Set of analytic functions that take functional variables as inputs. Since they run quickly, they can be used for testing of funGp functionalities as if they were black box computer models. They cover different situations (number of scalar inputs and complexity of the inputs-output mathematical relationship).
fgp_BB1(sIn, fIn, n.tr)fgp_BB2(sIn, fIn, n.tr)
fgp_BB3(sIn, fIn, n.tr)
fgp_BB4(sIn, fIn, n.tr)
fgp_BB5(sIn, fIn, n.tr)
fgp_BB6(sIn, fIn, n.tr)
fgp_BB7(sIn, fIn, n.tr)
An object of class "matrix" with the values of the output at the specified input coordinates.
Object with class "matrix". The scalar input
points. Variables are arranged by columns and coordinates by rows.
Object with class "list". The functional inputs.
Each element of the list must be a matrix containing the set of curves
corresponding to one functional input.
Object with class "numeric". The number of
input points provided and correspondingly, the number of observations
to produce.
For all the functions, the \(d_s\) scalar inputs \(x_i\) are in the real interval \([0,\,1]\) and the \(d_f\) functional inputs \(f_i(t_i)\) are defined on the interval \([0,\,1]\). Expressions for the values are as follows.
fgp_BB1 With \(d_s = 2\) \(d_f = 2\)
x1 * sin(x2) + x1 * mean(f1) - x2^2 * diff(range(f2))fgp_BB2 With \(d_s = 2\) and \(d_f = 2\)
x1 * sin(x2) + mean(exp(x1 * t1) * f1) - x2^2 * mean(f2^2 * t2)fgp_BB3 With \(d_s = 2\) and \(d_f = 2\)
is the first analytical example in Muehlenstaedt et al (2017)
x1 + 2 * x2 + 4 * mean(t1 * f1) + mean(f2)fgp_BB4 With \(d_s = 2\) and \(d_f = 2\) is the
second analytical example in preprint of Muehlenstaedt et al (2017)
(x2 - (5 / (4 * pi^2)) * x1^2 + (5 / pi) * x1 - 6)^2 +
10 * (1 - (1 / (8 * pi))) * cos(x1) + 10 +
(4 / 3) * pi * (42 * mean(f1 * (1 - t1)) +
pi * ((x1 + 5) / 5) + 15) * mean(t2 * f2))fgp_BB5 With \(d_s=2\) and \(d_f=2\) is
inspired by the second analytical example in final version of Muehlenstaedt et al (2017)
(x2 - (5 / (4 * pi^2)) * x1^2 + (5 / pi) * x1 - 6)^2 +
10 * (1 - (1 / (8 * pi))) * cos(x1) + 10 +
(4 / 3) * pi * (42 * mean(15 * f1 * (1 - t1) - 5) +
pi * ((x1 + 5) / 5) + 15) * mean(15 * t2 * f2))fgp_BB6 With \(d_s = 2\) and \(d_f = 2\)
is inspired by the analytical example in Nanty et al (2016)
2 * x1^2 + 2 * mean(f1 + t1) + 2 * mean(f2 + t2) + max(f2) + x2fgp_BB7 With \(d_s = 5\) and \(d_f = 2\) is
inspired by the second analytical example in final version of Muehlenstaedt et al (2017)
(x2 + 4 * x3 - (5 / (4 * pi^2)) * x1^2 + (5 / pi) * x1 - 6)^2 +
10 * (1 - (1 / (8 * pi))) * cos(x1) * x2^2 * x5^3 + 10 +
(4 / 3) * pi * (42 * sin(x4) * mean(15 * f1 * (1 - t1) - 5) +
pi * (((x1 * x5 + 5) / 5) + 15) * mean(15 * t2 * f2))Muehlenstaedt, T., Fruth, J., and Roustant, O. (2017), "Computer experiments with functional inputs and scalar outputs by a norm-based approach". Statistics and Computing, 27, 1083-1097. [SC]
Nanty, S., Helbert, C., Marrel, A., Pérot, N., and Prieur, C. (2016), "Sampling, metamodeling, and sensitivity analysis of numerical simulators with functional stochastic inputs". SIAM/ASA Journal on Uncertainty Quantification, 4(1), 636-659. tools:::Rd_expr_doi("10.1137/15M1033319")