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objective function, one will have discovered the best-performing gene
subset. This procedure can be made more sophisticated by introducing
weighting factors to increase the importance of user-specified samples in
training sets, as well as using other forms of the distance formula between
one subset and another.
5.3.3.
Search space reduction for global search
With local optimization, a fast method for a large number of genes, the
program finds the nearest minimum and stops. For some so-called global
optimization procedures, the algorithm not only finds a local minimum,
but can also find some neighboring minima. The processes, however, is a
hit-and-miss situation because starting at a different place can result in
different solutions.
The global algorithm in SDL repeatedly narrows the region where the
global minimum is known to lie by using a special OA sampling that oper-
ates simultaneously in all orthogonal dimensions (one for each gene in the
gene subset) to find the optimum solution. As the process runs, one can
observe the range of genes for each gene variable in an
n
-dimensional
subset being reduced.
The SDL global optimization algorithm operates to discover the opti-
mum solution. An analogy illustrates the principles involved: assume
plotting the objective function against 2000 genes in the colon cancer data
with a goal of finding a gene or a gene subset corresponding to the maxi-
mum objective function value
F
(or 1/
F
for the minimum value, for con-
venience in the illustration). See Fig. 5.10 for a one-dimensional (1D)
analogy showing local and global optimization processes.
As discussed earlier, a single objective function number can be used
to describe the classification performance of a current gene subset. By
plotting a multi-dimensional graph with objective function as one of
the axes, one can visualize the process. One requires as many orthogo-
nal axes as the number of variables (genes) plus one for the objective
function. Thus, for a two-gene problem, a three-dimensional (3D) plot
is required. To see the process used in a simplified form, imagine a
two-dimensional (2D) array along the
x
- and
y
-axes, which corresponds
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