Table 19.1 Test functions used in the studies of this section. The shift parameters are set to u i = 0. 3 for all dimensions i = 1, …, d, while the weight parameters are selected as w i = C∕i with normalization constant C = 1∕∑ i = 1 d i −1 to ensure ∑ i = 1 d w i = 1. The variance formula for the product-peak function is \(v(\boldsymbol{u},\boldsymbol{w}) =\prod _{ i=1}^{d}w_{i}^{4}\left ( \frac{1-u_{i}} {2(1+w_{i}^{2}(1-u_{i})^{2})} + \frac{u_{i}} {2(1+w_{i}^{2}u_{i}^{2})} + \frac{1} {2w_{i}}\left (\text{arctan}\left (w_{i}(1 - u_{i})\right ) + \text{arctan}\left (w_{i}u_{i}\right )\right )\right )-m(\boldsymbol{u},\boldsymbol{w})^{2}\)
From: Surrogate Models for Uncertainty Propagation and Sensitivity Analysis
Id | Function | Formula | Exact mean | Exact variance |
|---|---|---|---|---|
| Â | Â | \(f_{\boldsymbol{u},\boldsymbol{w}}(\boldsymbol{\lambda })\) | \(m(\boldsymbol{u},\boldsymbol{w}) =\int _{ 0}^{1}f_{\boldsymbol{u},\boldsymbol{w}}(\boldsymbol{\lambda })d\boldsymbol{\lambda }\) | \(v(\boldsymbol{u},\boldsymbol{w}) =\int _{ 0}^{1}(f_{\boldsymbol{u},\boldsymbol{w}}(\boldsymbol{\lambda }) - m(\boldsymbol{u},\boldsymbol{w}))^{2}d\boldsymbol{\lambda }\) |
1 | Oscillatory | \(\cos \left (2\pi u_{1} + \sum _{i=1}^{d}w_{ i}\lambda _{i}\right )\) | \(\left (\cos 2\pi u_{1} + \frac{1} {2}\sum _{i=1}^{d}w_{ i}\right )\prod _{i=1}^{d}\frac{2\sin (w_{i}/2)} {w_{i}}\) | \(\frac{1} {2} + \frac{1} {2}m(2\boldsymbol{u},2\boldsymbol{w}) - m(\boldsymbol{u},\boldsymbol{w})^{2}\) |
2 | Gaussian | \(\exp \left (-\sum _{i=1}^{d}w_{ i}^{2}(\lambda _{ i} - u_{i})^{2}\right )\) | \(\prod _{ i=1}^{d} \frac{\sqrt{\pi }} {2w_{i}}\left (\text{erf}\left (w_{i}(1 - u_{i})\right ) + \text{erf}\left (w_{i}u_{i}\right )\right )\) | \(m(\boldsymbol{u},\sqrt{2}\boldsymbol{w}) - m(\boldsymbol{u},\boldsymbol{w})^{2}\) |
3 | Exponential | \(\exp \left (\sum _{i=1}^{d}w_{ i}(\lambda _{i} - u_{i})\right )\) | \(\prod _{i=1}^{d} \frac{1} {w_{i}}\left (\exp \left (w_{i}(1 - u_{i})\right ) -\exp \left (-w_{i}u_{i}\right )\right )\) | \(m(\boldsymbol{u},2\boldsymbol{w}) - m(\boldsymbol{u},\boldsymbol{w})^{2}\) |
4 | Continuous | \(\exp \left (-\sum _{i=1}^{d}w_{ i}\vert \lambda _{i} - u_{i}\vert \right )\) | \(\prod _{i=1}^{d} \frac{1} {w_{i}}\left (2 -\exp \left (-w_{i}u_{i}\right ) -\exp \left (w_{i}(u_{i} - 1)\right )\right )\) | \(m(\boldsymbol{u},2\boldsymbol{w}) - m(\boldsymbol{u},\boldsymbol{w})^{2}\) |
5 | Corner peak | \(\left (1 + \sum _{i=1}^{d}w_{ i}\lambda _{i}\right )^{-(d+1)}\) | \(\frac{1} {d!\prod _{i=1}^{d}w_{i}}\:\sum _{r\in \{0,1\}^{d}} \frac{(-1)^{\vert \vert r\vert \vert _{1}}} {1 +\sum _{ i=1}^{d}w_{i}r_{i}}\) | Sampling estimator (19.34) with M = 107 |
6 | Product peak | \(\prod _{i=1}^{d} \frac{w_{i}^{2}} {1 + w_{i}^{2}(\lambda _{i} - u_{i})^{2}}\) | \(\prod _{i=1}^{d}w_{ i}\left (\text{arctan}\left (w_{i}(1 - u_{i})\right ) + \text{arctan}\left (w_{i}u_{i}\right )\right )\) | See the caption |