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 in his paper "Sensitivity Estimates for Nonlinear Mathematical Models" as

f

(

x

)

=

f

0

+

∑

i

=

1

n

f

i

(

x

i

)

+

∑

i

,

j

=

1

i

<

j

n

f

i

j

(

x

i

,

x

j

)

+

⋯

+

f

12

…

n

(

x

1

,

…

,

x

n

)

.

{\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

f

0

{\displaystyle f_{0}}

represents the mean response of the model

f

{\displaystyle f}

. It can be considered as measuring what is left from the model after stripping down all variable effects. The uni-variate functions

f

i

(

x

i

)

{\displaystyle f_{i}(x_{i})}

, however represents the "individual" contributions of the variables. For instance,

f

1

(

x

1

)

{\displaystyle f_{1}(x_{1})}

is the portion of the model that can be controlled only by the variable

x

1

{\displaystyle x_{1}}

. For this reason,

f

1

(

x

1

)

{\displaystyle f_{1}(x_{1})}

cannot contain constant terms, because constant terms are expressed in

f

0

{\displaystyle f_{0}}

.