High-dimensional model representation

High-dimensional model representation is a finite expansion for a given multivariable function. The expansion was first described by Ilya M. Sobol[1] as

${\displaystyle f(\mathbf {x} )=f_{0}+\sum _{i=1}^{n}f_{i}(x_{i})+\sum _{i,j=1 \atop i<j}^{n}f_{ij}(x_{i},x_{j})+\cdots +f_{12\ldots n}(x_{1},\ldots ,x_{n}).}$

The method, used to determine the right hand side functions, is given in Sobol's paper. A review can be found here: High Dimensional Model Representation (HDMR): Concepts and Applications.

The underlying logic behind the HDMR is to express all variable interactions in a system in a hierarchical order. For instance ${\displaystyle f_{0}}$ represents the mean response of the model ${\displaystyle f}$. It can be considered as measuring what is left from the model after stripping down all variable effects. The uni-variate functions ${\displaystyle f_{i}(x_{i})}$, however represents the "individual" contributions of the variables. For instance, ${\displaystyle f_{1}(x_{1})}$ is the portion of the model that can be controlled only by the variable ${\displaystyle x_{1}}$. For this reason, there can not be any constant in ${\displaystyle f_{1}(x_{1})}$ because all constants are expressed in ${\displaystyle f_{0}}$. Going further into higher interactions,the next stop is bivariate functions ${\displaystyle f_{ij}(x_{i},x_{j})}$ which represents the cooperative effect of variables ${\displaystyle x_{i}}$ and ${\displaystyle x_{j}}$ together. Similar logic applies here: the bivariate functions do not contain univarite functions nor constants as it violates the construction logic of HDMR. As we go into higher interactions, the number of interactions are increasing and at last we reach the residual term ${\displaystyle f_{12n}(x_{1},\ldots ,x_{n})}$ representing the contribution only if all variable act together.