Page 38 - Academic Press Encyclopedia of Physical Science and Technology 3rd Analytical Chemistry

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Encyclopedia of Physical Science and Technology En001f25 May 7, 2001 13:58

Analytical Chemistry 577

for the species of interest. The pattern p is assigned and method originally described must employ steps of fixed
values of d are determined so that a decision vector V can size, which can result in excessive experimentation when
be calculated. When the calculated vector V is mathemat- step size is small or in poor precision for large steps. A
ically combined with a new set of experimental data, a more efficient solution employs variable step size through-
pattern p is calculated for the experimental species and out the entire procedure, allowing expansion (accelera-
can be fitted to previous classifications. tion) of the simplex in favorable directions and contrac-
tion in zones that produce poor results. The distance to be
moved is controlled by constant arbitrarily chosen multi-
2. Optimization by the Simplex Method
plication factors, which are multiplied with the distance of
Numerous mathematical techniques exist for solving a se- movement obtained on reflection. Eventually, the simplex
ries of simultaneous equations given defined boundary contractsasmovementtotheoptimumoccurs.Theprocess
constraints in order to maximize or minimize a particu- halts when the distance of movement has dropped below
lar parameter. The general acceptance and implementa- some predetermined value, which is generally governed
tion of techniques such as linear programming attest to by experimental uncertainty or time limitations. Certain
the power of optimization strategies. The simplex method difficulties exist in the application of simplex methods
is an “evolutionary operations” method that has been used when considering error sources:
systematically in many problems. A simplex is a geomet-
ric figure whose vertices are defined by the number of ex- 1. The method cannot be used if discontinuous variables
perimental parameters plus 1. Each point of the simplex are chosen.
represents the actual measured analytical response at a set 2. Movement to a local optimum may occur if numerous
of chosen experimental parameters. Represented in some optima exist.
n-dimensional space, one vertex of the simplex always 3. Parameters must be judiciously selected to ensure that
represents the case of worst response in the experimental. nontrivial analyses occur.
A mirror reflection through a symmetry plane away from 4. As many significant parameters as possible should be
the point of worst response (assuming the response will be included in the simplex so that no important factors
greater at a point opposite to the worst case) generates an- are overlooked. This subsequently increases the
other simplex. An experiment is then performed using the experimental work for each step in the simplex
new parameters to determine which vertex represents the generation.
new worst case response. A reiteration process following
four well-defined rules allows movement along the “re-
sponse surface,” resulting in eventual convergence to the
3. Selectivity vs Specificity
optimal experimental conditions.
The basic rules are as follows: Adefinition of terminology has been attempted, where the
upper limit of the concept of selectivity implies specificity.
1. Rejection of the point with the worst result is followed A fully selective system can measure one component in the
by replacement with its mirror image across a line or presence of many others, while fully specific implies that
plane generated by the other remaining points. in all situations only one component is measured and other
2. If the new point has the worst response, the previous components in the experiment do not produce any signal.
simplex is regenerated and the process applied in rule A nonselective system produces an analytical signal due to
1 is repeated for the second worst case point. all components in the experiment. For any of these cases,
3. If one point is common to three successive simplexes, the measured signal x is a function of concentration c of
it represents the optimum, provided that the point the available component and is related to the latter by a
represented the best response in each case. If this is normalization parameter γ , where
false, the entire process must be repeated using new
x = γ c.
initial starting points.
4. Boundary conditions are defined so that if a point falls The element γ is determined by the sensitivity parameter
outside accepted bounds, it is assigned an artificially dx/dc, and the sensitivity of the method is numerically
low value, which forces the simplex to move into the determined by the value of γ . This example can readily
useful calculation area. be expanded to consider a multicomponent case, where
x, γ , and c become matrix representations and a partial
A variation of the latter optimization procedure known sensitivity ∂x/∂c is employed. A mathematical rearrange-
as “modified simplex optimization” has evolved to elimi- ment of the γ ij matrix to place the largest γ value in each
nate the limitations imposed by the simplex method. The row on the main diagonal results in a useful “calibration