1. Data Selection In Ad-Hoc Wireless Sensor Networks Olawoye Oyeyele 11/24/2003

2. Outline Objectives Data Selection Problem Randomization Spatial Selection Discussions

3. Objectives Outline the salient aspects of the Randomization algorithm [1] Indicate important parameters in the analysis of data selection Discuss properties of Randomization and highlight improvements possible with spatial selection Present further results in spatial selection

4. Sensor Selection Problem Densely deployed wireless sensor networks consume energy through communications Multiple communications may lead to insufficient bandwidth Computational burden of processing all available data may be prohibitive. Not all measured data necessarily required for detection Subset of data may provide acceptable detection

5. Notation For N network nodes, each indexed by i, data available in a single time slot m is denoted by The decision to select a measurement depends on the outcome of an indicator variable Select K out of N sensors.

6. Notation Each measurement in the current time slot is selected with probability i.e. has the probability mass function This rule reduces the expected complexity of the detector by a factor of. is chosen to be a small value.

7. Notation The randomly selected vector can be represented by the equation G[m] is a diagonal matrix where the i-th entry on the leading diagonal determines whether or not a data element in x[m] is selected Since the detector receives only a portion of the data, the algorithms are based on the conditional density for given G[m] = G denoted by

8. Application to Distributed Signal Processing Random selection with a small value of can lead to acceptable detector performance Energy dissipated by communication may also be limited Avoids computational and communicational overhead that may be incurred from more complicated iteration procedures or centralized coordination Compatible with ad-hoc networking, clustered or unclustered networks

9. Randomized Selection in Detection For a binary hypothesis test for signals in additive gaussian noise, The canonical detector for a binary hypothesis test is the likelihood ratio test given by is a fixed threshold. If L(x) is greater than, then otherwise

10. Randomized Selection in Detection A signal with probability density function such that can be defined as an even signal e.g. a sinusoid with unknown uniformly distributed phase, a zero-mean gaussian random vector Since the detector has access to the random variables in G[m] and processes the data in the likelihood ratio under randomized data selection is given as The simplification is because the random variables are independent of the hypotheses H i

11. Randomized Selection in Detection Likelihood ratio compared to a fixed threshold is optimal under the Neyman-Pearson criterion Two problems: Determining the threshold can be computationally complex – requires inversion of: Requires 2 N terms for N samples of data and approximations can be troublesome G fluctuates although threshold is constant Solution is to fix false alarm rate for each G (Constant False Alarm Rate detection).

12. Parameters used in Detection Likelihood Ratio Test Useful for discriminating between different hypothesis Receiver Operating Characteristics Depicts the performance of an algorithm with respect to the assumed models of signals sensed.

13. Signal Detection and Operating Characteristics Given two distributions, with different mean and variance such that they overlap A classifier(detector) threshold can be defined as x*, where x represents possible values of a sensed random variable. a hit – sensed signal is above x* a false alarm – sensed signal above x* but it is noise a miss – sensed signal is below x* and it is target signature a correct rejection – signal below x* and target not present.

14. Signal Detection and Operating Characteristics is the discriminability A hit False Alarm A miss Correct rejection

15. Consider data set v i generated by sampling a sinusoid; under a binary hypothesis test n i is a Gaussian random variable with zero mean and variance. Given that, probability density is where u() denotes the unit step function, K is number of selected data Example – Detecting a Sinusoidal Signal

16. For a selected subset K, the notation becomes Probability density under H 1 can be obtained by convolving PDF of the two terms in the summation Since x K [m] under Ho is Gaussian, the Likelihood ratio is

17. Example – Detecting a Sinusoidal Signal Thus the Likelihood ratio is of the form Plot of the log of likelihood shows that it is symmetric and increasing thus the LRT simplifies to a threshold test of the form

18. Example – Detecting a Sinusoidal Signal Plot of Log Likelihoods for different A: shows symmetry and monotonicity

19. Example – Detecting a Sinusoidal Signal - Performance The ROC can be determined by integrating the conditional densities over the decision region for ROC calculated from the LRT gives the maximum achievable P D for each false alarm rate

20. Example – Detecting a Sinusoidal Signal - ROC

21. Spatial Selection Variograms Use of Transect Estimate Variogram along straight line drawn through a region Such an estimate may exhibit directionality except if underlying process is isotropic

22. Spatial Selection Location of target May allow variable amount of data to be used in estimate 2D May lead different variogram estimates Different data arrangements possible

23. Spatial Selection Natural Estimator where And is the number of distinct pairs in Robust Estimator Robust to contamination by outliers [2]

24. Variogram Experiments Using a transect through the center of the area (assumes target is at center of sensor area) The figure on the left is a truncated version of figure on right.

25. Variogram Experiments Using 2D Data set – Different data arrangements possible The figure on the left is a truncated version of figure on right.

26. Discussions/Comparisons (Randomization) High dependence on base station/manager node All nodes are active all the time Data may arrive in out-of-order fashion at the base station Random selection may result in biased selection hence a one-sided view of phenomenon In multihop routing some of the selected data may be lost Potential for simple and practical implementation May consume surprising amount of energy (short network lifetime) Does not collect network state

27. References Charles K. Sestok, Maya R. Said, Alan V. Oppenheim, Randomized Data Selection in Detection with applications to distributed Signal Processing, IEEE Proceedings, Nov. 2003. Cressie Noel A. C., Statistics for Spatial Data, Wiley 1991