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.

Motivation

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 nth sensor encodes x_n by a function m_n(x_n). The application processing the data applies a pre-defined estimation rule \hat{\theta}=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{E}\|\theta-\hat{\theta}\|^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{\theta} = \frac{1}{N}\sum_{n=1}^N x_n is an unbiased estimator whose MSE is \mathbb{E}\|\theta-\hat{\theta}\|^2 = \text{var}(\hat{\theta}) = \frac{\sigma^2}{N} assuming a white Gaussian noise 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 m_n(x_n)=0 or 1.

Known noise PDF

We begin with an example of a Gaussian noise w_n\sim\mathcal{N}(0,\sigma^2), in which a suggestion for a system design is as follows

[1]
m_n(x_n)=I(x_n-\tau)= \begin{cases} 1 & x_n > \tau \\ 0 & x_n\leq \tau \end{cases}
\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 \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{q} 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{\pi\sigma^2}{4} 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.[2] In this setting it is assumed that both \theta and the noise w_n are confined to some known interval [-U,U]. The estimator of [2] also reaches an MSE which is a constant factor times \frac{\sigma^2}{N}. 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 [3] 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{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)
\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 \tau_1,\tau_2 to have an MSE with a reasonable factor of the unconstrained MLE variance.

Unknown noise PDF

We now describe the system design of [2] 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{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

m_n(x_n)= \begin{cases} 1 & x\in S_n \\ 0 & x \notin S_n \end{cases}

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

The design should set the decision intervals S_n and the coefficients \alpha_n. Intuitively, we would allocate N/2 sensors to encode the first bit of \theta 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]\cup[U,2U] 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{E}(\theta-\hat{\theta})|<\delta for every possible value of \theta\in[-U,U] and for every realization of 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 N\geq\lceil\log\frac{8U}{\delta}\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{2U}{\delta}\rceil, that is: the number of sensors is nearly optimal. It is also argued in [2] that if the targeted MSE \mathbb{E}\|\theta-\hat{\theta}\|\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 [4] 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.[5]

External links

References

  1. ↑ Ribeiro, Alejandro; Georgios B. Giannakis (March 2006). "Bandwidth-constrained distributed estimation for wireless sensor Networks-part I: Gaussian case". IEEE Trans. On Sig. Proc.
  2. ↑ 2.0 2.1 2.2 2.3 Luo, Zhi-Quan (June 2005). "Universal decentralized estimation in a bandwidth constrained sensor network". IEEE Trans. On Inf. Th.
  3. ↑ Ribeiro, Alejandro; Georgios B. Giannakis (July 2006). "Bandwidth-constrained distributed estimation for wireless sensor networks-part II: unknown probability density function". IEEE Trans. On Sig. Proc.
  4. ↑ Xiao, Jin-Jun; Andrea J. Goldsmith (June 2005). "Joint estimation in sensor networks under energy constraint". IEEE Trans. On Sig. Proc.
  5. ↑ Xiao, Jin-Jun; Zhi-Quan Luo (August 2005). "Universal decentralized detection in a bandwidth-constrained sensor network". IEEE Trans. On Sig. Proc.