908



119888= 09

Parameters 119901119908= 01 119901

119908= 02

1199011

1199012

1198751

1198752

119888 119889 1198751

1198752

119888 119889

07

03 066 034 1865 0202 069 041 2139 029804 066 042 1526 0234 069 048 1674 033005 066 050 1302 0268 069 055 1378 036306 066 058 1135 0302 069 062 1166 0397

08

03 074 034 2114 0202 076 041 2484 029804 074 042 1730 0234 076 048 1944 033005 074 050 1476 0268 076 055 1600 036306 074 058 1287 0302 076 062 1354 0397

09

03 082 034 0241 0202 083 041 2907 029804 082 042 1981 0234 083 048 2275 033005 082 050 1690 0268 083 055 1873 036306 082 058 1474 0302 083 062 1584 0397





0andM

119873are


119901119911119873

119908(1 minus 119901

119908)(1minus119911119873)

119892120579(119901119908) 119889119901119908

intΠ1198731015840isin119877119871(1198731015840)

119873(120579) 119892 (120579) 119889120579 respectively

(19)






1and 119901


119888and 119901

119908)








Acknowledgments


References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation


















119901119908



119908= 00001 (not shown in






119873





1199011and 119901






2increases it





1 On


and 119901119888 Since 119901

2= 0 means 119875



When 119901119908= 001 effect of 119901

2on 1198783 1198784 and 119878

5and119873

3 1198734



119894rsquos and 119873

119894rsquos


2= 03 for 119901

119908= 001

but not 119901119908



3and 119878



3





(119871119873gt 119862 sdot 119871max)


5corresponds to


119908=


2= 03 04 05 06


2increases


and 1199012

6 Discussion



119873+





1and 1198752






1198751and 119888 increase as 119901


2and 119889 do not



2


2 Therefore the






0 119885119873follows Ber(119901

119908) for all 119873 isin 119877


119873given M

0 So we



119908


119908


119873under M

119873 So





1 Similar


119888as well




Let 120579119873

= (1199011119873 1199012119873 119901119888119873 119901119908119873


119873 119873 isin 119877 are iid from






1199011

1199012

119901119908

1198781

1198782

1198783

1198733

1198784

1198734

1198785

1198735

09 00 001 000 008 081 1816 081 5985 082 181709 03 001 000 047 069 544 070 1481 073 57009 04 001 000 060 079 511 078 1351 080 51209 05 001 000 070 085 464 085 1150 086 48809 06 001 000 079 090 382 091 0818 092 39009 00 0001 035 050 081 317 081 717 081 31809 03 0001 036 059 082 303 083 634 086 30609 04 0001 035 067 087 289 087 618 089 30309 05 0001 036 075 090 289 089 585 093 29609 06 0001 036 083 094 274 093 533 096 279099 00 001 000 008 089 1743 090 5620 089 1770099 03 001 000 051 073 518 073 1298 079 577099 04 001 000 062 081 509 082 1301 084 521099 05 001 000 073 088 497 088 1269 089 504099 06 001 000 081 092 366 092 735 093 357099 00 0001 035 057 089 319 089 669 089 320099 03 0001 035 062 088 299 087 600 090 302099 04 0001 036 070 090 291 091 583 093 297099 05 0001 036 079 093 282 094 567 095 283099 06 0001 036 084 095 275 095 531 097 268


1199011

1199012

119901119908

1198781

1198782

1198783

1198733

1198784

1198734

1198785

1198735

09 00 001 000 008 090 182 089 5562 090 175809 03 001 000 054 076 514 076 1308 079 56409 04 001 000 067 085 505 085 1292 087 51209 05 001 000 077 091 486 090 1210 091 50209 06 001 000 086 094 357 093 724 095 35709 00 0001 036 057 090 318 089 661 089 31909 03 0001 036 065 088 298 089 634 092 30209 04 0001 036 073 092 287 092 591 094 29109 05 0001 035 081 094 281 094 551 096 28209 06 0001 037 088 096 272 096 524 097 290099 00 001 000 009 098 169 098 5140 098 176099 03 001 000 058 083 566 083 1473 087 569099 04 001 000 069 090 543 091 1438 092 569099 05 001 000 080 094 461 094 1119 095 473099 06 001 000 087 096 326 097 626 096 335099 00 0001 035 062 098 320 098 632 098 328099 03 0001 036 069 094 290 094 588 097 300099 04 0001 036 076 095 289 096 559 098 285099 05 0001 036 083 097 270 097 548 098 280099 06 0001 036 089 098 268 098 519 099 269




Π119873isin119877

int119901119911119873

119908119873(1 minus 119901

119908119873)(1minus119911119873)

119892119908(119901119908119873

) 119889119901119908119873

(17)




Π1198731015840isin119877

int119871(1198731015840)

119873(120579119873) 119892 (120579119873) 119889120579119873 (18)


119873and 119871(119873

1015840)




119888= 1199011= 09


1199012

119901119908



119888= 09

Parameters 119901119908= 01 119901

119908= 02

1199011

1199012

1198751

1198752

119888 119889 1198751

1198752

119888 119889

07

03 066 034 1865 0202 069 041 2139 029804 066 042 1526 0234 069 048 1674 033005 066 050 1302 0268 069 055 1378 036306 066 058 1135 0302 069 062 1166 0397

08

03 074 034 2114 0202 076 041 2484 029804 074 042 1730 0234 076 048 1944 033005 074 050 1476 0268 076 055 1600 036306 074 058 1287 0302 076 062 1354 0397

09

03 082 034 0241 0202 083 041 2907 029804 082 042 1981 0234 083 048 2275 033005 082 050 1690 0268 083 055 1873 036306 082 058 1474 0302 083 062 1584 0397





0andM

119873are


119901119911119873

119908(1 minus 119901

119908)(1minus119911119873)

119892120579(119901119908) 119889119901119908

intΠ1198731015840isin119877119871(1198731015840)

119873(120579) 119892 (120579) 119889120579 respectively

(19)






1and 119901


119888and 119901

119908)








Acknowledgments


References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












1199011

1199012

119901119908

1198781

1198782

1198783

1198733

1198784

1198734

1198785

1198735

09 00 001 000 008 081 1816 081 5985 082 181709 03 001 000 047 069 544 070 1481 073 57009 04 001 000 060 079 511 078 1351 080 51209 05 001 000 070 085 464 085 1150 086 48809 06 001 000 079 090 382 091 0818 092 39009 00 0001 035 050 081 317 081 717 081 31809 03 0001 036 059 082 303 083 634 086 30609 04 0001 035 067 087 289 087 618 089 30309 05 0001 036 075 090 289 089 585 093 29609 06 0001 036 083 094 274 093 533 096 279099 00 001 000 008 089 1743 090 5620 089 1770099 03 001 000 051 073 518 073 1298 079 577099 04 001 000 062 081 509 082 1301 084 521099 05 001 000 073 088 497 088 1269 089 504099 06 001 000 081 092 366 092 735 093 357099 00 0001 035 057 089 319 089 669 089 320099 03 0001 035 062 088 299 087 600 090 302099 04 0001 036 070 090 291 091 583 093 297099 05 0001 036 079 093 282 094 567 095 283099 06 0001 036 084 095 275 095 531 097 268


1199011

1199012

119901119908

1198781

1198782

1198783

1198733

1198784

1198734

1198785

1198735

09 00 001 000 008 090 182 089 5562 090 175809 03 001 000 054 076 514 076 1308 079 56409 04 001 000 067 085 505 085 1292 087 51209 05 001 000 077 091 486 090 1210 091 50209 06 001 000 086 094 357 093 724 095 35709 00 0001 036 057 090 318 089 661 089 31909 03 0001 036 065 088 298 089 634 092 30209 04 0001 036 073 092 287 092 591 094 29109 05 0001 035 081 094 281 094 551 096 28209 06 0001 037 088 096 272 096 524 097 290099 00 001 000 009 098 169 098 5140 098 176099 03 001 000 058 083 566 083 1473 087 569099 04 001 000 069 090 543 091 1438 092 569099 05 001 000 080 094 461 094 1119 095 473099 06 001 000 087 096 326 097 626 096 335099 00 0001 035 062 098 320 098 632 098 328099 03 0001 036 069 094 290 094 588 097 300099 04 0001 036 076 095 289 096 559 098 285099 05 0001 036 083 097 270 097 548 098 280099 06 0001 036 089 098 268 098 519 099 269




Π119873isin119877

int119901119911119873

119908119873(1 minus 119901

119908119873)(1minus119911119873)

119892119908(119901119908119873

) 119889119901119908119873

(17)




Π1198731015840isin119877

int119871(1198731015840)

119873(120579119873) 119892 (120579119873) 119889120579119873 (18)


119873and 119871(119873

1015840)




119888= 1199011= 09


1199012

119901119908



119888= 09

Parameters 119901119908= 01 119901

119908= 02

1199011

1199012

1198751

1198752

119888 119889 1198751

1198752

119888 119889

07

03 066 034 1865 0202 069 041 2139 029804 066 042 1526 0234 069 048 1674 033005 066 050 1302 0268 069 055 1378 036306 066 058 1135 0302 069 062 1166 0397

08

03 074 034 2114 0202 076 041 2484 029804 074 042 1730 0234 076 048 1944 033005 074 050 1476 0268 076 055 1600 036306 074 058 1287 0302 076 062 1354 0397

09

03 082 034 0241 0202 083 041 2907 029804 082 042 1981 0234 083 048 2275 033005 082 050 1690 0268 083 055 1873 036306 082 058 1474 0302 083 062 1584 0397





0andM

119873are


119901119911119873

119908(1 minus 119901

119908)(1minus119911119873)

119892120579(119901119908) 119889119901119908

intΠ1198731015840isin119877119871(1198731015840)

119873(120579) 119892 (120579) 119889120579 respectively

(19)






1and 119901


119888and 119901

119908)








Acknowledgments


References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119888= 1199011= 09


1199012

119901119908



119888= 09

Parameters 119901119908= 01 119901

119908= 02

1199011

1199012

1198751

1198752

119888 119889 1198751

1198752

119888 119889

07

03 066 034 1865 0202 069 041 2139 029804 066 042 1526 0234 069 048 1674 033005 066 050 1302 0268 069 055 1378 036306 066 058 1135 0302 069 062 1166 0397

08

03 074 034 2114 0202 076 041 2484 029804 074 042 1730 0234 076 048 1944 033005 074 050 1476 0268 076 055 1600 036306 074 058 1287 0302 076 062 1354 0397

09

03 082 034 0241 0202 083 041 2907 029804 082 042 1981 0234 083 048 2275 033005 082 050 1690 0268 083 055 1873 036306 082 058 1474 0302 083 062 1584 0397





0andM

119873are


119901119911119873

119908(1 minus 119901

119908)(1minus119911119873)

119892120579(119901119908) 119889119901119908

intΠ1198731015840isin119877119871(1198731015840)

119873(120579) 119892 (120579) 119889120579 respectively

(19)






1and 119901


119888and 119901

119908)








Acknowledgments


References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation















Acknowledgments


References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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