Biology Reference
In-Depth Information
X
SEM i @
1
G
ð
guesses
;
X i Þ
Y i
G
ð
guesses
;
X i Þ
ð
answer j
guess j Þ
:
@
guess j
SEM i
j
(8-40)
Equation (8-40) expresses the desired optimal answers in terms of the
data points (Y i , SEM i ,X i ), the fitting equation (G), and the initial
estimates of the answers (guesses). The SEM i is now included to allow
each data point to have a different level of experimental uncertainty and
thus a different statistical weight.
It is important to note that when we neglect the higher order derivative
terms, represented by
in Eq. (8-38), Eq. (8-40) is only approximately
correct, and thus the iteration of Eq. (8-40) must be performed many
times so that it can converge to the optimal parameter values. Linear
least-squares is a special case where all of the higher-order derivatives
are exactly zero, so the solution to Eq. (8-40) is exact, and only a single
cycle of the algorithm is required. In this context, the term linear refers to
the form of Eq. (8-38), not the form of the fitting equation Y
...
¼
G (parameters; X). For example, fitting data to a straight line (i.e.,
Y
bX) is a linear fit, as is fitting to all of the higher degree
polynomials. In these cases, the problem of finding the minimum WSSR
leads to the problem of solving a system of equations that is linear
with respect to all of the unknowns. In the same way, fitting a single
Fourier wave (i.e., Y
¼
a
þ
c) is also a linear
fit if d is a constant while only a, b, and c are being estimated. If,
however, d is also being estimated, then the second- and higher-
order derivatives are not all zero, and the fitting process becomes
nonlinear.
¼
a sin(2
p
X/d)
þ
b cos(2
p
X/d)
þ
E XERCISE 8-6
(a) Show that the model Y
¼
G
ð
a
;
b
;
c
;
d
;
e
;
f
X
Þ¼
aX 5
þ
bX 4
þ
cX 3
þ
;
dX 2
f is a linear model as far as least-squares fit for the
parameters a,b,c,d,e,f is concerned.
þ
eX
þ
(b) Generalize for a polynomial model of arbitrary (but fixed) degree
with coefficients that are being estimated from the data.
E XERCISE 8-7
Verify that if d is fixed (and thus not a parameter to be estimated
from the data), the model Y
¼
a sin
ð
2
p
X
=
d
Þþ
b cos
ð
2
p
X
=
d
Þþ
c is a linear
model.
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