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. These measurements are acquired in a distributed manner by a set of sensors.

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

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Let   denote the position of interest. A set of   sensors acquire measurements   contaminated by an additive noise   owing some known or unknown probability density function (PDF). The sensors transmit measurements to a central processor. The  th sensor encodes   by a function  . The application processing the data applies a pre-defined estimation rule  . The set of message functions   and the fusion rule   are designed to minimize estimation error. For example: minimizing the mean squared error (MSE),  .

Ideally, sensors transmit their measurements   right to the processing center, that is  . In this settings, the maximum likelihood estimator (MLE)   is an unbiased estimator whose MSE is   assuming a white Gaussian noise  . The next sections suggest alternative designs when the sensors are bandwidth constrained to 1 bit transmission, that is  =0 or 1.

Known noise PDF

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A Gaussian noise   system can be designed as follows:

[2]
 
 

Here   is a parameter leveraging our prior knowledge of the approximate location of  . In this design, the random value of   is distributed Bernoulli~ . The processing center averages the received bits to form an estimate   of  , which is then used to find an estimate of  . It can be verified that for the optimal (and infeasible) choice of   the variance of this estimator is   which is only   times the variance of MLE without bandwidth constraint. The variance increases as   deviates from the real value of  , but it can be shown that as long as   the factor in the MSE remains approximately 2. Choosing a suitable value for   is a major disadvantage of this method since our model does not assume prior knowledge about the approximated location of  . 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   and the noise   are confined to some known interval  . The estimator of [3] also reaches an MSE which is a constant factor times  . In this method, the prior knowledge of   replaces the parameter   of the previous approach.

Unknown noise parameters

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A noise model may be sometimes available while the exact PDF parameters are unknown (e.g. a Gaussian PDF with unknown  ). The idea proposed in [4] for this setting is to use two thresholds  , such that   sensors are designed with  , and the other   sensors use  . The processing center estimation rule is generated as follows:

 
 

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

Unknown noise PDF

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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:

 
 
 

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

 

where each   is a subset of  . The fusion estimator is also restricted to be linear, i.e.  .

The design should set the decision intervals   and the coefficients  . Intuitively, one would allocate   sensors to encode the first bit of   by setting their decision interval to be  , then   sensors would encode the second bit by setting their decision interval to   and so on. It can be shown that these decision intervals and the corresponding set of coefficients   produce a universal  -unbiased estimator, which is an estimator satisfying   for every possible value of   and for every realization of  . In fact, this intuitive design of the decision intervals is also optimal in the following sense. The above design requires   to satisfy the universal  -unbiased property while theoretical arguments show that an optimal (and a more complex) design of the decision intervals would require  , that is: the number of sensors is nearly optimal. It is also argued in [3] that if the targeted MSE   uses a small enough  , 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

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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   as to meet the energy constraints. Another work employs a similar approach to address distributed detection in wireless sensor arrays.[6]

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

References

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  1. ^ "Archived copy". Archived from the original on 2008-04-30. Retrieved 2008-04-30.{{cite web}}: CS1 maint: archived copy as title (link)
  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. Bibcode:2006ITSP...54.1131R. doi:10.1109/TSP.2005.863009. S2CID 16223482.
  3. ^ a b c d 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. S2CID 11574873.
  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. Bibcode:2006ITSP...54.2784R. doi:10.1109/TSP.2006.874366. S2CID 11410878.
  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. Bibcode:2005ITSP...53.2617X. doi:10.1109/TSP.2005.850334. S2CID 8072065.