Given a pointp and a set S of sites si in the plane with associated values zi, the value at point p is computed as a weighted mean as follows. Let s1, . . . , sk be the Delaunay neighbors of p when we insert it in the Voronoi V D(S) diagram of S. The weight of neighbor sj for j ∈ {1, k} is the proportion of the area of the Voronoi region of p in V D(S∪p)that intersects the Voronoi region of sj inV D(S); see Figure 1.
1Partially supported by VA094A08, HP-2008-0060 and MTM2008-05043/MTM.
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178 Sensor recalibration with VD
Figure 1. Sibson’s interpolation assigns weights to the Delaunay neighbors according to the proportion of their area in the shaded Voronoi region.
In [3] we explored the effect of different synchronization schemes for this technique since circular references arise in the process. We simulated both synchronous and asyn-chronous processes. In the first, recalibrations happen after communication rounds, while in the latter, values are updated independently. We concluded that the synchronous pro-cess needs less rounds to reach stability and shows better correction levels. Another interesting result therein is that whether the simulated input shape was a circle or a square did not significantly affect the measured values. The results in the present work are thus performed with a synchronous scheme for simulated disc-shaped sensor networks.
2 Experimental results
The sensor network has been simulated following a uniform distribution on a disc. We have set the values of the sensors according to a normal distribution centered at the origin (see Figure 2). We have changed the values of some of the sensors by some proportion and marked which of the sensors were recalibrated afterwards. We call the proportion of corrected values the success ratio. On the other hand, the non-correct ratio is the proportion of the sensors that have been wrongly recalibrated.
Figure 2. Input sensor network and assigned data (left). 3D view of a degra-dated network with a 20% of faulty sensors that need recalibration (center).
3D view of a recalibrated network (right).
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We have performed 1500 simulations using a latin hypercube scheme, where the fol-lowing four parameters have been studied:
• Size: The number of sensors is a value between 100 and 1000.
• Resistance: A sensor accepts its neighbour’s value when it differs from its mea-sured value from 0.005 to 0.05.
• Errors: The proportion of sensors with wrong values varies between 10% and 50%.
• Strength: The measures of the sensors with errors are between 10% and 100%
of their original values.
Since wrongly calibrated sensors are chosen randomly, for each combination of these four parameters, 5 runs have been performed.
Our experiments show that the most significant parameter is strength, being the other three of little to none effect. Figure 3 shows the performance ratios and the translation for the measured values with respect to the strength. We can see that, for small perturbations (a strength value of less than 30%), the success ratio sometimes does not even reach 50%, but the miscalculated recalibrations are also very low. On the other hand, for values bigger than 30%, 99.62% of the wrongly working sensors are correctly detected and recalibrated.
With respect to the average translation of the measured values, we can say that, as expected, the higher the value of the strength parameter, the bigger the translation performed. Nevertheless, the translation of the successful recalibrations for those high values nearly doubles the one of the wrongly recalibrated sensors.
●
Figure 3. On the left, success ratio (dark) and non-correct calibrations ratio (light). On the right, average translation of correct calibrations (dark) and non-correct ones (light).
The number of communication rounds is less sensitive to the strength but tends to increase both with high strength values and bigger input sizes. Anyway, the average number of rounds is as low as 6.33, and 98.87% of the experiments have required 9 rounds or less. Figure 4 shows the histogram of the number of comunication rounds for all experiments performed.
180 Sensor recalibration with VD
percent
2 4 6 8 10 12 14
05101520
Figure 4. Histogram of the number of comunication rounds.
3 Conclusions
We have implemented a software that simulates a recalibration process based on Sibson’s interpolation. Among the four parameters used in order to launch the experiments using Latin hypercube sampling, the variation on the original values (strength) is the most important one. In fact, all measured values have a high dependence on this parameter and it affects from the success ratio to the number of comunication rounds. On the other hand, the threshold value that marks when a sensor has to be recalibrated (resistance) seemed to be of importance a priori, but the experimental results show that its effect is negligible. Similarly, the amount of wrong data (errors) was not important, at least with the reasonable assumption that it had to be less than 50%.
This is an ongoing work and we still seem to have more questions than answers. In particular, it would be very useful to address the same problem in a non-random setting, where sensors do not simply misfunction, but are attacked in a strategical way. Two interesting approaches appear: how to defend a sensor network from such an attack, and how to attack such a sensor network.
Acknowledgements
The author wants to acknowledge Raúl Santos for the implementation of the first version of the simulator, and Joaquín Aguilar, Diego R. Llanos and David Orden for the fruitful discussions.
References
[1] I. F. Akyildiz, Weilian Su, Y. Sankarasubramaniam, E. Cayirci, A survey on sensor networks,IEEE Communications Magazine40(8)(2002), 102–114.
[2] V. J. Hodge, J. Austin, A survey of outlier detection methodologies,Artificial Intelligence Review22 (2)(2004), 85–126.
[3] R. Santos, Estrategias de contagio de información en una red de sensores, Master’s Thesis, Universidad de Valladolid, 2010.
[4] R. Sibson, A vector identity for the Dirichlet tessellation,Mathematical Proceedings of the Cambridge Philosophical Society87(1980), 151–155.
XIV Spanish Meeting on Computational Geometry, 27–30 June 2011