a distributed algorithm for managing multi-target identities in wireless ad-hoc sensor networks...

A Distributed Algorithm for A Distributed Algorithm for Managing Multi-target Managing Multi-target

Identities in Wireless Ad-hoc Identities in Wireless Ad-hoc Sensor NetworksSensor Networks

Jaewon Shin, Leonidas J. Guibas, and Feng ZhaoJaewon Shin, Leonidas J. Guibas, and Feng Zhao

Presented byPresented by

Raghu Kiran GantiRaghu Kiran Ganti

For CS 851For CS 851

Tracking ProblemTracking Problem

Outline of talkOutline of talk

IntroductionIntroduction Mathematical formulation of problemMathematical formulation of problem Multi-target identity updateMulti-target identity update Implementation on Wireless Adhoc Sensor Implementation on Wireless Adhoc Sensor

Networks (WASN)Networks (WASN) EvaluationEvaluation Discussion & ConclusionsDiscussion & Conclusions

IntroductionIntroduction

Problem statementProblem statement: Distributed multi-target : Distributed multi-target identity management. Maintain information identity management. Maintain information about about who is who who is who over time given targets’ over time given targets’ position estimates.position estimates.

DifficultyDifficulty- exponential complexities in - exponential complexities in associating target position estimates with associating target position estimates with target identities.target identities.

Related WorkRelated Work

An Algorithm for Tracking Multiple Targets An Algorithm for Tracking Multiple Targets (1979!) – IEEE Transactions on Automatic (1979!) – IEEE Transactions on Automatic Control.Control.

Develops a method for calculating Develops a method for calculating probabilities.probabilities.

Primarily uses Bayesian formulation for Primarily uses Bayesian formulation for determining probabilities.determining probabilities.

Not related to WASN, but ideas are very Not related to WASN, but ideas are very similar => tracking is not old problem!similar => tracking is not old problem!

Contributions of the paperContributions of the paper

Main contributionMain contribution – A rigorous mathematical – A rigorous mathematical framework for solving Multi-target identity framework for solving Multi-target identity management problem.management problem.

A new distributed representation for maintaining A new distributed representation for maintaining identity information.identity information.

A distributed algorithm for updating the belief matrix A distributed algorithm for updating the belief matrix which is O(Nwhich is O(N22) complexity.) complexity.

Estimate non-local parameters of a physical Estimate non-local parameters of a physical phenomenon using local information – how feasible phenomenon using local information – how feasible is it?is it?

Some new conceptsSome new concepts

Identity Belief MatrixIdentity Belief Matrix: Matrix to maintain identity : Matrix to maintain identity information of each target.information of each target.

Ex: Ex: represents a belief matrix => represents a belief matrix => target 1 at position 1 withtarget 1 at position 1 withprobability 1, etc.probability 1, etc.

Mixing matrix:Mixing matrix: Matrix used to update local belief Matrix used to update local belief matrix.matrix.

Ex:Ex: => m=> m1212 is probability that is probability that target came from position 1 at target came from position 1 at

time time k-1 to position 2 at time k.k-1 to position 2 at time k.

1 0 00 1 00 0 1

1 0 00 1 00 0 1

A Little MathematicsA Little Mathematics

Doubly stochastic matrixDoubly stochastic matrix: A matrix : A matrix A = (aA = (aijij) ) such that asuch that aijij ≥ 0 and≥ 0 and

Statistical EntropyStatistical Entropy: A measure or variation : A measure or variation defined on the probability distribution of defined on the probability distribution of observed events. Specifically, if P is the observed events. Specifically, if P is the probability of an event a, the entropy H(A) for probability of an event a, the entropy H(A) for all events a in A is: all events a in A is:

i j

jiij 1aa

)(log)( 2 aa

a PPAH

A Mathematical Framework- Target A Mathematical Framework- Target ConfigurationsConfigurations

Sparse (COLU) Sparse (COLU)Crowded (COHU)

Target configurationsTarget configurations

COLUCOLU- - CConfiguration onfiguration OOf f LLow ow UUncertainty.ncertainty. COHU- CCOHU- Configuration onfiguration OOf f HHigh igh UUncertainty.ncertainty. Actions/Decisions in COHU more uncertain Actions/Decisions in COHU more uncertain

than those of COLU.than those of COLU. Let Let I I = {1,2,…..,N} be target identities and = {1,2,…..,N} be target identities and

be the be the position estimates of N targets at time k.position estimates of N targets at time k.

},......,1|)({)( 2 NiRkxkX i

Identity Mass FlowIdentity Mass Flow

Identity management- compute correct Identity management- compute correct permutation of permutation of II given given XX(k).(k).

Maintain all possible permutations?- Maintain all possible permutations?- Exponential complexity!Exponential complexity!

Proposed idea- Identity Mass Flow (IMF).Proposed idea- Identity Mass Flow (IMF). IMF- The mass associated with identity flows IMF- The mass associated with identity flows

from from XX(k-1) to (k-1) to XX(k) in whole or partially.(k) in whole or partially.

t = k

t = k+1

t = k

t = k+1

Current state

New measurement

Current state

New measurement

t = k

t = k+1

t = k

t = k+1

p 1-p

Mass is neither created nor destroyed- Conservation of Mass

New measurement

Current state

New measurement

Current state

t = k

t = k+1

Sum of all masses arriving at xi(k) is one

ab

a+b = 1

Current state

t = k

t = k+1

Current state

New measurement

New measurement

DefinitionsDefinitions

The identity belief matrix B(k) is a N The identity belief matrix B(k) is a N ×× N N doubly stochastic matrix, whose entry bdoubly stochastic matrix, whose entry b ijij(k) (k)

represents the amount of identity mass from irepresents the amount of identity mass from iєєI I that arrives at xthat arrives at xjj(k). The j(k). The jthth column b column bjj(k) of B(k) (k) of B(k)

is called the identity belief vector of xis called the identity belief vector of x jj(k).(k).

wherewhere

bbii(k) = (k) =

p(xi(k))’s ID is 1p(xi(k))’s ID is 2 . . .P(xi(k))’s ID is N

NNN kbkbkbkB ]1,0[)](......)()([)( 21

NN ]1,0[

DefinitionsDefinitions

The mixing matrix M(k) is a N The mixing matrix M(k) is a N × N doubly × N doubly stochastic matrix, whose entry mstochastic matrix, whose entry m ijij(k) (k) represents the probability of xrepresents the probability of x jj(k) begin (k) begin originated from xoriginated from xii(k-1), and is statistically (k-1), and is statistically independent with M(l) for all l ≠ k.independent with M(l) for all l ≠ k.

Theorem 1:Theorem 1: Let B(k) and M(k) be the identity Let B(k) and M(k) be the identity belief matrix and the mixing matrix at time k belief matrix and the mixing matrix at time k as in the above 2 definitions, then:as in the above 2 definitions, then:

B(k+1) = B(k)M(k+1)B(k+1) = B(k)M(k+1) Matrix multiplication – compute intensive!Matrix multiplication – compute intensive!

Updating Updating B(k)B(k) using using M(k)M(k)M(k-1)

α

α

1-α

β

β

1-β

M(k)

1

0

0

1

t = k-1

B(k-1)

α

1-α

1-α

α

t = k

B(k)

β

t = k+1

α

1-α+ (1-β)

1-α

α

(1-β) α

1-α+ β

1-α

α

B(k+1)

Uncertainty change in systemUncertainty change in system

Lemma 1Lemma 1: Let : Let ππB(k) B(k) be the be the

probability mass function over all the possible probability mass function over all the possible identity associations in B(k), thenidentity associations in B(k), then

H(H(ππB(k)B(k)) ≥ H() ≥ H(ππB(k-1)B(k-1)))

where H(.) is the statistical entropy of a where H(.) is the statistical entropy of a probability mass function.probability mass function.

The proof follows from strict concavity of the The proof follows from strict concavity of the entropy function.entropy function.

!]1,0[ N

Uncertainty- How does it affect?Uncertainty- How does it affect?

Lemma 1 => Uncertainty will grow until every Lemma 1 => Uncertainty will grow until every identity association becomes equally like identity association becomes equally like without any additional information.without any additional information.

Uncertainty will be less in a COLU Uncertainty will be less in a COLU configuration.configuration.

Proper use of local information could make Proper use of local information could make uncertainty less in the IMF formulation.uncertainty less in the IMF formulation.

Computing Mixing Matrix M(k)Computing Mixing Matrix M(k)

M(k) M(k) – collection of marginal association – collection of marginal association probabilities, can be computed from joint association probabilities, can be computed from joint association probability, BUT computing is exponential.probability, BUT computing is exponential.

Proposed solution – Compute a matrix L(k), such that Proposed solution – Compute a matrix L(k), such that l(i,j) = p(s(k) = |xl(i,j) = p(s(k) = |xjj(k) – x(k) – xii(k-1)|/(k-1)|/ΔΔT); where s(k) T); where s(k)

indicates the speed information of the target. Since indicates the speed information of the target. Since M(k) is doubly-stochastic, L(k) must be transformed M(k) is doubly-stochastic, L(k) must be transformed into a doubly-stochastic matrix.into a doubly-stochastic matrix.

Computing Mixing MatrixComputing Mixing Matrix

Done by an Iterative Scaling Algorithm – Done by an Iterative Scaling Algorithm – Presented later.Presented later.

Issue to be addressedIssue to be addressed: How are the speeds of : How are the speeds of the targets calculated? Is it possible to estimate the targets calculated? Is it possible to estimate speed of targets correctly?speed of targets correctly?

Multi-target identity UpdateMulti-target identity Update

Local information must be used for updating Local information must be used for updating identity belief matrix.identity belief matrix.

IMF approach – natural setting for exploiting IMF approach – natural setting for exploiting local services.local services.

Sparse (COLU) Sparse (COLU)Crowded (COHU)

1 0

0 1

0 1

1 0

.5 .5

.5 .5

.5 .5

.5 .5

.1 .9

.9 .1

.9 .1

.1 .9

Sparse (COLU) Sparse (COLU)Sparse (COLU)

1 0

0 1

0 1

1 0

.9 .1

.1 .9

.1 .9

.9 .1

.9 .1

.1 .9

.1 .9

.9 .1

Bayesian NormalizationBayesian Normalization

Above 2 examples – 2 target case. Given 1 Above 2 examples – 2 target case. Given 1 target’s probability, other target’s probability target’s probability, other target’s probability can be calculated. (Doubly stochastic)can be calculated. (Doubly stochastic)

For N target case – Use Bayesian For N target case – Use Bayesian Normalization.Normalization.

Desirable properties of a Desirable properties of a perfect solutionperfect solution achieved by computing achieved by computing B(k)B(k) as Bayesian as Bayesian posterior belief distribution.posterior belief distribution.


Bayesian posterior belief distribution: Use a Bayesian posterior belief distribution: Use a priori belief and local data to get the new / priori belief and local data to get the new / posterior belief.posterior belief.

Assume joint probability distribution Assume joint probability distribution over all possible N! over all possible N!

associations at time associations at time k k is available for all is available for all k.k. Define Define to be set of indices to be set of indices

associated with | associated with | ππ(k(kii)| > 2, where k)| > 2, where kii is i is ith th

element in K.element in K.

i ii

N zzRZk }0,1|{)( 1!

},......,1,0{ NK


RRii’s are sequence of random variables ’s are sequence of random variables

associated with associated with ππ(k(kii), which takes values j ), which takes values j єє

{1,………, | {1,………, | ππ(k(kii)|} with probability of )|} with probability of ππjj(k(kii).). A specific permutation is chosen according to A specific permutation is chosen according to

the value of the value of RRii with some probability. with some probability. Single identity association – A point Single identity association – A point εε in the in the

joint event space joint event space with |S| = with |S| = ΠΠii||ππ(u(uii)|.)|.

)]}(|,1[|),......,,{( ||21 iiK uRRRRS


The probability of the above event is:The probability of the above event is:p((Rp((R11,…,R,…,R|K||K|) = ) = εε) = p() = p(RR11==εε11)…p(R)…p(R|K||K|= = εε|K||K|))

Given a local evidence L, the posterior belief Given a local evidence L, the posterior belief matrix B(k) can be calculated.matrix B(k) can be calculated.

Theorem 2Theorem 2: : Let ELet Eijij(k) be the subset of S (k) be the subset of S

satisfying ID(xsatisfying ID(xjj(k)) = i and L be the subset of S (k)) = i and L be the subset of S

satisfying the local observation, then satisfying the local observation, then

L l

LE l

ijij

l

ijl

p

pLEpLkbp

)(

)()|()|)((

Bayesian NormalizationBayesian Normalization Lemma 2Lemma 2: The local observation L does not increase : The local observation L does not increase

the entropies of the columns i.e.the entropies of the columns i.e.

Theorem 3Theorem 3: Let b: Let bpqpq(k) be the entry that becomes 1 (k) be the entry that becomes 1 from local evidence L, then the columns with zero at from local evidence L, then the columns with zero at ppthth entry and the rows at q entry and the rows at qthth entry do not change. entry do not change.

Theorem 3 => lesser number of updates for the belief Theorem 3 => lesser number of updates for the belief matrix.matrix.

Bayesian normalization not practical – introduce Bayesian normalization not practical – introduce Iterative Scaling.Iterative Scaling.

))(()|)(( kbHLkbH ii

Iterative ScalingIterative ScalingB := A;B_old := A;for k = 1 to maximum_number_of_iteration

for i = 1 to number_of_rowrow_sum := 0;for k = 1 to number_of_column

row_sum := row_sum + B(i,k);endfor j = 1 to number_of_column

B(i,j) := B(i,j)/row_sum;end

endfor i = 1 to number_of_column

column_sum := 0;for k = 1 to number_of_row column_sum := column_sum + B(i,k);endfor j = 1 to number_of_row B(j,i) := B(j,i)/column_sum;end

endif |B - B_old| < error

terminate;endB_old := B;

end

Something wrong?

Normalize rows

Normalize columns

Check for termination condition

Paper states: We present a version of the Iterative Scalingalgorithm to achieve a doubly-stochastic matrix A given a N × N non-negative matrix B.

Iterative ScalingIterative Scaling

Divide each element in iDivide each element in ithth row/column by the row/column by the

sum of that row/column.sum of that row/column. Repeat the normalization till Repeat the normalization till error error margin is margin is

small.small. Empirical observations:Empirical observations:

Algorithm converges to a unique doubly-stochastic matrix given an Algorithm converges to a unique doubly-stochastic matrix given an initial matrix.initial matrix.

The ordering of row/column normalization does not affect the The ordering of row/column normalization does not affect the convergence.convergence.

The total number of iterations are not affected by the size of the matrix.The total number of iterations are not affected by the size of the matrix.

Number of Iterations

log|

B(n)

-B(n

-

1)|

Convergence of iterative scaling

Iterative ScalingIterative Scaling

Convergence rate affected by how different the Convergence rate affected by how different the initial matrix is from being a doubly-stochastic initial matrix is from being a doubly-stochastic matrix.matrix.

Larger matrices are Larger matrices are closercloser to being doubly- to being doubly-stochastic than smaller ones.stochastic than smaller ones.

Q – First iteration gives a doubly-stochastic Q – First iteration gives a doubly-stochastic matrix, why do we need so many iterations?matrix, why do we need so many iterations?

Q – How does it do a correct update of B(k)?Q – How does it do a correct update of B(k)?

Implementation on WASNImplementation on WASN

Quantity to be distributed – Belief matrix B(k).Quantity to be distributed – Belief matrix B(k). Small number of nodes called Small number of nodes called leadersleaders are are

active and responsible for maintaining / active and responsible for maintaining / updating the information of interest.updating the information of interest.

When When leaders leaders are no longer “good” for are no longer “good” for information gathering, select a new leader and information gathering, select a new leader and handoff the information.handoff the information.

Each Each leader leader maintains B(k) and its position maintains B(k) and its position estimate xestimate xii(k).(k).

Implementation on WASNImplementation on WASN

When a leader observes local evidence about When a leader observes local evidence about the ID of a target, then it needs to send the the ID of a target, then it needs to send the information to other leaders to allow them to information to other leaders to allow them to update their belief matrices.update their belief matrices.

Question – Above communication is Question – Above communication is multicastmulticast, , no mention in paper of how to achieve it! They no mention in paper of how to achieve it! They assume that they have a good group assume that they have a good group management protocol for this purpose.management protocol for this purpose.

Leader?

Multiplemeasurements

Sleep

Wake-up

Do sensing

Y N

Mixing matrixavailable?

Y

Identity Sensing goodEnough?

N

Each meashas unique

leader?

N

Update B & select nextleader & hand off data

Y

Compute M & send infoto other leaders.

N

Ignore theother meas

Y

Initializenormalization

Y

N

Example – Single TargetExample – Single Target

1

1

1

1

1

1

1

1

Example – Multiple targetsExample – Multiple targets1 00 1

0 11 0

.1 .9

.9 .1

.9 .1

.1 .9

.9 .1

.1 .9

.1 .9

.9 .1

EvaluationEvaluation

Simulations are used, no real experiment setup.Simulations are used, no real experiment setup. Initial leaders are selected manually.Initial leaders are selected manually. Next leader is selected based on its Next leader is selected based on its

geographical position??geographical position?? Each node has a signal processing module for Each node has a signal processing module for

signal classification.signal classification. Localization is achieved and nodes know other Localization is achieved and nodes know other

nodes relative positions.nodes relative positions.

Target 1Target 2

Target 3

Target 4

Simulation Example – At t = 0

Target 1

Target 2

Target 3

Target 4


Target 1Target 2

Target 3

Target 4


Latest WorkLatest Work

Latest works – Latest works – Tracking a Moving Object with a Tracking a Moving Object with a Binary Sensor Network. (Sensys ’03)Binary Sensor Network. (Sensys ’03)

ClaimClaim: With 1 bit of information from a sensor node, : With 1 bit of information from a sensor node, we can track objects! 1 bit conveys whether object we can track objects! 1 bit conveys whether object moves away/towards a node.moves away/towards a node.

Uses geometry to determine target.Uses geometry to determine target. Tracking algorithm based on particle filtering method Tracking algorithm based on particle filtering method

– represent the location density function as a set of – represent the location density function as a set of random points which are updated based on sensor random points which are updated based on sensor readings.readings.

SiSj

X

αβ

Geometry of object moving away from Sj and towards Si

Latest WorkLatest Work

Paper has a simple solution – only 1 bit! But, Paper has a simple solution – only 1 bit! But, multi-target tracking not addressed.multi-target tracking not addressed.

How complex will it be if multi-target identity How complex will it be if multi-target identity management problem is addressed?management problem is addressed?

Only motion direction information is obtained. Only motion direction information is obtained. Exact location is not known using only 1 bit!Exact location is not known using only 1 bit!

But, a direction to reduce the complexity of But, a direction to reduce the complexity of tracking objects.tracking objects.

ConclusionConclusion

Tracking multiple targets with identity Tracking multiple targets with identity management not easy. Because of distributed management not easy. Because of distributed nature of sensor networks.nature of sensor networks.

Any higher level application in sensor Any higher level application in sensor networks requires good distributed algorithms.networks requires good distributed algorithms.

Minimal computation at nodes, but final result Minimal computation at nodes, but final result should be appreciable.should be appreciable.


Trend in previous papers and this paper – Trend in previous papers and this paper – Higher level programming paradigms Higher level programming paradigms presented. Moving away from lower layers.presented. Moving away from lower layers.

Moving towards achieving practical things, Moving towards achieving practical things, like tracking of objects.like tracking of objects.


Envirotrack is a general framework for Envirotrack is a general framework for tracking, this paper presents a specific solution tracking, this paper presents a specific solution and addresses multi-target identity.and addresses multi-target identity.

Mate is a virtual machine, unlike the solution Mate is a virtual machine, unlike the solution proposed here.proposed here.

Solution based on rigorous mathematical Solution based on rigorous mathematical background.background.

But, no experimental testing using sensor But, no experimental testing using sensor nodes.nodes.

Conclusion & DiscussionConclusion & Discussion

How is mapping between the sensor network How is mapping between the sensor network nodes and the targets achieved?nodes and the targets achieved?

How is next leader elected?How is next leader elected? How many leaders must be elected? If we have How many leaders must be elected? If we have

more leaders, there is more communication more leaders, there is more communication overhead, and if we have lesser number of overhead, and if we have lesser number of leaders, tracking can become difficult.leaders, tracking can become difficult.

a distributed algorithm for managing multi-target identities in wireless ad-hoc sensor networks...

Documents

identity information

multitarget identities

local belief matrix

identity flows

mixing matrix

conservation of mass

distributed algorithm

local information