Step7: Calculate the parameter uncertainty measure
var( ?† An) = ??2
z ||(MT
n Mn)??’1||p.
If ??–n+k is empty (the whole space has been sampled), or if var( ?† An+k) ?‰¤ ", and
|| ?†B
n+k ??’ ?†B
n|| ?‰¤ " (the parameter uncertainty measure is below a threshold,
and the parameter estimates B do not change), go to Step8. Otherwise, set n
to be n + k and repeat Step3.
Step8: If the residual error between the measurement vector Z and the
parametric field g(A,B) is smaller than " exit. Otherwise, pick another starting
guess for the parameter set B from its coarse grid and repeat Step2.
4.3 Closed-form estimation for a linear field with localization
uncertainty
Previously, we used a numerical ???closed-form??? calculation of the parameter
variance, in the absence of state uncertainty. The parameter variance calculation
relies on calculating the pseudo-inverse of a matrix whose entries are
assumed to be known. These entries depend on the position at the point where
measurements are taken, and in the context of MWSNs, this location is not
known exactly. Let??™s consider a very simple one-dimensional sampling example,
in which we add localization uncertainty. We set m = 2 and g(x) = x in
order to obtain an algebraic closed-form solution of the parameter variance.
If the set of measurements is given by:
y1 = a1 + a2x1
y2 = a1 + a2x2
...
yn = a1 + a2xn
, ??®???????????°
y1
y2
...
yn
??????????????»
= ??®???????????°
1 x1
1 x2
.
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