(28)
Here ??2 is a noise variance. |!(p)| denotes the total number of squares in
the partition p, and f(n) is some monotonically increasing function of n. Since
both the sum-of-squared errors and the penalty formula are additive functions,
it can be solved in O(n) time using a bottom-up tree pruning algorithm [3].
A remaining problem here is how to choose an appropriate function f(n).
Obviously, it depends on the definition of best pruned tree. The authors in [9]
set f(n) = 2/3 log n to reach a nearly optimal performance. They apply the
above scheme on a sensor network of size 4k for k = 2, 3, . . . , 8 in the simulation
operated in a environment with three di?®erent noise levels (??2 = 1, 10, 100).
The results agree very well with their theoretical predictions.
6 Conclusion and future work
The estimation of boundary between regions of distinct behavior in a large
physical space is highly desired in many domains (e.g., chemical monitoring).
172
Chapter 6 Boundary Detection for Sensor Networks
In this chapter, we presented recent techniques for detecting a boundary in
sensor networks. Techniques for boundary detection are classified into four
di?®erent classes: (1) centralized estimation; (2) localized estimation; (3) distributed
estimation; (4) hierarchical estimation.
All of these recent e?®orts only concerned static sensor fields. The main
drawback of a static sensor network is that estimation accuracy relies on the
density of sensor deployment.
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