Aggregation Matrix

Definition

Aggregation Matrix (also Summation Matrix) in the context of Input-Output Analysis is a Boolean Matrix that aims to produce a coarse-grained version of a more granular Input-Output Model.

Aggregation can be for example along sectoral or regional dimensions.

Vectors and Matrices can be aggregated by multiplying with the aggregation matrix. Mathematically a aggregation matrix S is a $K \times N$ matrix, where each value $s_{mn}$ is either zero or one. The aggregation matrix has in total N non-zero values.

$S=\left(\begin{matrix} s_{00} & s_{01} & \dots &s_{0n} & \dots & s_{0N} \\ s_{10} & s_{11} & \dots &s_{1n} & \dots & s_{1N} \\ \vdots & \vdots & \ddots &\vdots & \ddots & \vdots \\ s_{m0} & s_{m1} & \dots &s_{mn} & \dots & s_{mN} \\ \vdots & \vdots & \ddots & \vdots& \ddots & \vdots \\ s_{K0} & s_{K1} & \dots & s_{Kn} & \dots & s_{KN}\\ \end{matrix}\right).$

An aggregation matrix with K rows and N columns is used to aggregate a dimension of size N into a smaller dimension of size K. The N non-zero (unit) values select and group the elements of the vector or matrix that is to be aggregated.

Vector Quantity Aggregation

A vector Y of dimension N is aggregated into a vector K through its pre-multiplication with the aggregation matrix.

$\mathbf{Y}^{s} = \mathbf{S} \mathbf{Y}$

or more explicitly:

$Y^{s}_{i} = \sum_{j=1}^{N} s_{ij} Y_j$

Matrix Quantity Aggregation

A matrix of dimension N x N is aggregated into a K x K matrix through its pre-multiplication with the aggregation K x N matrix and the post-multiplication with the (N x K) aggregation matrix transpose.

$\mathbf{A}_{s} = \mathbf{S} \mathbf{A} \mathbf{S}^T$

or more explicitly:

$a^{s}_{ij} = \sum_{k=1}^{N} \sum_{l=1}^{N} s_{ik} a_{lk} s_{il}$

See Also

• Aggregation Bias
• Summation Vector

Further Resources

• Crash Course on Input-Output Model Mathematics
• Introduction to Input-Output Models using Python