Trend functions in Kriging models

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

  • The constant trend involves \(p = 1\) coefficient and \(\mathbf{f}(\mathbf{x})^\top\boldsymbol{\beta} = \beta\).

  • The linear trend involves \(p = d +1\) coefficients

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

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

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

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

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

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.