Page 83 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

72 PARAMETER ESTIMATION

T
two-dimensional: x ¼ [ Dy 0 ]. Since all other parameters are
assumed to be known, a radiometric model can be worked out to a
measurement function h(x) which quantitatively predicts the cross-
section, and thus also the measurement vector z for any value of the
parameter vector x.
With this measurement function it is straightforward to calculate the
2
LS norm e kk for a couple of values of x. Figure 3.8(c) is a graphical
2
2
representation of that. It appears that the minimum of e kk is obtained
2
^
if D LS ¼ 0:42 mm. The true diameter is D ¼ 0:40 mm. The thus
D
obtained fitted cross-section is also shown in Figure 3.8(b).
Note that the LS norm in Figure 3.8(c) is a smooth function of x.
Hence, the convergence region of a numerical optimizer will be large.
3.3.2 Fitting using a robust error norm

Suppose that the measurement vector in an LS estimator has a few
number of elements with large measurement errors, the so-called out-
liers. The influence of an outlier is much larger than the one of the others
because the LS estimator weights the errors quadraticly. Consequently,
the robustness of LS estimation is poor.
Much can be improved if the influence is bounded in one way or
another. This is exactly the general idea of applying a robust error norm.
Instead of using the sum of squared differences, the objective function of
(3.48) becomes:
N 1 N 1
X X
x
e kk ¼ ð" n Þ¼ ðz n h n ð^ xÞÞ ð3:55Þ
robust
n¼0 n¼0
(:) measures the size of each individual residual z n h n (^ x). This meas-
x
ure should be selected such that above a given level of " n its influence is
ruled out. In addition, one would like to have (:) being smooth so that
numerical optimization of ekk is not too difficult. A suitable choice
robust
(among others) is the so-called Geman–McClure error norm:
" 2
ð"Þ¼ ð3:56Þ
" þ 2
2
A graphical representation of this function and its derivative is shown in
Figure 3.9. The parameter is a soft threshold value. For values of e
smaller than about , the function follows the LS norm. For values larger
than , the function gets saturated. Consequently, for small values of e the