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.

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'' 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 $\theta$ denote the position of interest. A set of $N$ sensors
acquire measurements $x_n = \theta + w_n$ contaminated by an
additive noise $w_n$ owing some known or unknown probability density function (PDF). The sensors transmit measurements to a central processor. The $n$th sensor encodes
$x_n$ by a function $m_n(x_n)$. The application processing the data applies a pre-defined estimation rule
$\hat=f(m_1(x_1),\cdot,m_N(x_N))$. The set of message functions
$m_n,\, 1\leq n\leq N$ and the fusion rule $f(m_1(x_1),\cdot,m_N(x_N))$ are
designed to minimize estimation error.
For example: minimizing the mean squared error (MSE),
$\mathbb\, \theta-\hat\, ^2$.

Ideally, sensors transmit their measurements $x_n$
right to the processing center, that is $m_n(x_n)=x_n$. In this
settings, the maximum likelihood estimator (MLE) $\hat = \frac\sum_^N x_n$ is an unbiased estimator whose MSE is
$\mathbb\, \theta-\hat\, ^2 = \text(\hat) = \frac$ assuming a white Gaussian noise
$w_n\sim\mathcal(0,\sigma^2)$. The next sections suggest
alternative designs when the sensors are bandwidth constrained to
1 bit transmission, that is $m_n(x_n)$=0 or 1.

Known noise PDF

A Gaussian noise
$w_n\sim\mathcal(0,\sigma^2)$ system can be designed as follows:
:

: $m_n(x_n)=I(x_n-\tau)= \begin 1 & x_n > \tau \\ 0 & x_n\leq \tau \end$

: $\hat=\tau-F^\left(\frac\sum\limits_^m_n(x_n)\right),\quad F(x)=\frac \int\limits_^ e^ \, dw$

Here $\tau$ is a parameter leveraging our prior knowledge of the
approximate location of $\theta$. In this design, the random value
of $m_n(x_n)$ is distributed Bernoulli~$(q=F(\tau-\theta))$. The
processing center averages the received bits to form an estimate
$\hat$ of $q$, which is then used to find an estimate of $\theta$. It can be verified that for the optimal (and
infeasible) choice of $\tau=\theta$ the variance of this estimator
is $\frac$ which is only $\pi/2$ times the
variance of MLE without bandwidth constraint. The variance
increases as $\tau$ deviates from the real value of $\theta$, but it can be shown that as long as $, \tau-\theta, \sim\sigma$ the factor in the MSE remains approximately 2. Choosing a suitable value for $\tau$ is a major disadvantage of this method since our model does not assume prior knowledge about the approximated location of $\theta$. 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.
In this setting it is assumed that both $\theta$ and
the noise $w_n$ are confined to some known interval U,U/math>. The
estimator of also reaches an MSE which is a constant factor
times $\frac$. In this method, the prior knowledge of $U$ replaces
the parameter $\tau$ 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 for this setting is to use two
thresholds $\tau_1,\tau_2$, such that $N/2$ sensors are designed
with $m_A(x)=I(x-\tau_1)$, and the other $N/2$ sensors use
$m_B(x)=I(x-\tau_2)$. The processing center estimation rule is generated as follows:

: $\hat_1=\frac\sum\limits_^m_A(x_n), \quad \hat_2=\frac\sum\limits_^m_B(x_n)$

: $\hat=\frac,\quad F(x)=\frac\int\limits_^e^dw$

As before, prior knowledge is necessary to set values for
$\tau_1,\tau_2$ to have an MSE with a reasonable factor
of the unconstrained MLE variance.

Unknown noise PDF

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

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

:
\theta\in U,U

:
w_n\in\mathcal, \text: w_n \text
U,U \mathbb(w_n)=0

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

: $m_n(x_n)= \begin 1 & x\in S_n \\ 0 & x \notin S_n \end$

where each $S_n$ is a subset of 2U,2U/math>. The fusion estimator is also restricted to be linear, i.e.
$\hat=\sum\limits_^\alpha_n m_n(x_n)$.

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

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 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 $m_n(\cdot)$ as to meet the energy
constraints. Another work employs a similar approach to address
distributed detection in wireless sensor arrays.

External links

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

References

{{Wireless Sensor Network

Estimation theory
Detection theory
Wireless sensor network