Biology Reference
In-Depth Information
parameters exists corresponding to the maximum likelihood
optimization criteria. However, this cannot be demonstrated for all
nonlinear models, and the possibility exists that multiple sets of
parameters could be optimal for some fitting equations. Consider again
Figure 8-6 depicting the topographical map of the WSSR as a function of
two parameters for a nonlinear fitting equation. For linear fitting
equations, it can be demonstrated this plot contains only a single
minimum.
Figure 8-8 presents one method of detecting whether multiple minima
exist for a nonlinear least-squares procedure. This method is to simply
start the iterative nonlinear fitting procedure at several different
locations. In the example in Figure 8-8, we show initial guesses at
locations A, B, and C. When the iterative least-squares procedure is
started at either location A or B, the algorithm converges to the same
minimum, but if the procedure is started at location C, the algorithm
converges to a different minimum in the topographical map. Also note
that a minimum exists that was not found when starting at these
positions.
C
A
B
The potential for multiple minima always exists when fitting to
nonlinear equations but, unfortunately, no computational method
exists that will guarantee locating all of these minima. It is, however,
common for some of the multiple minima to have parameter values
that are physically unrealistic. For example, a negative molecular
weight has no physical meaning. If multiple, physically meaningful
minima are found, they must all be described in your report of the
results.
K
21
FIGURE 8-8.
This figure presents a topographical contour
map of the variance-of-fit as a function of two
parameters, K
21
and K
22
. In this example, three
minimal points exist. Note that this is a
nonlinear fitting equation and thus multiple
minima can exist. For linear models, only a
single minimum will exist.
VI. APPENDIX: BASIC MATRIX ARITHMETIC
In this chapter, we expressed a system of equations as a matrix equation
and used matrix algebra to solve the system of equations. This is a
convenient and common technique because hand-held calculators and
computers are equipped to do matrix computations. In this appendix,
we outline some basic matrix arithmetic that the reader needs to be
familiar with in order to follow the matrix computations presented in the
chapter.
A matrix is a rectangular array of numbers. An m
n matrix is one that
is a
1
10
245
has m rows and n columns. For example, the matrix
2
3 matrix. Matrices are equal when they have the same dimensions
and each corresponding entry is equal. The following arithmetic
operations are fundamental to matrix arithmetic: addition,
multiplication by a number, and multiplication of a matrix by a matrix
(matrix multiplication). For our purposes, matrix multiplication is the
most important operation.
