Difference between revisions of "Aggregation Matrix"
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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 T is a <math>K \times N</math> matrix, where each value <math> | + | Vectors and Matrices can be aggregated by multiplying with the aggregation matrix. Mathematically a aggregation matrix T is a <math>K \times N</math> matrix, where each value <math>S_{mn}</math> is either zero or one. |
| − | :<math> | + | :<math>S=\left(\begin{matrix} |
T^{00} & T^{01} & \dots &T^{0n} & \dots & T^{0N} \\ | T^{00} & T^{01} & \dots &T^{0n} & \dots & T^{0N} \\ | ||
T^{10} & T^{11} & \dots &T^{1n} & \dots & T^{1N} \\ | T^{10} & T^{11} & \dots &T^{1n} & \dots & T^{1N} \\ | ||
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=== Vector Quantity Aggregation === | === Vector Quantity Aggregation === | ||
| − | A vector | + | A vector of dimension N is aggregated through its pre-multiplication with the aggregation matrix. |
:<math>\mathbf{y}_{s} = \mathbf{S} \mathbf{y}</math> | :<math>\mathbf{y}_{s} = \mathbf{S} \mathbf{y}</math> | ||
| + | |||
| + | === 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. | ||
:<math>\mathbf{A}_{s} = \mathbf{S} \mathbf{A} \mathbf{S}^T</math> | :<math>\mathbf{A}_{s} = \mathbf{S} \mathbf{A} \mathbf{S}^T</math> | ||
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== See Also == | == See Also == | ||
* [[Aggregation Bias]] | * [[Aggregation Bias]] | ||
| + | * [[Summation Vector]] | ||
== Further Resources == | == Further Resources == | ||
Revision as of 17:37, 20 November 2023
Contents
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 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 T is a 
matrix, where each value 
is either zero or one.
Vector Quantity Aggregation
A vector of dimension N is aggregated through its pre-multiplication with the aggregation matrix.
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.
