If the Gaussian centers and variance are known, then the unknown
SOG coe?±cients aj can be estimated using equations 4-6 above. If,
however, the centers and variances of the Gaussians are unknown, then the
field to be estimated becomes nonlinear-in-parameters. For m = 2, the measurement
equations are: y = a0+a1e??’(x??’x1)2+(y??’y1)2
2??2 +a2e??’(x??’x2)2+(y??’y2)2
2??2 , and
we are estimating a0, a1, a2, x1, y1, x2, y2, ??. Instead of the linear least-square
solution (6), we can use a numerical nonlinear least- square solution provided
by MATLAB??™s fmincon, however, we cannot obtain a closed form measure of
uncertainty for the whole parameter vector.
44
Chapter 2 Algorithms for Robotic Deployment of WSN
4.2 Example: Monte-Carlo simulations with a SOG field
We have run simulations using a 2D nonlinear field (though still linear in the
parameters), consisting of a sum of two Gaussian distributions:
G(x, y) = a0 + a1g1(x, y) + a2g2(x, y), g(x, y) = e??’
???(x??’x0)2+(y??’y0)2
2??2 .
First, we assume that the centers of Gaussians and the covariance for g1(x, y),
g2(x, y) are known, and they are located at (30, 30), (65, 45), ?? = 10. The sampling
sequences generated are shown in Figure 4 (a)??’(d). The second sampling
sequence is obtained for a SOG field assuming that the location of the Gaussian
centers are not known. In this case we use a numerical constrained optimization
solver from MATLAB (fmincon) to obtain the sampling sequence.
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