Location estimation in 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.

Use[edit]

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[edit]

Let ${\displaystyle \theta }$ denote the position of interest. A set of ${\displaystyle N}$ sensors acquire measurements ${\displaystyle x_{n}=\theta +w_{n}}$ contaminated by an additive noise ${\displaystyle w_{n}}$ owing some known or unknown probability density function (PDF). The sensors transmit measurements to a central processor. The ${\displaystyle n}$th sensor encodes ${\displaystyle x_{n}}$ by a function ${\displaystyle m_{n}(x_{n})}$. The application processing the data applies a pre-defined estimation rule ${\displaystyle {\hat {\theta }}=f(m_{1}(x_{1}),\cdot ,m_{N}(x_{N}))}$. The set of message functions ${\displaystyle m_{n},\,1\leq n\leq N}$ and the fusion rule ${\displaystyle f(m_{1}(x_{1}),\cdot ,m_{N}(x_{N}))}$ are designed to minimize estimation error. For example: minimizing the mean squared error (MSE), ${\displaystyle \mathbb {E} \|\theta -{\hat {\theta }}\|^{2}}$.

Ideally, sensors transmit their measurements ${\displaystyle x_{n}}$ right to the processing center, that is ${\displaystyle m_{n}(x_{n})=x_{n}}$. In this settings, the maximum likelihood estimator (MLE) ${\displaystyle {\hat {\theta }}={\frac {1}{N}}\sum _{n=1}^{N}x_{n}}$ is an unbiased estimator whose MSE is ${\displaystyle \mathbb {E} \|\theta -{\hat {\theta }}\|^{2}={\text{var}}({\hat {\theta }})={\frac {\sigma ^{2}}{N}}}$ assuming a white Gaussian noise ${\displaystyle w_{n}\sim {\mathcal {N}}(0,\sigma ^{2})}$. The next sections suggest alternative designs when the sensors are bandwidth constrained to 1 bit transmission, that is ${\displaystyle m_{n}(x_{n})}$=0 or 1.

Known noise PDF[edit]

A Gaussian noise ${\displaystyle w_{n}\sim {\mathcal {N}}(0,\sigma ^{2})}$ system can be designed as follows:

^[2]

${\displaystyle m_{n}(x_{n})=I(x_{n}-\tau )={\begin{cases}1&x_{n}>\tau \\0&x_{n}\leq \tau \end{cases}}}$

${\displaystyle {\hat {\theta }}=\tau -F^{-1}\left({\frac {1}{N}}\sum \limits _{n=1}^{N}m_{n}(x_{n})\right),\quad F(x)={\frac {1}{{\sqrt {2\pi }}\sigma }}\int \limits _{x}^{\infty }e^{-w^{2}/2\sigma ^{2}}\,dw}$

Here ${\displaystyle \tau }$ is a parameter leveraging our prior knowledge of the approximate location of ${\displaystyle \theta }$. In this design, the random value of ${\displaystyle m_{n}(x_{n})}$ is distributed Bernoulli~${\displaystyle (q=F(\tau -\theta ))}$. The processing center averages the received bits to form an estimate ${\displaystyle {\hat {q}}}$ of ${\displaystyle q}$, which is then used to find an estimate of ${\displaystyle \theta }$. It can be verified that for the optimal (and infeasible) choice of ${\displaystyle \tau =\theta }$ the variance of this estimator is ${\displaystyle {\frac {\pi \sigma ^{2}}{4}}}$ which is only ${\displaystyle \pi /2}$ times the variance of MLE without bandwidth constraint. The variance increases as ${\displaystyle \tau }$ deviates from the real value of ${\displaystyle \theta }$, but it can be shown that as long as ${\displaystyle |\tau -\theta |\sim \sigma }$ the factor in the MSE remains approximately 2. Choosing a suitable value for ${\displaystyle \tau }$ is a major disadvantage of this method since our model does not assume prior knowledge about the approximated location of ${\displaystyle \theta }$. 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 ${\displaystyle \theta }$ and the noise ${\displaystyle w_{n}}$ are confined to some known interval ${\displaystyle [-U,U]}$. The estimator of ^[3] also reaches an MSE which is a constant factor times ${\displaystyle {\frac {\sigma ^{2}}{N}}}$. In this method, the prior knowledge of ${\displaystyle U}$ replaces the parameter ${\displaystyle \tau }$ of the previous approach.

Unknown noise parameters[edit]

A noise model may be sometimes available while the exact PDF parameters are unknown (e.g. a Gaussian PDF with unknown ${\displaystyle \sigma }$). The idea proposed in ^[4] for this setting is to use two thresholds ${\displaystyle \tau _{1},\tau _{2}}$, such that ${\displaystyle N/2}$ sensors are designed with ${\displaystyle m_{A}(x)=I(x-\tau _{1})}$, and the other ${\displaystyle N/2}$ sensors use ${\displaystyle m_{B}(x)=I(x-\tau _{2})}$. The processing center estimation rule is generated as follows:

${\displaystyle {\hat {q}}_{1}={\frac {2}{N}}\sum \limits _{n=1}^{N/2}m_{A}(x_{n}),\quad {\hat {q}}_{2}={\frac {2}{N}}\sum \limits _{n=1+N/2}^{N}m_{B}(x_{n})}$

${\displaystyle {\hat {\theta }}={\frac {F^{-1}({\hat {q}}_{2})\tau _{1}-F^{-1}({\hat {q}}_{1})\tau _{2}}{F^{-1}({\hat {q}}_{2})-F^{-1}({\hat {q}}_{1})}},\quad F(x)={\frac {1}{\sqrt {2\pi }}}\int \limits _{x}^{\infty }e^{-v^{2}/2}dw}$

As before, prior knowledge is necessary to set values for ${\displaystyle \tau _{1},\tau _{2}}$ to have an MSE with a reasonable factor of the unconstrained MLE variance.

Unknown noise PDF[edit]

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:

${\displaystyle x_{n}=\theta +w_{n},\quad n=1,\dots ,N}$

${\displaystyle \theta \in [-U,U]}$

${\displaystyle w_{n}\in {\mathcal {P}},{\text{ that is }}:w_{n}{\text{ is bounded to }}[-U,U],\mathbb {E} (w_{n})=0}$

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

${\displaystyle m_{n}(x_{n})={\begin{cases}1&x\in S_{n}\\0&x\notin S_{n}\end{cases}}}$

where each ${\displaystyle S_{n}}$ is a subset of ${\displaystyle [-2U,2U]}$. The fusion estimator is also restricted to be linear, i.e. ${\displaystyle {\hat {\theta }}=\sum \limits _{n=1}^{N}\alpha _{n}m_{n}(x_{n})}$.

The design should set the decision intervals ${\displaystyle S_{n}}$ and the coefficients ${\displaystyle \alpha _{n}}$. Intuitively, one would allocate ${\displaystyle N/2}$ sensors to encode the first bit of ${\displaystyle \theta }$ by setting their decision interval to be ${\displaystyle [0,2U]}$, then ${\displaystyle N/4}$ sensors would encode the second bit by setting their decision interval to ${\displaystyle [-U,0]\cup [U,2U]}$ and so on. It can be shown that these decision intervals and the corresponding set of coefficients ${\displaystyle \alpha _{n}}$ produce a universal ${\displaystyle \delta }$-unbiased estimator, which is an estimator satisfying ${\displaystyle |\mathbb {E} (\theta -{\hat {\theta }})|<\delta }$ for every possible value of ${\displaystyle \theta \in [-U,U]}$ and for every realization of ${\displaystyle w_{n}\in {\mathcal {P}}}$. In fact, this intuitive design of the decision intervals is also optimal in the following sense. The above design requires ${\displaystyle N\geq \lceil \log {\frac {8U}{\delta }}\rceil }$ to satisfy the universal ${\displaystyle \delta }$-unbiased property while theoretical arguments show that an optimal (and a more complex) design of the decision intervals would require ${\displaystyle N\geq \lceil \log {\frac {2U}{\delta }}\rceil }$, that is: the number of sensors is nearly optimal. It is also argued in ^[3] that if the targeted MSE ${\displaystyle \mathbb {E} \|\theta -{\hat {\theta }}\|\leq \epsilon ^{2}}$ uses a small enough ${\displaystyle \epsilon }$, 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[edit]

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 ${\displaystyle m_{n}(\cdot )}$ as to meet the energy constraints. Another work employs a similar approach to address distributed detection in wireless sensor arrays.^[6]

External links[edit]

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

References[edit]