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Adaptive quantized target tracking in wireless sensor networks

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Abstract

This paper addresses target tracking in wireless sensor networks (WSN) where the observed system is assumed to evolve according to a probabilistic state space model. We propose to improve the use of the variational filtering (VF) by optimally quantizing the data collected by the sensors. Recently, VF has been proved to be suitable to the communication constraints of WSN. Its efficiency relies on the fact that the online update of the filtering distribution and its compression are executed simultaneously. However, this problem has been used only for binary sensor networks neglecting the transmission energy consumption in a WSN and the information relevance of sensor measurements. Our proposed method is intended to jointly estimate the target position and optimize the quantization level under fixed and variable transmitting power. At each sampling instant, the adaptive method provides not only the estimate of the target position by using the VF but gives also the optimal number of quantization bits per observation. The adaptive quantization is achieved by minimizing the predicted Cramér–Rao bound if the transmitting power is constant for all sensors, and optimizing the power scheduling under distortion constraint if this power is variable. The computation of the predicted Cramér–Rao bound is based on the target position predictive distribution provided by the VF algorithm. The proposed adaptive quantization scheme suggests that the sensors with bad channels or poor observation qualities should decrease their quantization resolutions or simply become inactive in order to save energy.

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Authors and Affiliations

  1. ICD/LM2S, University of Technology of Troyes, Troyes, France

    Majdi Mansouri & Hichem Snoussi

  2. Insitut Supérieur d’Informatique et des Technologies de Communication Hammam Sousse, Hammam Sousse, Tunisia

    Ouachani Ilham

  3. Laboratoire FIZEAU UMR CNRS 6525, Université de Nice Sophia–Antipolis, Valbonne, France

    Cédric Richard

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  1. Majdi Mansouri
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  2. Ouachani Ilham
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  3. Hichem Snoussi
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Corresponding author

Correspondence to Majdi Mansouri.

Appendix: Variational calculus

Appendix: Variational calculus

Assuming that the approximate distribution for the mean \(\varvec{\mu}_{t-1}\) follows a gaussian model \( \left(q (\varvec{\mu}_{t-1}) \sim \mathcal{N}\left(\varvec{\mu}_{t-1}^*, \varvec{\lambda}_{t-1}^*\right)\right)\) and taking into account the Gaussian transition of the mean (\(p\left(\varvec{\mu}_{t}|\varvec{\mu}_{t-1}) \sim \mathcal{N}(\varvec{\mu}_{t-1},\bar{\varvec{\lambda}}\right)\)), the predictive distribution of μ t is given by:

$$ \begin{aligned} q_{p}(\varvec{\mu}_{t})&=\int p(\varvec{\mu}_{t}|\varvec{\mu}_{t-1})q(\varvec{\mu}_{t-1})d\varvec{\mu}_{t-1}\\ &\sim {\mathcal{N}}\left(\varvec{\mu}_{t-1}^*, {\left({\varvec{\lambda}_{t-1}^{*}}^{-1}+{\bar{\varvec{\lambda}}}^{-1}\right)}^{-1}\right). \end{aligned}$$
(29)

Let denote \(\varvec{\mu}_t^p\) and \(\varvec{\lambda}_t^p\) respectively the mean and the precision of the Gaussian distribution \(q_{p}(\varvec{\mu}_{t}): \,\,q_{p}(\varvec{\mu}_{t})\sim \mathcal{N}\left(\varvec{\mu}_t^p, \varvec{\lambda}_t^p\right).\) According to the (6), the approximate distribution \(q(\varvec{\mu}_t)\) is expressed as:

$$ \begin{aligned} q(\varvec{\mu}_t) & \propto \exp \langle\log p(\user2{z}_{1:t},\varvec{\alpha}_t)\rangle _{q(\user2{x}_t)q(\varvec{\lambda}_t)}\\ & \propto \exp \langle\log p(\varvec{\alpha}_t|\user2{z}_t)\rangle _{q(\user2{x}_t)q(\varvec{\lambda}_t)}\\ & \propto \exp \langle\log p(\user2{z}_t|\user2{x}_t)+\log p(\user2{x}_{t}|\varvec{\mu}_t, \varvec{\lambda}_t)+\log p(\varvec{\lambda}_t)+\log q_{p}(\varvec{\mu}_{t})\rangle _{q(\user2{x}_t)q(\varvec{\lambda}_t)}. \end{aligned}$$
(30)

Therefore,

$$ \begin{aligned} q(\varvec{\mu}_t) &\propto q_{p}(\varvec{\mu}_{t})\exp \langle\log p(\user2{x}_{t}|\varvec{\mu}_t, \varvec{\lambda}_t)\rangle _{q(\user2{x}_t)q(\varvec{\lambda}_t)}\\ & \propto q_{p}(\varvec{\mu}_{t})\exp \langle -\frac{1}{2}(\user2{x}_t-\varvec{\mu}_t)^T\varvec{\lambda}_t(\user2{x}_t-\varvec{\mu}_t)\rangle _{q(\user2{x}_t)q(\varvec{\lambda}_t)}\\ & \propto q_{p}(\varvec{\mu}_{t})\exp -\frac{1}{2}\left\{tr\left[\langle\varvec{\lambda}_t\rangle _{q(\varvec{\lambda}_t)} \langle(\user2{x}_t-\varvec{\mu}_t)^T(\user2{x}_t-\varvec{\mu}_t)\rangle _{q(\user2{x}_t)}\right]\right\}\\ & \propto \exp -\frac{1}{2}\left[\left(\varvec{\mu}_t-\varvec{\mu}_t^p\right)^T\varvec{\lambda}_t\left(\varvec{\mu}_t- \varvec{\mu}_t^p\right)-2\varvec{\mu}_t^T\langle\varvec{\lambda}_t\rangle \langle\user2{x}_t\rangle +{\varvec{\mu}_t}^T\langle\varvec{\lambda}_t\rangle \varvec{\mu}_t\right], \end{aligned}$$
(31)

yielding a Gaussian distribution \(q(\varvec{\mu}_t) = \mathcal{N}\left(\varvec{\mu}_t^*,\varvec{\lambda}_t^*\right).\) The first and the second derivatives of the logarithm of \(q(\varvec{\mu}_t)\) are expressed as:

$$ \begin{aligned} \frac{\partial{\log(q(\varvec{\mu}_t))}}{\partial{\varvec{\mu}_t}} &=-\frac{1}{2}\left[2\varvec{\lambda}_t^p(\varvec{\mu}_t-\varvec{\mu}_t^p)-2 \langle\varvec{\lambda}_t\rangle \langle\user2{x}_t\rangle +2\langle\varvec{\lambda}_t\rangle \varvec{\mu}_t\right],\\ \frac{\partial^2 {\log(q(\varvec{\mu}_t))}}{\partial {\varvec{\mu}_t}\partial{{\varvec{\mu}_t}^T}}&=- \varvec{\lambda}_t^p-\langle\varvec{\lambda}_t\rangle, \end{aligned}$$

the precision \(\varvec{\lambda}_t^*\) and the mean \(\varvec{\mu}_t^*\) of \(q(\varvec{\mu}_t)\) are obtained as follows:

$$ \varvec{\lambda}_t^* = \langle \varvec{\lambda}_t\rangle + \varvec{\lambda}_t^p,\quad\hbox{and}\quad \varvec{\mu}_t^* = \varvec{\lambda}_t^{* -1}\left(\langle \varvec{\lambda}_t \rangle \langle \user2{x}_t \rangle + \varvec{\lambda}_t^p\varvec{\mu}_t^p\right).$$
(32)

The approximate separable distribution corresponding to \(\varvec{\lambda}_t\) can be computed following the same reasoning as above:

$$ \begin{aligned} q(\varvec{\lambda}_t) &\propto \exp \langle\log p(\varvec{\alpha}_t|\user2{z}_t)\rangle _{q(\user2{x}_t)q(\varvec{\mu}_t)}\\& \propto \exp \langle\log p(\user2{z}_t|\user2{x}_t)+\log p(\user2{x}_{t}|\varvec{\mu}_t, \varvec{\lambda}_t)+\log p(\varvec{\lambda}_t)+\log q_{p}(\varvec{\mu}_{t})\rangle _{q(\user2{x}_t)q(\varvec{\mu}_t)}\\& \propto p(\varvec{\lambda}_t)\exp \langle\log p(\user2{x}_{t}|\varvec{\mu}_t, \varvec{\lambda}_t)\rangle _{q(\user2{x}_t)q(\varvec{\mu}_t)}\\& \propto {\mathcal{W}}_2(\bar{\user2{V}},\bar{n})|\varvec{\lambda}_t|^{\frac{1}{2}}\exp -\frac{1}{2}\left\{tr\left[\varvec{\lambda}_t\langle(\user2{x}_t- \varvec{\mu}_t)^T(\user2{x}_t-\varvec{\mu}_t)\rangle_{q(\user2{x}_t)q(\varvec{\mu}_t)}\right]\right\}\\& \propto |\varvec{\lambda}_t|^{\frac{\bar{n}+1-(2+1)}{2}}\exp -\frac{1}{2}\left\{tr\left[\varvec{\lambda}_t\left(\langle \user2{x}_t \user2{x}_t^T \rangle -\langle \user2{x}_t \rangle \langle \varvec{\mu}_t \rangle ^T-\langle \varvec{\mu}_t \rangle \langle \user2{x}_t \rangle ^T + \langle \varvec{\mu}_t\varvec{\mu}_t^T \rangle + {\bar{\user2{V}}}^{-1}\right)\right]\right\}, \end{aligned}$$
(33)

which yields a Wishart distribution \(\mathcal{W}_2(\user2{V}^*,n^*)\) for the precision matrix \(\varvec{\lambda}_t\) with the following parameters:

$$ \left\{\begin{array}{l} n^* = \bar{n} + 1,\\ \user2{V}^*= \left(\langle \user2{x}_t \user2{x}_t^T \rangle -\langle \user2{x}_t \rangle \langle \varvec{\mu}_t \rangle ^T-\langle \varvec{\mu}_t \rangle \langle \user2{x}_t \rangle ^T + \langle \varvec{\mu}_t\varvec{\mu}_t^T \rangle + {\bar{\user2{V}}}^{-1} \right)^{-1}.\\ \end{array}\right.$$
(34)

Finally, the approximate distribution \(q(\user2{x}_t)\) has the following expression:

$$ \begin{aligned} q(\user2{x}_t) &\propto \exp \langle\log p(\varvec{\alpha}_t|\user2{z}_t)\rangle _{q(\varvec{\mu}_t)q(\varvec{\lambda}_t)}\\&\propto \exp \langle\log p(\user2{z}_t|\user2{x}_t)+\log p(\user2{x}_{t}|\varvec{\mu}_t, \varvec{\lambda}_t)+\log p(\varvec{\lambda}_t)+\log q_{p}(\varvec{\mu}_{t})\rangle _{q(\varvec{\mu}_t)q(\varvec{\lambda}_t)} \\&\propto p(\user2{z}_t|\user2{x}_t)\exp \langle\log p(\user2{x}_{t}|\varvec{\mu}_t, \varvec{\lambda}_t)\rangle _{q(\varvec{\mu}_t)q(\varvec{\lambda}_t)}\\&\propto p(\user2{z}_t|\user2{x}_t)\exp -\frac{1}{2}\left\{tr\left[\langle\varvec{\lambda}_t\rangle _{q(\varvec{\lambda}_t)}\langle(\user2{x}_t-\varvec{\mu}_t)^T(\user2{x}_t-\varvec{\mu}_t)\rangle _{q(\varvec{\mu}_t)}\right]\right\}\\&\propto p(\user2{z}_t|\user2{x}_t){\mathcal{N}}(\langle\varvec{\mu}_t\rangle ,\langle\varvec{\lambda}_t\rangle ),\end{aligned}$$
(35)

which does not have a closed form. Therefore, contrary to the cases of the mean \(\varvec{\mu}_t\) and the precision \(\varvec{\lambda}_t,\) in order to compute the expectations relative to the distribution \(q(\user2{x}_t),\) one has to resort to the importance sampling method where samples are generated according to the Gaussian \(\mathcal{N}(\langle\varvec{\mu}_t\rangle ,\langle\varvec{\lambda}_t\rangle )\) and then weighted according to the likelihood \(p(\user2{z}_t|\user2{x}_t).\)

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Mansouri, M., Ilham, O., Snoussi, H. et al. Adaptive quantized target tracking in wireless sensor networks. Wireless Netw 17, 1625–1639 (2011). https://doi.org/10.1007/s11276-011-0368-1

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