| unidTab {mixtox} | R Documentation |
Uniform Design Table
Description
The good lattice point method with a power generator was used to construct the uniform experimental tables . The centered L2-discrepancy (CD2) and the symmetric discrepancy algorithms (sd2) were employed to measure the uniformity and find the one with lowest discrepancy.
Usage
unidTab(lev, fac, algo = "cd2", sav = FALSE)Arguments
lev |
the number of runs (levels or pseudo-levels) |
fac |
the number of factors |
algo |
algorithms used to calculate the discrepancy. "cd2": the centered L2-discrepancy algorithm. "sd2": the symmetric discrepancy algorithm. |
sav |
TRUE: save output to a default file; FALSE: output will not be saved; a custom file directory: save output to the custom file directory. |
Details
Uniform design (UD) was proposed by Fang et al (Acta Math Appl Sin 3:363-372 (1980)).
An appropriate uniform design table is constructed
according to the factor (the number mixture
components) and level (the number of experiments need to run).
Many methods can be used to
construct the uniform table. In the past decades many methods have been proposed for
constructing (nearly) uniform designs, such as the good lattice point (glp) method, the
glp method with a power generator (pglp method) (Fang 1980; Fang andWang 1994),
the cutting method (Maand Fang 2004), the optimization method (Winker and Fang 1998).
However, when s is large, the glp method has a large computational cost. And the pglp
method has the lowest computation complexity among various methods in quasi-Monte Carlo
methods and a good performance when n or n + 1 is a prime number
and s is small (Fang 1980;
Fang and Wang 1994), while the pglp method may have a poor performance when s is large.
Here, we choose the glp method with a power generator to construct the uniform table. The
centered L2-discrepancy (cd2) is set as default over the symmetric discrepancy algorithm
for its accuracy. The cd2 algorithm is defined as follows:
C{D_2}(P) = {\left[ {{{\left( {\frac{{13}}{{12}}} \right)}^s} - \frac{{{2^{1 - s}}}}
{n}\sum\limits_{k = 1}^n {\prod\limits_{i = 1}^s {{\theta _{ki}} + \frac{1}{{{n^2}}}
\sum\limits_{k,l = 1}^n {\prod\limits_{i = 1}^s {{\phi _{k,li}}} } } } }
\right]^{\frac{1}{2}}}
With the definition of \theta _{ki} and \phi _{k,li} as follows:
{\theta _{ki}} = 2 + \left| {{x_{ki}} - \frac{1}{2}} \right| -
{\left| {{x_{ki}} - \frac{1}{2}} \right|^2}
{\phi _{k,li}} = 1 + \frac{1}{2}\left( {\left| {{x_{ki}} - \frac{1}{2}}
\right| + \left| {{x_{li}} - \frac{1}{2}} \right| - \left| {{x_{ki}} -
{x_{li}}} \right|} \right)
where n, s are the number of runs (levels or multiple of levels) and
the number of input variables (factors), respectively.
Value
T |
the selected uniform table(s) . It may contain one or more uniform tables with the same discrepancy |
D |
the discrepancy of the constructed uniform table |
sav |
TRUE: save output to a default file; FALSE: output will not be saved; a custom file directory: save output to the custom file directory. |
References
Wang, Y., Fang, K.-T., 1996. Uniform design of experiments with mixtures. Sci. China Ser.
A-Mathematics Phys. Astron. 39, 264-275.
Hickernell, F.J., 1996. A generalized discrepancy and quadrature error bound. Math. Comput.
67, 299-322.
Examples
## example 1
# construct uniform table with 11 runs and 7 factors using the default centered L2-discrepancy
# algorithm
unidTab(11, 7)
## example 2
# construct uniform table with 37 runs and 13 factors using the symmetric discrepancy algorithm
unidTab(lev = 37, fac = 13, algo = "sd2")
## example 3
# construct uniform table with 37 runs and 13 factors using default centered L2-discrepancy
# algorithm
unidTab(lev = 37, fac = 13, algo = "cd2")