Trend functions in Kriging models

Essai $$ \Z $$

The possible trend functions in libKriging are as follow, by increasing level of complexity.

The constant trend involves $p = 1$ coefficient and $\m{f}(\m{x})^\top\bs{\beta} = \beta$.
The linear trend involves $p = d +1$ coefficients

\[ \m{f}(\m{x})^\top \bs{\beta} = \beta_0 + \sum_{i=1}^d \beta_i \, x_i. \]
The interactive trend involves $1 + d + d (d-1) /2$ coefficients

\[ \m{f}(\m{x})^\top \bs{\beta} = \beta_0 + \sum_{i=1}^d \sum_{j=1}^{i-1} \beta_{ji} \, x_j x_i. \]
The quadratic trend involves $p = 1 + d + d(d+1) /2$ coefficients

\[ \m{f}(\m{x})^\top \bs{\beta} = \beta_0 + \sum_{i=1}^d \sum_{j=1}^i \beta_{ji} \, x_j x_i. \]

Starting from the constant trend, the other forms come by adding the $d$ linear terms $x_i$, adding the $d \times (d-1) / 2$ interaction terms $x_i x_j$ with $j <i$, and finally adding the squared input terms $x_i^2$.

For instance with $d=3$ inputs the four possible trends are in order of complexity

\[ \begin{aligned} \textsf{constant} \qquad & \m{f}(\m{x})^\top = [1] \\ \textsf{linear} \qquad & \m{f}(\m{x})^\top = [1, \: x_1, \: x_2,\: \:x_3] \\ \textsf{interaction} \qquad & \m{f}(\m{x})^\top = [1, \: x_1, \: x_2,\: x_1x_2, \:x_3,\: x_1x_3,\: x_2x_3] \\ \textsf{quadratic} \qquad & \m{f}(\m{x})^\top = [1, \: x_1, \: x_1^2, \: x_2,\: x_1x_2, \: x_2^2, \: x_3,\: x_1x_3,\: x_2x_3, \:x_3^2] \\ \end{aligned} \]

Mind that the coefficients relate to a specific order of the inputs.

Note The number of coefficients required in the interactive and quadratic trend increases quadratically with the dimension. For $d = 10$ the quadratic trend involves $66$ coefficients.