Add the Two-Dimensional Test Function from Bastos and O'Hagan (2009)

[damar-wicaksono]

The two-dimensional function from Bastos and O'Hagan (2009)¹ is a test function used in the context of metamodeling (specifically, Gaussian process metamodels). The function reads as follows:

$$ \mathcal{M}(\boldsymbol{x}) = \left( 1 - \exp{\left( - \frac{1}{2 x_2} \right)} \right) \left( \frac{2300 x_1^3 + 1900 x_1 ^ 2 + 2092 x_1 + 60}{100 x_1^3 + 500 x_1^2 + 4 x_1 + 20} \right) $$

where $\boldsymbol{x} = \{ x_1, x_2 \}$ is the vector of input variables modeled as an independent uniform random variable in $[0, 1]^2$. The function is highly curved with respect to some input variables and less so with respect to some others.

Footnotes

1. Leonardo Bastos and Anthony O'Hagan, "Diagnostics for Gaussian process emulators," Technometrics, vol. 51, no. 4, pp. 425-438, 2009. doi: 10.1198/TECH.2009.08019. See Eq. (23), Section 4.1. ↩

[damar-wicaksono]

This function actually already appeared in Currin et al. (1991)¹. While the report by the same authors was cited in Bastos and O'Hagan (2009), the paper did not specifically cite the report when the function was introduced.

I would opt to name the function referring to the original paper of Currin et al.

Footnotes

1. C. Currin, T. Mitchell, M. Morris, and D. Ylvisaker, “Bayesian Prediction of Deterministic Functions, with Applications to the Design and Analysis of Computer Experiments,” Journal of the American Statistical Association, vol. 86, no. 416, pp. 953–963, 1991. ↩