Location estimation in sensor networks

Short description: Estimating objects's location in wireless sensor networks

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

## Contents

* 1 Use
* 2 Setting
* 3 Known noise PDF
* 4 Unknown noise parameters
* 5 Unknown noise PDF
* 6 Additional information
* 7 External links
* 8 References

## Use

Many civilian and military applications require monitoring that can identify objects in a specific area, such as monitoring the front entrance of a private house by a single camera. Monitored areas that are large relative to objects of interest often require multiple sensors (e.g., infra-red detectors) at multiple locations. A centralized observer or computer application monitors the sensors. The communication to power and bandwidth requirements call for efficient design of the sensor, transmission, and processing.

The CodeBlue system[1] of Harvard University is an example where a vast number of sensors distributed among hospital facilities allow staff to locate a patient in distress. In addition, the sensor array enables online recording of medical information while allowing the patient to move around. Military applications (e.g. locating an intruder into a secured area) are also good candidates for setting a wireless sensor network.

## Setting

Let θ denote the position of interest. A set of N sensors acquire measurements xn=θ+wn contaminated by an additive noise wn owing some known or unknown probability density function (PDF). The sensors transmit measurements to a central processor. The nth sensor encodes xn by a function mn(xn). The application processing the data applies a pre-defined estimation rule θ^=f(m1(x1),⋅,mN(xN)). The set of message functions mn,1≤n≤N and the fusion rule f(m1(x1),⋅,mN(xN)) are designed to minimize estimation error. For example: minimizing the mean squared error (MSE), 𝔼‖θ−θ^‖2.

Ideally, sensors transmit their measurements xn right to the processing center, that is mn(xn)=xn. In this settings, the maximum likelihood estimator (MLE) θ^=1N∑n=1Nxn is an unbiased estimator whose MSE is 𝔼‖θ−θ^‖2=var(θ^)=σ2N assuming a white Gaussian noise wn∼𝒩(0,σ2). The next sections suggest alternative designs when the sensors are bandwidth constrained to 1 bit transmission, that is mn(xn)=0 or 1.

## Known noise PDF

A Gaussian noise wn∼𝒩(0,σ2) system can be designed as follows:

[2]

mn(xn)=I(xn−τ)={1xn>τ0xn≤τ

θ^=τ−F−1(1N∑n=1Nmn(xn)),F(x)=12πσ∫x∞e−w2/2σ2dw

Here τ is a parameter leveraging our prior knowledge of the approximate location of θ. In this design, the random value of mn(xn) is distributed Bernoulli~(q=F(τ−θ)). The processing center averages the received bits to form an estimate q^ of q, which is then used to find an estimate of θ. It can be verified that for the optimal (and infeasible) choice of τ=θ the variance of this estimator is πσ24 which is only π/2 times the variance of MLE without bandwidth constraint. The variance increases as τ deviates from the real value of θ, but it can be shown that as long as |τ−θ|∼σ the factor in the MSE remains approximately 2. Choosing a suitable value for τ is a major disadvantage of this method since our model does not assume prior knowledge about the approximated location of θ. A coarse estimation can be used to overcome this limitation. However, it requires additional hardware in each of the sensors.

A system design with arbitrary (but known) noise PDF can be found in.[3] In this setting it is assumed that both θ and the noise wn are confined to some known interval [−U,U]. The estimator of [3] also reaches an MSE which is a constant factor times σ2N. In this method, the prior knowledge of U replaces the parameter τ 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 σ). The idea proposed in [4] for this setting is to use two thresholds τ1,τ2, such that N/2 sensors are designed with mA(x)=I(x−τ1), and the other N/2 sensors use mB(x)=I(x−τ2). The processing center estimation rule is generated as follows:

q^1=2N∑n=1N/2mA(xn),q^2=2N∑n=1+N/2NmB(xn)

θ^=F−1(q^2)τ1−F−1(q^1)τ2F−1(q^2)−F−1(q^1),F(x)=12π∫x∞e−v2/2dw

As before, prior knowledge is necessary to set values for τ1,τ2 to have an MSE with a reasonable factor of the unconstrained MLE variance.

## Unknown noise PDF

The system design of [3] for the case that the structure of the noise PDF is unknown. The following model is considered for this scenario:

xn=θ+wn,n=1,…,N

θ∈[−U,U]

wn∈𝒫, that is :wn is bounded to [−U,U],𝔼(wn)=0

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

mn(xn)={1x∈Sn0x∉Sn

where each Sn is a subset of [−2U,2U]. The fusion estimator is also restricted to be linear, i.e. θ^=∑n=1Nαnmn(xn).

The design should set the decision intervals Sn and the coefficients αn. Intuitively, one would allocate N/2 sensors to encode the first bit of θ by setting their decision interval to be [0,2U], then N/4 sensors would encode the second bit by setting their decision interval to [−U,0]∪[U,2U] and so on. It can be shown that these decision intervals and the corresponding set of coefficients αn produce a universal δ-unbiased estimator, which is an estimator satisfying |𝔼(θ−θ^)|<δ for every possible value of θ∈[−U,U] and for every realization of wn∈𝒫. In fact, this intuitive design of the decision intervals is also optimal in the following sense. The above design requires N≥⌈log⁡8Uδ⌉ to satisfy the universal δ-unbiased property while theoretical arguments show that an optimal (and a more complex) design of the decision intervals would require N≥⌈log⁡2Uδ⌉, that is: the number of sensors is nearly optimal. It is also argued in [3] that if the targeted MSE 𝔼‖θ−θ^‖≤ϵ2 uses a small enough ϵ, then this design requires a factor of 4 in the number of sensors to achieve the same variance of the MLE in the unconstrained bandwidth settings.

## Additional information

The design of the sensor array requires optimizing the power allocation as well as minimizing the communication traffic of the entire system. The design suggested in [5] incorporates probabilistic quantization in sensors and a simple optimization program that is solved in the fusion center only once. The fusion center then broadcasts a set of parameters to the sensors that allows them to finalize their design of messaging functions mn(⋅) as to meet the energy constraints. Another work employs a similar approach to address distributed detection in wireless sensor arrays.[6]

## External links

* CodeBlue Harvard group working on wireless sensor network technology to a range of medical applications.

## References

1. ↑ "Archived copy". Archived from the original on 2008-04-30. https://web.archive.org/web/20080430133030/http://www.eecs.harvard.edu/~mdw/proj/codeblue/. Retrieved 2008-04-30.
2. ↑ Ribeiro, Alejandro; Georgios B. Giannakis (March 2006). "Bandwidth-constrained distributed estimation for wireless sensor Networks-part I: Gaussian case". IEEE Transactions on Signal Processing 54 (3): 1131. doi:10.1109/TSP.2005.863009. Bibcode: 2006ITSP...54.1131R.
3. ↑ 3.0 3.1 3.2 3.3 Luo, Zhi-Quan (June 2005). "Universal decentralized estimation in a bandwidth constrained sensor network". IEEE Transactions on Information Theory 51 (6): 2210–2219. doi:10.1109/TIT.2005.847692.
4. ↑ Ribeiro, Alejandro; Georgios B. Giannakis (July 2006). "Bandwidth-constrained distributed estimation for wireless sensor networks-part II: unknown probability density function". IEEE Transactions on Signal Processing 54 (7): 2784. doi:10.1109/TSP.2006.874366. Bibcode: 2006ITSP...54.2784R.
5. ↑ Xiao, Jin-Jun; Andrea J. Goldsmith (June 2005). "Joint estimation in sensor networks under energy constraint". IEEE Transactions on Signal Processing.
6. ↑ Xiao, Jin-Jun; Zhi-Quan Luo (August 2005). "Universal decentralized detection in a bandwidth-constrained sensor network". IEEE Transactions on Signal Processing 53 (8): 2617. doi:10.1109/TSP.2005.850334. Bibcode: 2005ITSP...53.2617X.

* v
* t
* e

Wireless sensor network

Operating
systems|

* Contiki
* ERIKA Enterprise
* LiteOS
* NanoQplus
* Nano-RK
* OpenTag
* RIOT
* TinyOS
* OpenWSN

Communications
protocols|

* 6LoWPAN
* IEEE 802.15.4
* ANT
* Bluetooth
* Bluetooth Low Energy (Wibree)
* DASH7
* ISA100.11a
* MiWi
* Near-field communication
* OCARI
* One-Net
* OSIAN
* Sidewalk
* Thread
* TIBUMAC
* TSMP
* WirelessHART
* Zigbee
* Z-Wave

Programming
languages|

* C
* LabVIEW

Hardware|

* Arduino
* Iris Mote
* Sun SPOT
* XBee

Software|

* NS-2
* LinuxMCE
* OPNET
* TinyDB-TOSSIM

Applications|

* Key distribution
* Location estimation
* Sensor network query processor
* Sensor web
* Wireless powerline sensor
* Telemetry

Routing
protocols|

* Ad hoc On-Demand Distance Vector Routing (AODV)
* Dynamic Source Routing (DSR)

Conferences,
journals|

* European Conference on Wireless Sensor Networks (EWSN)
* International Conference on Information Processing in Sensor Networks (IPSN)
* Conference on Embedded Networked Sensor Systems (SenSys)

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