Applications of the SDL Global Optimization Method in DNA Microarray Data Analysis - DNA Microarray Technology and Data Analysis in Cancer Research - page 78

function value) bounded by the user-defined maximum and minimum
values of x and y . If he just walks upwards until he can go no further
uphill, he will have found the local maximum. The “best guess”
approach is based on a starting design (position) that may be based on
many years of experience. There are several mathematical methods
available, such as the gradient method, that alter several gene variables
simultaneously, see what happens to the objective function, and move
the gene subset in the direction of a maximum of the objective function.
These find the local optimum, and the end subset solution is completely
dependent on the starting guess. A derivation of this is the GA method
which, by use of evolution in populations and random numbers, is able
to find better maxima, exploring other subsets by “jumping away” from
the nearest peak.
The GA optimization methods use a completely different technique
to optimize the gene subsets. While it has the advantage of being capa-
ble of producing complex gene subsets with minimum user interaction,
the solution found is unlikely to be the global optimum. The final
solution is still dependent on the initial starting gene selection, and
many more genes than are necessary are often required for a given
performance.
SDL optimization operates by a process of searching for all regions in
the gene space where a height greater than a specific level is located. This
is akin to creating a contour map by slicing parameter space at a constant
value of objective function. The levels (choices) of the genes were equally
spaced within every reduced search space.
In Fig. 5.11(a), one sees a representation of a two-gene-subset prob-
lem. Using a search analogy, to find the peak, the region in which the
highest peak must lie is narrowed each time the plane is raised. This
occurs until there is only one peak left. Its coordinates correspond to the
gene IDs of the optimum solution. This is the equivalent of a plane paral-
lel to the x - y plane at height z [see Fig. 5.11(b)]. This plane intersects the
topography and identifies the entire region within which the peak is
known to lie. By raising the slicing plane repeatedly, the region within
which the peak must lie is made smaller and smaller until only the high-
est peak remains [see Fig. 5.11(c)]. Its coordinates correspond to the gene
IDs of the optimum gene subset. In practice, the surface is a mathematical