The metafunction randomly combines the following functions in a metafunction of dimension \(k\):
\(f(x) = x ^ 3\) (cubic).
\(f(x) = 1~\mbox{if}(x > 0.5), 0~\mbox{otherwise}\) (discontinuous).
\(f(x) = \frac{e ^ x}{e - 1}\) (exponential).
\(f(x) = \frac{10 - 1}{1.1} ^ {-1} (x + 0.1) ^ {-1}\) (inverse).
\(f(x) = x\) (linear)
\(f(x) = 0\) (no effect).
\(f(x) = 4(x - 0.5) ^ 2\) (non-monotonic).
\(f(x) = \frac{\sin (2 \pi x)}{2}\) (periodic).
\(f(x) = x ^ 2\) (quadratic).
\(f(x) = \cos(x)\) (trigonometric).
It is constructed as follows:
$$y=\sum_{i=1}^{k}\alpha_i f^{u_i}(x_i) \\
+ \sum_{i=1}^{k_2}\beta_i f^{u_{V_{i,1}}}(x_{V_{i,1}}) f^{u_{V_{i,2}}} (x_{V_{i,2}}) \\
+ \sum_{i=1}^{k_3}\gamma_i f^{u_{W_{i,1}}}(x_{W_{i,1}}) f^{u_{W_{i,2}}}(x_{W_{i,2}}) f^{u_{W_{i,3}}} (x_{W_{i,3}})$$
where \(k\) is the model dimensionality, \(u\) is a \(k\)-length vector formed by randomly
sampling with replacement the ten functions mentioned above, \(V\) and \(W\) are two matrices specifying the
number of pairwise and three-wise interactions given the model dimensionality,
and \(\mathbf{\alpha}, \mathbf{\beta}, \mathbf{\gamma}\) are three
vectors of length \(k\) generated by sampling from a mixture of two normal distributions
\(\Psi=0.3\mathcal{N}(0, 5) + 0.7\mathcal{N}(0, 0.5)\).
See Puyj;textualsensobol and Becker2020;textualsensobol for a full
mathematical description of the metafunction approach.