Location estimation in sensor networks

Location estimation in wireless sensor networks is the problem of estimating the location of an object from a set of noisy measurements, when the measurements are acquired in a distributedmanner by a set of sensors.

Motivation

In many civilian and military applications it is required tomonitor a specific area in order to identify objects within itsboundaries. For example: monitoring the front entrance of aprivate house by a single camera. When the physical dimensions ofthe monitored area are very large relatively to the object ofinterest, this task often requires a large number of sensors (e.g.infra-red detectors) at several locations. The location estimationis then carried out in a centralized fusion unit based oninformation gathered from all the sensors. The communication tothe fusion center costs power and bandwidth which are scarceresources of the sensor, thus calling for an efficient design ofthe main tasks of the sensor: sensing, processing andtransmission.

The "CodeBlue system" [http://www.eecs.harvard.edu/~mdw/proj/codeblue/] of Harvard university is an example where avast number of sensors distributed among hospital facilitiesallow to locate a patient under distress. In addition, the sensorarray enables online recording of medical information whileallowing the patient to move around. Military applications (e.g.locating an intruder into a secured area) are also good candidatesfor setting a wireless sensor network.

etting

Let $heta$ denote the position of interest. A set of $N$ sensorsacquire measurements $x_n = heta + w_n$ contaminated by anadditive noise $w_n$ owing some known or unknown probability density function (PDF). The sensors transmit messages (based ontheir measurements) to a fusion center. The $n$th sensor encodes$x_n$ by a function $m_n\left(x_n\right)$. The fusion center applies apre-defined estimation rule$hat\left\{ heta\right\}=f\left(m_1\left(x_1\right),cdot,m_N\left(x_N\right)\right)$. The set of message functions$m_n,, 1leq nleq N$ and the fusion rule $f\left(m_1\left(x_1\right),cdot,m_N\left(x_N\right)\right)$ aredesigned in order to minimize the estimation error in some sense.For example: minimizing the mean squared error (MSE),$mathbb\left\{E\right\}| heta-hat\left\{ heta\right\}|^2$.

Ideally, the sensors would transmit their measurements $x_n$exactly to the fusion center, that is $m_n\left(x_n\right)=x_n$. In thissettings, the maximum likelihood estimator (MLE) $hat\left\{ heta\right\} =frac\left\{1\right\}\left\{N\right\}sum_\left\{n=1\right\}^N x_n$ is an unbiased estimator whose MSE is$mathbb\left\{E\right\}| heta-hat\left\{ heta\right\}|^2 = ext\left\{var\right\}\left(hat\left\{ heta\right\}\right) =frac\left\{sigma^2\right\}\left\{N\right\}$ assuming a white Gaussian noise$w_nsimmathcal\left\{N\right\}\left(0,sigma^2\right)$. The next sections suggestalternative designs when the sensors are bandwidth constrained to1 bit transmission, that is $m_n\left(x_n\right)$=0 or 1.

Known noise PDF

We begin with an example of a Gaussian noise$w_nsimmathcal\left\{N\right\}\left(0,sigma^2\right)$, in which a suggestion for asystem design is as follows [cite journal
last = Ribeiro
first = Alejandro
coauthors = Georgios B. Giannakis
title = Bandwidth-constrained distributed estimation for wireless sensor Networks-part I: Gaussian case
journal = IEEE Trans. on Sig. Proc.
date = March 2006
] :

:

: $hat\left\{ heta\right\}= au-F^\left\{-1\right\}left\left(frac\left\{1\right\}\left\{N\right\}sumlimits_\left\{n=1\right\}^\left\{N\right\}m_n\left(x_n\right) ight\right),quadF\left(x\right)=frac\left\{1\right\}\left\{sqrt\left\{2pi\right\}sigma\right\} intlimits_\left\{x\right\}^\left\{infty\right\}e^\left\{-w^2/2sigma^2\right\} , dw$

Here $au$ is a parameter leveraging our prior knowledge of theapproximate location of $heta$. In this design, the random valueof $m_n\left(x_n\right)$ is distributed Bernoulli~$\left(q=F\left( au- heta\right)\right)$. Thefusion center averages the received bits to form an estimate$hat\left\{q\right\}$ of $q$, which is then used to find an estimate of $heta$. It can be verified that for the optimal (andinfeasible) choice of $au= heta$ the variance of this estimatoris $frac\left\{pisigma^2\right\}\left\{4\right\}$ which is only $pi/2$ times thevariance of MLE without bandwidth constraint. The varianceincreases as $au$ deviates from the real value of $heta$, but it can be shown that as long as $| au- heta|simsigma$ the factor in the MSE remains approximately 2. Choosing a suitable value for $au$ is a major disadvantage of this method since our model does not assume prior knowledge about the approximated location of $heta$. A coarse estimation can be used to overcome this limitation. However, it requires additional hardware in each ofthe sensors.

A system design with arbitrary (but known) noise PDF can be found in cite journal
last = Luo
first = Zhi-Quan
title = Universal decentralized estimation in a bandwidth constrained sensor network
journal = IEEE Trans. on Inf. Th.
date = June 2005
] . In this setting it is assumed that both $heta$ andthe noise $w_n$ are confined to some known interval $\left[-U,U\right]$. Theestimator of also reaches an MSE which is a constant factortimes $frac\left\{sigma^2\right\}\left\{N\right\}$. In this method, the prior knowledge of $U$ replacesthe parameter $au$ of the previous approach.

Unknown noise parameters

A noise model may be sometimes available while the exact PDF parameters are unknown (e.g. a Gaussian PDF with unknown $sigma$). The idea proposed in [cite journal
last = Ribeiro
first = Alejandro
coauthors = Georgios B. Giannakis
title = Bandwidth-constrained distributed estimation for wireless sensor networks-part II: unknown probability density function
journal = IEEE Trans. on Sig. Proc.
date = July 2006
] for this setting is to use twothresholds $au_1, au_2$, such that $N/2$ sensors are designedwith $m_A\left(x\right)=I\left(x- au_1\right)$, and the other $N/2$ sensors use$m_B\left(x\right)=I\left(x- au_2\right)$. The fusion center estimation rule is generated as follows:

: $hat\left\{q\right\}_1=frac\left\{2\right\}\left\{N\right\}sumlimits_\left\{n=1\right\}^\left\{N/2\right\}m_A\left(x_n\right), quadhat\left\{q\right\}_2=frac\left\{2\right\}\left\{N\right\}sumlimits_\left\{n=1+N/2\right\}^\left\{N\right\}m_B\left(x_n\right)$

: $hat\left\{ heta\right\}=frac\left\{F^\left\{-1\right\}\left(hat\left\{q\right\}_2\right) au_1-F^\left\{-1\right\}\left(hat\left\{q\right\}_1\right) au_2\right\}\left\{F^\left\{-1\right\}\left(hat\left\{q\right\}_2\right)-F^\left\{-1\right\}\left(hat\left\{q\right\}_1\right)\right\},quadF\left(x\right)=frac\left\{1\right\}\left\{sqrt\left\{2piintlimits_\left\{x\right\}^\left\{infty\right\}e^\left\{-v^2/2\right\}dw$

As before, prior knowledge is necessary to set values for$au_1, au_2$ in order to have an MSE with a reasonable factorof the unconstrained MLE variance.

Unknown noise PDF

We now describe the system design of for the case that the structure of the noisePDF is unknown. The following model is considered for this scenario:

: $x_n= heta+w_n,quad n=1,dots,N$

: $hetain \left[-U,U\right]$

: $w_ninmathcal\left\{P\right\}, ext\left\{ that is \right\}: w_n ext\left\{ is bounded to \right\} \left[-U,U\right] , mathbb\left\{E\right\}\left(w_n\right)=0$

In addition, the message functions are limited to have the form

:

where each $S_n$ is a subset of $\left[-2U,2U\right]$. The fusion estimator is also restricted to be linear, i.e.$hat\left\{ heta\right\}=sumlimits_\left\{n=1\right\}^\left\{N\right\}alpha_n m_n\left(x_n\right)$.

The design should set the decision intervals $S_n$ and thecoefficients $alpha_n$. Intuitively, we would allocate $N/2$ sensors to encode the first bit of $heta$ by setting their decision interval to be $\left[0,2U\right]$, then $N/4$ sensors would encode the second bit by setting their decision interval to$\left[-U,0\right] cup \left[U,2U\right]$ and so on. It can be shown that these decisionintervals and the corresponding set of coefficients $alpha_n$produce a universal $delta$-unbiased estimator, which is anestimator satisfying

The design of the sensor array requires optimizing the powerallocation as well as minimizing the communication traffic of theentire system. The design suggested in [cite journal
last = Xiao
first = Jin-Jun
coauthors = Shuguang Cui
coauthors = Zhi-Quan Luo
coauthors = Andrea J. Goldsmith
title = Joint estimation in sensor networks under energy constraint
journal = IEEE Trans. on Sig. Proc.
date = June 2005
] incorporates probabilistic quantization insensors and a simple optimization program that is solved in thefusion center only once. The fusion center then broadcasts a setof parameters to the sensors that allows them to finalize theirdesign of messaging functions $m_n\left(cdot\right)$ as to meet the energyconstraints. Another work employs a similar approach to addressdistributed detection in wireless sensor arrays [cite journal
last = Xiao
first = Jin-Jun
coauthors = Zhi-Quan Luo
title = Universal decentralized detection in a bandwidth-constrained sensor network
journal = IEEE Trans. on Sig. Proc.
date = August 2005
] .

* [http://www.eecs.harvard.edu/~mdw/proj/codeblue/ CodeBlue] Harvard group working on wireless sensor network technology to a range of medical applications.

References

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