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					ECE 776

Information-theoretic Approaches for Sensor
Selection and Placement in Sensor Networks for
Target Localization and Tracking

                          Renita Machado

Wireless sensor networks and their applications

   Wireless sensor networks (WSNs) are networks of large number
    of nodes deployed over a region to sense, gather and process
    data about their environment.

   The self organizing capabilities of WSNs enable their use in
    applications ranging from surveillance, ecology monitoring, bio-
    monitors and various other applications for developing smart

Key challenges in WSNs

Performance measures
 Coverage

 Connectivity

 Optimal redundancy

 Reliability of network operation

 Power limited nodes

 Economic constraints for dense deployments

Application: Target localization and tracking

In localization/ tracking, each sensor updates the probability
   distribution of the target location estimation.

Each observation reduces the uncertainty about target
  location or equivalently, gains information about the
  target location.

Preliminaries and problem formulation

Given that we have the
 Prior target location distribution p (x)

 Set of candidate sensors for selection S

 Locations of candidate sensors xi

 Observation models of candidate sensors p (zi|x)

 The sensor whose observation minimizes the expected

   conditional entropy of the posterior target location distribution, i.e.

      Equivalently, observation of this sensor maximizes the expected
       reduction of the target location entropy

Entropy difference in minimizing the
uncertainty of localization
   Reduction of localization uncertainty attributable to a sensor depends
    on the difference between
    A. Entropy of noise free sensor observation
    B. Entropy of that sensor observation model corresponding to the true
       target location

A. Sensor observation model

   Sensor observation model corresponding to the true target
         Probability distribution of the sensor observation conditioned
        on true target location
       Incorporates observation error from all sources, including
           Target
           Signal modeling error in estimation algorithm used by the sensor
           Inaccuracy of the sensor hardware

   Amount of uncertainty in sensor observation model may
    depend on the target location.

Determination of the sensor observation
   Since true target location is unknown during the process of target
    localization and tracking, we have to use an estimated target location to
    approximate the true target location to determine the sensor observation

Single-modal target location

    For a single model target location distribution p(x) that has a single
     peak, we can use the maximum likelihood estimate (MLE) estimate
     of the target location x’ to estimate the true target location and the
     approximate sensor observation model is

       H  Zi x '   p  zi x ' ln p  zi x ' dzi

    • When p(x) is a single-modal distribution, H(Zi|x’) is the entropy of the
      sensor observation model for the most likely target location estimate x’.

Multimodal target location distribution

   For a multimodal target location distribution p(x), viz., x’(m),
    m=1, 2…M, the entropy of the observation model of sensor i can be
    approximated by a weighted average as follows

                                     p  x '( m ) H Z i x '( m )           

    H  Zi x '              m 1

                                           px' 

                                          m 1

    When p(x) is a multi-modal distribution, the entropy of the sensor
    observation model is averaged over all target locations with local maximum

Relationship of H (Zi|x) to H(Zi|x’)

   H (Zi|x) is actually the entropy of the sensor observation
    model averaged over all possible target locations.

   When the entropy of the sensor observation model H(Zi|x)
    changes slowly with respect to the target location x,
    H(Zi|x’) reasonably approximates H (Zi|x).

B. Noise free sensor observation

   Noise free sensor observation
     No error is introduced in the sensor observation

   Let Ziv = noise-free observation of sensor i.
   Ziv assumes no randomness in the process of observation regarding the
    target location.
   Hence it is a function of target location X and sensor location xi.
   The target location X is a random variable, sensor location xi is a
    deterministic quantity.
   Hence the noise free sensor observation is a random variable.

    Distribution of the noise free sensor
    The target location X could be three-dimensional.
    The noise-free sensor observation Ziv could be two-dimensional.
    The distribution of the noise-free sensor observation Ziv is

           P ( Z iv  ziv )                            p ( x )dx
                                    f x , xi  ziv   
      where the observation perspective of sensor i largely depends on the sensor
      location xi.

Computing the noise free sensor observation
distribution and its entropy
   Let X be the set of target location grid values with non-trivial probability
   Let Z be the set of noise-free sensor observation grid values of non-trivial
    probability density
   For each grid point ziv є Z, initialize p(ziv) to zero;
   For each grid point x є X, the corresponding grid point ziv є Z is calculated
                    Ziv = f (X, xi)
   The probability is updated as
                   p (ziv)= p (ziv) + p (x)
   Normalize p (ziv) to make the total probability of Z to be 1.
   From p (ziv), we calculate the noise-free sensor observation entropy H(Ziv).

Relationship of H(Ziv) to H(Zi)

    H(Zi) is the entropy of the predicted sensor observation
                            p( zi )  p  zi x  p  x 
     The predicted sensor observation distribution p(Zi) becomes the noise-free
     sensor observation p(ziv) when the sensor observation model p(zi|x) is
      deterministic without any uncertainty.

     The uncertainty in the sensor observation model p(zi|x) makes the
    predicted sensor observation entropy H(Zi) larger than the noise-free
    sensor observation entropy H(Ziv) .

Approximations to the mutual-information calculation

  When the sensor observation model has only a small amount of uncertainty,
                       H     Ziv   H ( Zi )

     Since     H  Ziv   H (Zi )    and    H (Zi x ')  H (Zi x)

                  I ( X ; Z i )  H ( Zi )  H ( Zi | X )
                                 H ( Ziv )  H ( Zi x ')
      Thus, the sensor with the maximum entropy difference

                I ( X ; Zi )  H (Ziv )  H (Zi x ')
      probably also has the maximum mutual information.

Why not select the mutual information
     For target location X and the predicted sensor observation Zi

                                                          p ( x, zi )
                        I ( X ; Z i )   p ( x, zi ) ln                  dxdzi
                                                         p ( x ) p ( zi )

    The target location could be 3-dimensional and the sensor observation
    could be 2-dimensional.

    Then I(X;Zi) could be a complex integral in the joint state space of 5 dimensions.

    Thus, the total cost to select one of K candidate sensors is O(n5).

Complexity of the entropy approach

   H(Ziv) can be computed from p(ziv) with complexity O(n2)

    P ( Z iv  ziv )                         p ( x ) dx
                          f x , xi  ziv   

     Computing H(Zi|x’) (from single and multi-modal) distributions also
     requires complexity of O(n2).

     Thus the cost to compute the entropy difference for one candidate
     sensor is O(n3).

     Thus the total cost to select one sensor out of K candidate sensors is

Reduction in complexity

   The computational complexity of the mutual information
    approach is greater than that of the entropy difference

   With power constraints and processing complexity
    constraints, the entropy difference approach fares better
    than the mutual information approach for selecting a
    sensor for target localization.


   TDOA sensors   TDOA, range and DOA sensors


   The entropy difference approach is simpler to calculate
    than the mutual information criterion for sensor selection.

   A sub-optimal sensor can be selected without retrieving
    sensor data.

   Reference
   H.Wang, K.Yao and D.Estrin, “Information-theoretic approaches for sensor selection
    and placement in sensor networks for target localization and tracking”, CENS Technical
    Report #52, 2005.


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