Geology Reference
In-Depth Information
1
k
w
j
þ
1
¼ w
j
g
j
ð
4
:
30
Þ
This is now a steepest descent.
Now the LM algorithm adjusts
s require-
ment, whether E need to be increased or decreased. The following four steps show
how the Levenberg algorithm works:
k
value in accordance with the user
'
1. Carry out an update as mentioned in the rule above (
4.29
or
4.30
).
2. Evaluate the error at the new weight vector.
3. If the error has increased as a result of the update, then retract the step (i.e., reset
the weights to their previous values) and increase
k
by a factor of 10 or some
cant factor. Then follow Step 1 to update again.
4. If the error has decreased as a result of the update, then accept the step (i.e., keep
the weights at their new values) and decrease
such signi
k
by a factor of 10 or so.
The user can always apply an intuition; i.e., if error is increasing, our quadratic
approximation is not working well and we are probably not near a minimum, so we
should increase
in order to blend more towards simple gradient descent. The
above-mentioned Marquardt method was later improved with a clever incorporation
of estimated local curvature information which resulted in a new concept called the
LM method. The insight of Marquardt was that, when
k
is high, we are doing
essentially gradient descent. Thus he replaced the identity matrix in Levenberg
k
'
s
original equations with the diagonal of the Hessian matrix.
½H
Þ
1
g
j
w
j
þ
1
¼ w
j
ð
H
þ k
diag
ð
4
:
31
Þ
The working of the algorithm is as mentioned above in the case of the Levenberg
algorithm.
4.5 Discrete Wavelet Transforms
The wavelet was introduced by Morlet and Grossman [
63
] as a time-scale analysis
tool for non-stationary signals for analyzing irregular data with discontinuities and
nonlinearity. The wavelet transform could be used as a tool to split up data or
functions into different frequency components, and then one could study each
component with a certain resolution. Flexibility is the key advantage of wavelets
over Fourier analysis. This helps to study the signal of interest at various resolu-
tions. Fourier transform techniques are widely used for linear decomposition of
static time series. The problem with the Fourier approach is that information in a
given time series is represented as a function of frequency with no speci
c time
information. On the other hand, a wavelet analysis works with translation and
dilation of a single local function
w
, generally referred to as a mother wavelet.
