Figure 5 Griewank (s variables), given by:\(f\left(X\right)=\ \sum_{r=1}^{s}{\frac{x_{r}^{2}}{4000}-\prod_{r=1}^{s}{\cos\left(\frac{x_{r}}{\sqrt{r}}\right)+1,\ \ -50\leq x_{r}\leq 70,}}\)and quadratic function.
The best objective value found, SSE value and WMS are reported in Table
3 for each of the test functions. Although estimated metamodel
parameters are omitted to emphasize analysis on window results, the
Sphere’s, Griewank’s, and Schwefel’s quadratic function,\(\mathbf{Z\ =\ f}\left(\mathbf{R}\right)\), followed the test
function shape, as is shown in Figure 5 for Griewank case. The WMS
generated for functions Griewank and Schwefel potentially detected a
zone of maximum similarity. The global solution for each global test
function is additionally included. The solution is in all cases
contained within the window of maximum similarity in, at least, one of
the independent variables. For three other cases: Sphere, Griewank and
Easom, the global solution is contained within the window for all the
independent variables. The simplest case, the Sphere, is the most
evident case where the quadratic function is a good descriptor of the
‘data at hand’, unlike the results of the remaining test functions which
displayed quadratic functions of varied shapes and consequently their
WMS were adjusted in varied zones.
Table 3 Results for optimization test functions in literature and
(movable) metamodel.