Increasingly Correct Message Passing Algorithmsfor Heat Source Detection in Sensor Networks1

K. Plarre and P. R. KumarElectrical and Computer Engineering Department

and Coordinated Science Laboratory1308 W. Main St., Urbana, IL, 61801

Email: {plarre, prkumar}@control.csl.uiuc.edu

T. SeidmanDepartment of Mathematics and StatisticsUniversity of Maryland, Baltimore County

Baltimore, MD, 21250Email: seidman@math.umbc.edu

Abstract— Solving complex source inference and esti-mation problems in the distributed environment of sensornetworks is a difficult task. Data is distributed, as iscomputational power, and energy is limited.

We consider the problem of detecting and locating aheat source appearing in a region monitored by a sensornetwork. This task requires complex computations thatmust be shared by all sensors.

One significant difficulty we overcome is the problemof local minima. By using a two step procedure, where inthe first step nodes estimate their distances to the source,and in the second step localize it, we avoid the problemof having erroneous estimates of location that local min-ima can produce. Further, to organize the computationsinvolved, we draw on ideas from graphical models. Wedevelop an algorithm that has a property which we call“increasing correctness,” that at any time the algorithmcan be stopped and it nevertheless provides the correctanswer for the problem defined by the information thathas been fed into the algorithm up to that time.

I. INTRODUCTION

There are a number of issues that make the designof applications for sensors networks a difficult task.With nodes powered by batteries, power limitation isa central issue in sensor networks. To reduce powerconsumption, nodes are provided with a “sleep” mode inwhich they consume only a fraction of the power thanin “awake,” or active mode. Management of sleepingand awake modes in sensor networks is an importantproblem. Since transmission of messages requires power,any application for sensor networks has to be designed to

1This material is based upon work partially supported by NSFunder Contract Nos. NSF ANI 02-21357 and CCR-0325716, USAROunder Contract Nos. DAAD19-00-1-0466 and DAAD19-01010-465,DARPA/AFOSR under Contract No. F49620-02-1-0325, DARPAunder Contact Nos. N00014-0-1-1-0576 and F33615-0-1-C-1905, andAFOSR under Contract No. F49620-02-1-0217.

operate with minimal communications. The joint designof computation and communications constitutes anotherchallenging problem in sensor networks; see [1].

Computational power in sensor networks is also lim-ited, making complex computations prohibitively slow.Moreover information in the network is distributed, andso computations cannot be centralized on one sensor,even if its computational power would allow it. Evenalgorithms that could be trivially programmed in a cen-tralized environment, prove to be difficult to implementin sensor networks. The organization of such distributedalgorithms is another challenging task, (e.g., [2]–[4]).

In this paper we explore these issues through a veryspecific problem, namely the detection and localizationof a heat source. A number of motes with omnidirec-tional temperature sensors are located in a certain region.At some unknown time a heat source appears in theregion. The tasks of the network is to detect the onset ofthe heat source, determine its time of appearance, itsamplitude and location, and then issue an alarm. Wewill see that this problem involves a large number ofthe issues mentioned above.

We design a distributed algorithm to detect an localizethe heat source. The detection of the source is castas hypothesis testing, while its localization leads to anonconvex optimization problem. To avoid local minima,which can give erroneous estimates of the source locationwe use a two step procedure where in the first step eachnode estimates its distance to the source, and in thesecond step the source is localized by message passing.

To organize the computations in-network, we draw onideas from graphical models and message passing [5]–[8]. We provide an algorithm that has a property whichwe call “increasing correctness.” This property allows usto stop the algorithm at any given time and still obtaina result that is correct for the problem defined by all the

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information included in the computation up to that time.Graphical models and message passing for

computation in sensor and ad-hoc networks havebeen considered before, ( [3], [4]), although in differentcontexts and with different approaches than the one wetake here.

II. CONTRIBUTIONS

We consider the problem of detecting and localizing aheat source appearing at some random time and position.

The source localization requires the global minimiza-tion of a nonconvex cost function.

To overcome this difficulty we devise a multistepalgorithm that can find the global minimum of the costfunction by first estimating the distances of the sensors tothe source, and then forming an approximation of its lo-cation trough a triangulation process. This approximationis then used as initial condition for a descent algorithm.

To organize the last step of the algorithm we useideas from graphical models. We introduce the notion of“increasing correctness.” We show that message passingfor marginalization in trees satisfies the property, that thealgorithm can be terminated at any time and still returnan answer that is correct in a certain precisely definablesense.

Graphical models and message passing have beenconsidered before for inference in sensor and ad-hocnetworks, for example, in [3], [4]. The approach we takehere is different from those.

In [9] the in-network optimization of a certain class offunctions is considered. A distributed algorithm is pro-posed, in which sensors cooperate to find the minimumof a sum of convex functions.

The ideas used here are not limited to the applica-tion we consider, but can be used for designing otherdistributed computation algorithms.

III. PROBLEM FORMULATION

We consider a collection of sensors s1, s2, . . . dis-tributed on the plane. Let xi ∈ IR2 denote the locationof sensor si. We assume that each sensor knows its ownlocation.

After being deployed, sensors form a wireless ad-hoc network; see Figure 1. We assume that any pairof sensors can communicate, either directly, or via anunderlying multihop protocol. The goal of the networkis to detect and locate a heat source appearing at somelocation on the plane. Before the source appears, thetemperature at any point of the plane is some u0. At some

Fig. 1. Connectivity graph of a network of sensors.

random time τ , a heat source appears at some position x.The presence of the source should be detected as quicklyas possible, with minimal false alarms.

There are high costs associated with a delay or failurein detecting the source, as well as with false alarms.In addition to this, sensors run on batteries, so thatcomputation and communication are costly in terms ofnetwork lifetime.

To deal with these issues we propose a three phasealgorithm:

1) The first phase corresponds to a setup phase inwhich sensors are deployed, set up the communi-cation network, and perform some initial compu-tations.

2) During the second phase most sensors are in a“sleep” state, a low power, inactive state. The restof the sensors, which we call “sentinels” are awakeand periodically sense the temperature. When asentinel sensor detects the presence of a heatsource, it issues an alert signal, waking up othersensors in its vicinity.

3) Third phase: After a source has been detected,the sensors that have been woken up cooperate toestimate the location of the source.

The set of sentinel sensors is periodically changed, sothat no sensor is continually awake.

We assume the following approximate heat equationmodel for the behavior of the temperature:

u(x, t) ={

u0 if t ≤ τ ,aµ(||x− x||, t− τ) + u0 if t > τ ,

where

µ(∆x, ∆t) :=∫ ∆t

0

14κπθ

e−∆x2

4κθ dθ .

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Here κ is the diffusion coefficient, which we assumeknown.

If node si senses the temperature at time tj , itsmeasurement is given by

ui(tj) = u(xi, tj) + wi(tj) ,

where wi(t), t ≥ 0 is a white Gaussian noise process,independent of the noise processes at other nodes.

IV. GRAPHICAL MODELS AND MESSAGE PASSING

Graphical models, e.g., [5], [6], [8], represent thestructure of marginalization problems through a graph,and allow us to construct efficient algorithms for comput-ing marginals. A marginalization problem [5] refers to aproblem where the goal is to compute the “marginals,”

β(xS) =∑xSc

m∏j=1

αj(xSj) . (1)

where S, and {Sj}mj=1 denote index sets, and we haveused the usual notational convention that variables in-dexed by a set denote a set of variables, for example,x{1,2,3} = {x1, x2, x3}. Note that S need not be oneof the sets Sj , but usually one assumes that it is. Inour example, x1, x2, . . . xn are discrete variables, andthe sum is taken over all combinations of values ofall variables in the complement of S. The functions{αj(xSj

)|1 ≤ j ≤ m} are called “kernels” and are inputdata to the problem.

The graphical structure of a marginalization problemlike (1) can be represented by a graph G = (V, E),where V = {1, 2, . . . , n}, and (i, j) ∈ E, if and onlyif Si ∩ Sj �= φ. For example, if we let S1 = {1, 2},S2 = {2, 3}, S3 = {3, 4}, and S4 = {3, 5}, weobtain the graphical representation shown in Figure 2(a).Message passing algorithms [5] solve marginalizationproblems efficiently by constructing algorithms that workon the corresponding graph. These algorithms solve amarginalization problem by letting nodes in the graphsend “messages” to their neighbors, and computing “be-liefs,” which are estimates of the marginals.

Let Ni denote the set of neighbors of i in G, andNi,j := Ni \ {j}. Denote Si,j := Si ∩ Sj . The messagethat node i sends to node j is a function of xSi,j

,

mi,j(xSi,j) =

∑xSi\Sj

αi(xSi)

∏k∈Ni,j

mk,i(xSk,i) . (2)

the product over the empty set being unity.

1, 2 2, 3

3, 4

3, 5

(c)

S3

m3,2

S4

S2S1

m1,2

(b)

S3

m3,2

m2,4

m1,2

S1 S2

S4

(a)

S1 S2

S3

S4

m4,2

Fig. 2. Graphical representation of marginalization problem.

The belief at node i, i.e., the estimate of the marginalβ(xSi

) is

b(xSi) = α(xSi

)∏

k∈Ni

mk,i(xSk,i) . (3)

This message passing process is shown in Figure 2(b),and the computation of the belief is shown in Figure2(c). Note that message passing allows us to constructestimates b(xSi

), for the marginal at each node β(xSi)

at the same time. Note also that this algorithm is asyn-chronous, hence little or no synchronization is necessarywhen implemented in a distributed environment.

It is well known that when G is a tree, messagepassing will converge after a finite number of steps, anddeliver the correct marginals; see for example [5], [7].When G has cycles (i.e., “is a loopy graph”), messagepassing might not converge, and, even if it converges,produces only approximations to the marginals; see [5],[7]).

We will construct a message passing algorithm toestimate the location, amplitude, and onset time of thesource. We will not use message passing for marginal-ization directly, but construct an algorithm that followssimilar ideas. We will design our algorithm to operate ona spanning subtree of the communication graph; hencewe do not need to worry about cycles.

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One of the properties of message passing in trees,which we will attempt to replicate, is that, at eachtime, the belief at node i is the correct solution of asubproblem of (1), which includes all the nodes that i“has heard of” up to that time. This set can be formallydefined in the following way: Let Hi ⊂ V be the set ofnodes i has heard of at time t. Then, if i sends a messageto j, Hj is updated according to the rule

Hj ← Hj ∪Hi ∪ {i} .

Initially Hi = {i}, for i = 1, 2, . . . , n. We call thisproperty “increasing correctness,” and an algorithm withthis property “increasingly correct.”

Increasingly correct algorithms are of interest to usbecause they allow us to control the time of operation ofthe algorithm in a large network. An increasingly correctalgorithm can be stopped at any given time, and node iwould have the correct answer to a problem that includesall the nodes in Hi.

V. INCREASINGLY CORRECT ALGORITHMS

In this section we generalize the notion of increasingcorrectness. We begin with some definitions.

Definition 1: Let G = (V, E) be a tree, with V ={1, 2, . . . , n}. To each i we associate a vector of parame-ters ai. Let a := {a1, a2, . . . , an}. For each i, let σi(G, a)be a function of G and a, and let β := {β1, β2, ..., βn}.We let P [G, a, β] denote the problem of computingβi(G, a), for each i = 1, 2, . . . , n.

Definition 2: Given G, a, and β, a message passingalgorithm for P [G, a, β] is an algorithm of the form:

mi,j ← fi,j

({mk,i}k∈Ni,j, ai

),

bi ← fi ({mk,i}k∈Ni, ai) ,

Hi,j ← ⋃k∈Ni,j

Hk,i ∪ {i} ,

Hi ← ⋃k∈Ni

Hk,i ∪ {i} ,

such that bi = βi(G, a), when Hi = {1, 2, . . . , n}.As above Ni denotes the set of neighbors of i in G,Ni,j := Ni \{j}, while fi,j and fi are given functionals.

Note that it is not necessary to include Hi,j and Hi

in the definition of a message passing algorithm, sincemany computations can be performed without them. Weinclude them here because these sets carry information

about the identity of the nodes that have already partic-ipated in the computations, which is useful information,especially in sensor networks, as we will see.

Let G = (V, E) be a tree, with V = {1, 2, . . . , n}.Let a and β be given. For any S ⊂ V , let GS denotethe subgraph of G induced by S. Assume that for everyS ⊂ V , we can define a restriction of βi to (GS , aS),which we denote by βi(GS , aS). We can then define afamily of problems PS [GS , aS , βS ].

Definition 3: A message passing algorithm for a prob-lem P [G, a, β] is called increasingly correct if its re-striction to GS is a message passing algorithm forPS [GS , aS , βS ] for each S ⊂ V .

We state the following lemma here and give the proofin Appendix A.

Lemma 1: Message passing for marginalization ontrees is increasingly correct.

A. Example: Computation of the maximum

As a small example, consider the computation of themaximum of a set of numbers. Let G be as before,and a = {a1, a2, . . . , an} ∈ IR. We wish to computethe maximum of the ai. We then have βi(G, a) =max{ai|i = 1, 2, . . . , n}. For S ⊂ V , the restrictionof βi to (GS , aS) is βi(GS , aS) = max{ai|i ∈ S} Wecan accomplish this with the following message passingalgorithm

mi,j ← max{max{mk,i|k ∈ Ni,j}, ai}

bi ← max{max{mk,i|k ∈ Ni}, ai}

Hi,j ← ⋃k∈Ni,j

Hk,i ∪ {i} ,

Hi ← ⋃k∈Ni

Hk,i ∪ {i} .

It is easy to see that this algorithm will indeed computethe maximum of the ai, and also that it is increasinglycorrect.

VI. ALGORITHM

We present here the algorithm to estimate the locationand amplitude of the source, and the time it appears. Itis a simple algorithm, but implementing it in a sensornetwork presents a large number of difficulties.

As was already mentioned, the algorithm is dividedinto three phases. We now describe in detail the tasksthat the sensor network has to perform in each phase,and their implementation.

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A. First phase: Setup

During the setup phase, all sensors are awake, andcooperate to estimate the background temperature and amodel for the noise. Sensors also precompute and storesome of the computations they will require later.

Let us first consider the estimation of u0 and themeasurement noise variance. For this the sensors firstfind a spanning subtree of the communication graph,and collect temperature measurements. Let Ui :={ui,1, ui,2, . . . , ui,mi

} be the set of temperature measure-ments taken by sensor si.

To compute the average temperature and noise vari-ance we use a message passing algorithm:

mµk,i ←

∑k∈Ni,j

mµk,i +

mi∑j=1

ui,j

bµi ← ∑

k∈Ni

mµk,i +

mi∑j=1

ui,j

mσk,i ←

∑k∈Ni,j

mσk,i +

mi∑j=1

u2i,j ,

bσi ← ∑

k∈Ni

mσk,i +

mi∑j=1

u2i,j ,

Hi,j ← ⋃k∈Ni,j

Hk,i ∪ {i} ,

Hi ← ⋃k∈Ni

Hk,i ∪ {i} .

Then, si can estimate the background temperature andnoise variance as

u0 = bµi

|Hi|

σ2 = bσi

|Hi| − u20 .

During this phase, sensors also precompute some of thevalues they will require during the third phase, and storethem. In particular, the values of µ(∆x, ∆t) are needed.These values are computationally intensive to compute.They are thus especially beneficial to precompute ini-tially, when no other computational load is present, ratherthan in real-time when the computational resources areneeded for other purposes.

In order to reduce the number of values to be stored,we exploit the structure of the function µ(∆x, ∆t). Itcan be rewritten as

µ(∆x, ∆t) =∫ ∆t0

14κπθ e−

∆x2

4κθ dθ

= 14kπη(∆t/∆x2) ,

where

η(z) :=∫ z

0

1θ

e−1

4κθ dθ .

This is a one dimensional function. Each sensor com-putes a table of values of η(zl), for zl = (l−1)h, whereh is a fixed small number. Let

f(θ) :=1θ

e−1

4κθ .

To compute the integrals we can use any numericalintegration algorithm. When a value of µ(∆x, ∆t) isneeded, the values in the table can be interpolated,for example by cubic splines, and the needed valuecomputed.

B. Second phase: Sentinels

Each sentinel sensor si periodically senses thetemperature, and uses these data to perform a statisticaltest to either declare the presence of a source or continuesampling.

Let H0 be the hypothesis “no source present,” andH1 be “source present.” Then under each hypothesis wehave:

H0 : E[ui(ti,j)] = u0 ,

H1 : E[ui(ti,j)] > u0 .

where ui(ti,j) is the temperature measured by sensor si

at time ti,j , and E[·] denotes expectation.

To decide between H0, and H1, si keeps a recordof its m + 1 last measurements. Let u(tk), u(tk−1), ...,u(tk−M ) be these measurements. A Student t-test is usedto make the decision. For this, si also keeps record ofthe following running sums

Ak =m∑

j=0u(tk−j) ,

Bk =m∑

j=0u(tk−j)2 .

These values are updated at each time according to therules:

Ak+1 = Ak − u(tk−m) + u(tk+1) ,

Bk+1 = Sk − u(tk−m)2 + u(tk+1)2 .

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With these values we can estimate the noise variance atnode si as

σ2 =1

m + 1Bk − 1

m2A2

k .

The statistics for the Student t-test is given by

ttest =Ak −mµ0√

Bk.

which has the t-distribution wirth m degrees of freedom.

Since the cost of failing to detect the source is higherthan the cost of a false alarm, we reject H0 at a lowsignificance level. For example, for m = 15, with asignificance level of 0.05, we reject H0, and hencedeclare an alert, if ttest ≥ 1.735.

C. Third phase: Location detection

The sentinel sensor that issues the warning wakesup other sensors in its vicinity. After waking up, thesesensors quickly take their own temperature samples. Let{ui(ti,1), ui(ti,2), ..., ui(ti,mi

)} denote the set of mea-surements taken at sensor si.

We define the following local cost functions:

Ji(a, τ, x) :=mi∑j=1

[aµ(||x− xi||, ti,j − τ)− ui(ti,j)]2

Ji(a, τ, di) :=m1∑j=1

[aµ(di, ti,j − τ)− ui(ti,j)]2 .

Given any set of sensors S′ ⊂ S, we define the followingcost functions:

JS′(a, τ, x) :=∑

si∈S′Ji(a, τ, x) ,

JS′(a, τ, dS′) :=∑

si∈S′Ji(a, τ, di) .

The estimates of the amplitude, time, and location of thesource, will be obtained from:

(a, τ , x) = argmina,τ,x

JS′(a, τ, x) , (4)

where S′ is the set of sensors that have time to participatein above the computations.

Before discussing the implementation of this phaseof the algorithm, we briefly mention the difficulties weexpect to encounter.

Besides the already mentioned difficulties, such aspower, time constraints, and false alarms, we see thatthe algorithm presents the following challenges:

Fig. 3. Typical landscape of cost function to be minimized to findlocation of the source. Note the presence of local minima, and thatthe locations of the local minima are scattered all over the domain.

1) Computational complexity: The algorithm we pro-pose requires computationally intensive operations.These operations are performed in-network, on therelatively slow processors of the sensors.

2) Organization: It is a challenging task to organizethe computations in the sensor network. It is herethat the message passing approach is helpful.

3) Nonconvex optimization: We see that (4) is anonconvex optimization problem. We are forcedto find a global minimum of this problem becauselocal minima will indicate false positions of thesource. Even if a and τ are known exactly, theoptimization problem (4) is nonconvex in x. Atypical landscape for this cost function is shownin Figure 3. Note that the locations of the localminima are scattered all over the domain and soa localization procedure which finds only a localminimum can be completely useless.

The general approaches we will follow to solve theseproblems are:

1) Power constraints: Besides the use of sentinelsensors, we will reduce the power consumption byadequate coding of data transmitted in the network.This will reduce the lengths of the messages to betransmitted.

2) Time constraints: By reducing the sizes of themessages transmitted, we also reduce the timerequired for transmission and the probability ofcollisions. The fact that a large part of the nec-essary computations have been precomputed andstored during the setup phase, and can be retrievedinstead of recomputed, largely reduces the reaction

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time of the network to the presence of a source.3) Complexity: Computational complexity is greatly

reduced by the use of precomputation. We comple-ment this with the use of parallel implementationsof the most computationally demanding portions ofthe algorithm, to reduce the workload of individualsensors. Note that the use of parallelism also helpsin reducing the reaction time of the network.

4) Organization: We make use of the message pass-ing approach to facilitate the organization of thealgorithm. We will design the algorithm in such away that it is increasingly correct. This will alsohelp us in managing the time constraint, becausethe algorithm can be stopped at any time, andyet it is guaranteed to produce the correct answergiven all the information that has been fed into thecomputation up to that point in time.

5) Local minima: To avoid local minima in the op-timization problems we will adopt a two stepprocedure. First, each sensor estimates its distanceto the source and then, in the second step, thenodes collaborate to find the location of the source.

Now we address the above issues (2-5) in detail.To define a message passing algorithm to solve (4),

we need to define the messages and beliefs. Assume firstthat, at the beginning of the third phase, all the sensorsposses a “good enough” estimate of a, τ , and x. Wediscuss how to obtain this initial estimate in Section 6.

For each i, we let

Ui := {ui(ti,1), ui(ti,2), . . . , ui(ti,mi)} ,

Ti := {ti,1, ti,2, . . . , ti,mi} .

The message that si sends to sj has the form:

mi,j = (Hi,j , Ui,j , Ti,j , Xi,j , Pi,j) ,

where Pi,j = (ai,j , τi,j , xi,j). The update rules for thesemessages are:

Hi,j ← ⋃k∈Ni,j

Hk,i ∪ {i} ,

Ui,j ← ⋃k∈Ni,j

Uk,i ∪ Ui ,

Ti,j ← ⋃k∈Ni,j

Tk,i ∪ Ti ,

Xi,j ← ⋃k∈Ni,j

Xk,i ∪ {xi} ,

sk

si

sj

Gk,i

Fig. 4. The message that sk sends to si contains information comingonly from the subtree Gk,i.

The computation of Pi,j is more involved. First note thatHk,i contains the indices of all sensors sk has heard fromin the subtree Gk,i; see Figure 4. From the way we willconstruct the algorithm, we will also have

(ak,i, τk,i, xk,i) = argmina,τ,x

JHk,i(a, τ, x) .

To compute Pi,j , sensor i then solves

(ai,j , τi,j , xi,j) = argmina,τ,x

JHi,j(a, τ, x) . (5)

Note that the information that si receives from itsneighbors allows it to do these computations.

To reduce the time to solve (5), sensor si uses thereceived estimated parameters Pk,i, k ∈ Ni,j as initialconditions for a descent algorithm, for example Newton’smethod. Note that si starts Newton iterations at |Ni,j |different points. In the case that these iterations do notconverge to the same point, si chooses the minimum ofthe resulting costs.

The belief at node i is given by

bi = (Hi, Ui, Ti, Xi, Pi) ,

where Pi = (ai, τi, xi). The update of these beliefs isdone according to the following rules:

Hi ←⋃

k∈Ni

Hk,i ∪ {i} ,

Ui ← ⋃k∈Ni

Uk,i ∪ Ui ,

Ti ← ⋃k∈Ni

Tk,i ∪ Ti ,

Xi ←⋃

k∈Ni

Xk,i ∪ {xi} ,

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Pi is obtained from

(ai, τi, xi) = argmina,τ,x

JHi(a, τ, x) .

Again, si uses the received Pk,i as initial conditionsfor a descent method, and chooses the minimum of theobtained results.

Figure 5 shows how the information from sensors in avicinity of si is incorporated progressively into the beliefat si. It illustrates the property of increasing correctness.

We have not yet specified how to obtain the initialestimate of (a, τ , x). We do so now.

1) Obtaining the initial estimate: Let s0 be the firstsensor that detects the source. Let N0 be the set ofneighbors of s0. After detecting the source, s0 wakes upits neighbors. Together they form a crude, initial estimateof the source location.

Let N0 := N0∪{0}. The sensors sN0obtain the initial

estimate by solving

(a0, τ0, x0) = argmina,τ,x

JN0(a, τ, x) . (6)

This is however a nonconvex problem in four variables,to overcome it the sensors first solve the related problem

(a0, τ0, d0N0

) = argmina,τ,dN0

JN0(a, τ, dN0

) . (7)

Note that (7) involves more variables than (6). It mightseem that this is then a harder problem. But notice thatwe can rewrite (7) as

(a0, τ0, d0N0

) = argmina,τ

∑i∈N0

argmindi

Ji(a, τ, di) .

We can thus divide the optimization problem into an“outer loop” in a and τ , and an “inner loop” in eachdi, i ∈ N0. These last series of optimizations can beperformed in parallel by each si in N0. Specifically, eachsensor si, i ∈ N0, computes a table

di,l,m = argmindi

Ji(al, τm, di) ,

for every (al, τm) on a grid:

0 = a0 < a1 < . . . < ana= amax ,

0 = τ0 < τ1 < . . . < τnτ= τmax ,

where amax, and τmax are the maximum possible valuesof a, and τ , respectively.

Each sensor sends its corresponding table to s0.For each (l, m), the following triangulation problem issolved:

xl,m = argminx

∑i∈N0

[||x− xi||2 − di,l,m

]2.

si si

si si

Fig. 5. Illustration of how the information from the vicinity of si

is progressively included into the belief at si. Black disks representsensors in Hi as time passes.

To reduce the workload of each single sensor, s0 reas-signs these problems to its neighbors, which solve themin parallel.

In the last step of this process, each (al, τm, xl,m)is used as an initial condition for a descent algorithmfor solving (6). Again, this step can be performed inparallel. The minimizing value found in this process isthen (a0, τ0, x0).

2) Coding: We notice that the computation of theinitial parameter estimate involves the communication ofa large amount of data. Although this communication islocal, i.e., it does not involve multi-hop communication,we find it convenient to reduce the length of the messagessent, by coding them in an appropriate way.

For doing so, we view di,l,m as a two dimensionalfunction of l and m. We can thus construct a linearregression

di,l,m = αl + βm + γ + εl,m .

We can therefore encode the values of α, β, γ, andεl,m. Since, in general |ε(l, m)| |di,l,m|, this encodingreduces the number of required bits.

VII. SIMULATIONS

In this section, we present simulation results, where,given the space constraints we present simulations of thedetection and triangulation algorithms only.

A. Detection

To study the behavior of the detection algorithm welocate five sensors at distances 0.2, 0.3, 0.4, 0.5, and0.6 from the location of the source. The source appearsat time τ = 0.6. Figure 6 shows the beahvior of ttest,

0-7803-8797-X/04/$20.00 (C) 2004 IEEE

0.6 0.62 0.64 0.66 0.68 0.7 0.720

0.5

1

1.5

2

time

t test

0.6 0.62 0.64 0.66 0.68 0.7 0.720

0.5

1

1.5

2

time

t test

0.6 0.62 0.64 0.66 0.68 0.7 0.720

0.5

1

1.5

2

time

t test

distance

distance

Fig. 6. behavior of ttest with distance from the source

parametrized by the distance to the source. The top plotshows the noisless case, while the bottom plot shows thecase with noise. The noise variance is σ2 = 0.001.

We can observe that the sensors that are far away fromthe source will detect it much later than sensors that areclose to it.

B. Localization

We let 10 sensors located in the unit square, detectand determine the location of a heat source. The sourceappears at time τ = 0.6. Sensors have available tempera-ture measurements at times {1, 1.2, 1.4, 1.6, 1.8, 2}. Theamplitude of the source is a = 1. The noise variance isσ2 = 0.001. We for these simulations we use a valueκ = 1. The value of u0 is irrelevant to the results,therefore we set u0 = 0.

In the upper left plot in Figure 7, we show theconfiguration of sensors. Sensors are represented bycircles, while the true heat source location is shown asa cross.

The first approximation of the source location, givenby the triangulation step is shown in the top right plotin Figure 7. The diamond shows the estimated location.

The previous location estimate is then used as initialcondition to minimize the global cost function (4). Therefined estimated location is shown in Figure 7 (bottomleft).

Figure 7 (bottom right) shows a close-up of the regionsurrounding the source. It can be seen that the our twostep procedure converges to the source location.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1Configuration

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1First estimate

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1Refinement

0.2 0.3 0.4 0.50.3

0.4

0.5

0.6

Close−up

Fig. 7. Two step location process: First a rough estimate of thesource location is found, then this estimate is refined.

VIII. CONCLUSIONS

The notion of incresing correctness we introducedproves to be useful in the design of algorithm forcomputations in distributed environments, especially inlarge networks and time constrained situations, and withunreliable communication. In fact, increasingly correctalgorithms always give the correct answer to the problemdefined on the data they have been fed with. Thisallows each node to gracefully incorporate new arrivinginformation into the computations, in a graceful way.

We have used these ideas to construct a multistagealgorithm for the detection and localization of a heatsource in a sensor network.

APPENDIX

A. Proof of lemmma 1

Consider the marginalization problem defined by (1),and the message passing iteration defined by (2) and(3). Suppose that at some given time the algorithm isstopped. Call this time t. Let bt

i be the last belief at node icomputed before the algorithm stopped. Likewise, let Ht

i

be the set of nodes i “heard from,” before the algorithmwas stopped.

We can show that bti is the correct marginal for the

marginalization problem:

βti(xSi

) =∑

xHi\Si

∏j∈Hi

αj(xSj) .

Let GHibe the subgraph of G induced by the nodes in

Hi. It is easy to see that GHiis connected: If j ∈ Hi,

0-7803-8797-X/04/$20.00 (C) 2004 IEEE

j

i

pj

Fig. 8. Subtree of G, induced by Hi, and rooted at i.

then there exists a path from j to i in G, otherwise iwould “never have heard from j.” Then GHi

is a subtreeof G containing i. Let Ti be the corresponding tree,rooted at i, and for each j ∈ Hi, j �= i, let pj denote theparent if j in Ti. For each j ∈ Hi, let Cj be the set ofchildren of j in Ti. See Figure 8.

Let MT be the set of all messages sent during messagepassing up to time t. We note that not all messages inM t contributed to the computed value of bt

i. We nowconstruct the minimal set of messages that determinedthe value of bt

i. We call this set M ti

First note that no message from a node j ∈ Hi to anode in Cj can be in M t

i . This is so because a messagefrom j to k never uses the information sent to j by k,and Ti is cycle free. So, only messages from a node jto pj can be in M t

i .We also note that, for each j ∈ Hi, only one message

from j to pj can be in M ti . We call this message mt

j .We now construct M t

i recursively:

1) If j ∈ Ci, then mtj is the last message that j sent

to i before i computed bti.

2) For every j ∈ Hi, j /∈ Ci, mtj is the last message

that j sent to pj before pj sent mtpj

.

Given M t, M ti is unique by construction.

Let j be a leaf node of Ti. Then mtj contains only

information that is local to j. This is so because:

• mtj cannot contain information from any of its

ancestors, because Ti is cycle free.• mt

j cannot carry information from any k ∈ Nj,pj.

Otherwise k ∈ Hi, which is a contradiction.

In fact,

mtj(xSj,pj

) =∑

xSj\Spj

αj(xSj) .

For any j ∈ Hi, j not a leaf of Ti, we then have

mtj,pj

(xSj,pj) =

∑xSj\Spj

αj(xSj)

∏k∈Cj

mk,j(xSk,j) .

We also have:

bti(xSi

) = αi(xSi)∏

k∈Ci

mk,i(xSk,i) .

These are exactly the messages and belief that a syn-chronous message passing (started at the leaves, then theparents of the leaves, etc.) produced. We know that, sinceTi is a tree, this set of messages produces the correctmarginal bt

i. So, message passing for marginalization intrees is increasingly correct.

REFERENCES

[1] A. Giridhar and P. R. Kumar, “Data fusion over sensor networks:Computing and communicating functions of measurements,”December 2003,” To appear in Journal on Selected Areas inCommunications, Special Issue on Distributed and CollaborativeSensor Networks.

[2] D. Scherber and H. Papadopoulos, “Locally constructed al-gorithms for distributed computations in ad-hoc networks,” inProceedings of the 3rd International Symposium on InformationProcessing in Sensor Networks, 2004, pp. 11–19.

[3] A. Ihler, J. Fisher, R. Moses, and A. Willsky, “Nonparametricbelief propagation for self-calibration in sensor networks,” inProceedings of the 3rd International Symposium on InformationProcessing in Sensor Networks, 2004, pp. 225–233.

[4] R. Biswas, L. Guibas, and S. Thrun, “A probabilistic approachto inference with limited information in sensor networks,” inProceedings of the 3rd International Symposium on InformationProcessing in Sensor Networks, 2004, pp. 269–276.

[5] M. Jordan and C. Bishop, An introduction to graphical models.Preprint, October 2001.

[6] B. Frey, Graphical models for machine learning and digitalcommunication. Cambridge, MA: MIT Press, 1998.

[7] S. M. Aji and J. McEliece, “The generalized distributive law,”IEEE Transactions on Information Theory, vol. 46, pp. 325–343,1999.

[8] F. R. Kschiang, B. J. Frey, and H.-A. Loeliger, “Factor graphs andthe sum-product algorithm,” IEEE Transactions on InformationTheory, vol. 47, no. 2, pp. 498–519, 2001.

[9] M. Rabbat and R. Novak, “Distributed optimization in sensornetworks,” in Proceedings of the 3rd International Symposiumon Information Processing in Sensor Networks, 2004, pp. 20–27.

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