Difference between revisions of "Aggregation Matrix"
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== Definition == | == Definition == | ||
| − | '''Aggregation 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 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. | 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 <math>K \times N</math> matrix, where each value <math>s_{mn}</math> is either zero or one. | + | Vectors and Matrices can be aggregated by multiplying with the aggregation matrix. Mathematically a aggregation matrix S is a <math>K \times N</math> matrix, where each value <math>s_{mn}</math> is either zero or one. The aggregation matrix has in total N non-zero values. |
:<math>S=\left(\begin{matrix} | :<math>S=\left(\begin{matrix} | ||
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</math> | </math> | ||
| − | 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. | + | 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 === | === Vector Quantity Aggregation === | ||
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=== Matrix Quantity Aggregation === | === Matrix Quantity Aggregation === | ||
| − | A matrix of dimension N is aggregated through its pre-multiplication with the aggregation matrix and the post-multiplication with the aggregation matrix transpose. | + | 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. |
:<math>\mathbf{A}_{s} = \mathbf{S} \mathbf{A} \mathbf{S}^T</math> | :<math>\mathbf{A}_{s} = \mathbf{S} \mathbf{A} \mathbf{S}^T</math> | ||
| + | |||
| + | or more explicitly: | ||
| + | |||
| + | :<math> | ||
| + | a^{s}_{ij} = \sum_{k=1}^{N} \sum_{l=1}^{N} s_{ik} a_{lk} s_{il} | ||
| + | </math> | ||
| + | |||
== See Also == | == See Also == | ||
Latest revision as of 18:50, 20 November 2023
Contents
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 
matrix, where each value 
is either zero or one. The aggregation matrix has in total N non-zero values.
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.
or more explicitly:
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.
or more explicitly:
