Aggregate FCMs

Generate an aggregate adj. matrix from a list of adj. matrices. FCM aggregation works by calculating the mean/median edge weight for all edges across the input adj. matrices (i.e. the mean/median of the edge weight connecting A -> B across all maps, the mean/median of the edge weight connecting B -> C across all maps, and so on). The user may dictate whether to incorporate 0-valued edge weights in the mean/median calculations.

All input adj. matrices must have the same dimensions and concept names to generate an aggregate. Call standardize_adj_matrices to make sure all adj. matrices have the same dimensions.

Usage

aggregate_fcms(
  adj_matrices = list(),
  agg_function = c("mean", "median"),
  include_zeroes_in_sampling = FALSE,
  skip_checks = FALSE
)

Arguments

adj_matrices: [list()]
A single adjacency matrix or a list of adjacency matrices (n x n) representing FCMs. Matrices can have conventional edge weights, IVFN edge weights or TFN edge weights.
agg_function: [character(1)]
Choice of aggregation method (mean, median) for producing a single "collective" FCM from a group of individual FCMs. Omit this argument when analyzing a single FCM.
include_zeroes_in_sampling: [logical(1)]
If TRUE, incorporate zero-weighted edges during FCM aggregation.
skip_checks: [logical(1)]
FOR DEVELOPER USE ONLY. If TRUE, skip call to check_simulation_inputs (internal function)

Value

[aggregate]
An an aggregate adj. matrix with edges represented as the same data types as the inputs (i.e. Numerics for, IVFNs, or TFNs)

Details

This function implements FCM aggregation methods used in Aminpour 2020 (https://doi.org/10.1038/s41893-019-0467-z) for use with conventional FCMs, and expands upon them for IVFN- and TFN-FCMs.

Conventional FCMs

The mean aggregate FCM of a set of conventional FCMs is: $$ \begin{array} {lcl} A_{ij}^{aggregate} & = & \frac{\sum _{p\ =\ 1}^{N} A_{ij}^{FCM_{p}}}{N} \end{array} $$ where $A_{ij}^{aggregate}$ is the weight in the aggregate adjacency matrix for the element in row i and column j, and $FCM_{p}$ is the weight for the element in row i and column j in the adjacency matrix of the p-th FCM.

If not including zero-valued edges in aggregation calculations, the mean aggregate equation becomes: $$ \begin{array} {lcl} A_{ij}^{aggregate} & = & \frac{\sum _{p\ =\ 1}^{N} A_{ij}^{FCM_{p}}}{\sum _{p\ =\ 1}^{N} 1|A_{ij}^{FCM_{p}} \neq 0} \end{array} $$

The median aggregate FCM of a set of conventional FCMs is: $$ \begin{array} {lcl} A_{ij}^{aggregate} & = & \mathrm{Median} (A_{ij}^{FCM_{1}} ,A_{ij}^{FCM_{2}} ,...,A_{ij}^{FCM_{N}} )\ \end{array} $$ If not including zero-valued edges in aggregation calculations, the median aggregate equation becomes: $$ \begin{array} {lcl} A_{ij}^{aggregate} & = & \mathrm{Median} (A_{ij}^{FCM_{1}} ,A_{ij}^{FCM_{2}} ,...,A_{ij}^{FCM_{N}} )\ |\ A_{ij}^{FCM_{p}} \neq 0 \end{array} $$

IVFN-FCMs

The mean aggregate FCM of a set of IVFN-FCMs is: $$ \begin{array} {lcl} A_{ij}^{aggregate} & = & \left[ A^{L} ,A^{U} \ \right]_{ij}^{FCM_{aggregate}} \\ \\ & = & \left[\frac{\sum _{p\ =\ 1}^{N}\left[ A^{L}\right]_{ij}^{FCM_{p}}}{N} ,\ \frac{\sum _{p\ =\ 1}^{N}\left[ A^{U}\right]_{ij}^{FCM_{p}}}{N}\right] \end{array} $$ where $\left[ A^{L} ,A^{U} \ \right]_{ij}^{FCM_{aggregate}}$ is the IVFN weight (with lower bound $A^{L}$ and upper bound $A^{U}$) of the element in row i and column j of the aggregate, and $\left[ A^{L}\right]_{ij}^{FCM_{p}}$ and $\left[ A^{L}\right]_{ij}^{FCM_{p}}$ are the lower and upper bounds of the IVFN weight of the element in row i and column j for the p-th FCM adjacency matrix.

If not including zero-valued edges in aggregation calculations, the mean aggregate equation becomes: $$ \begin{array} {lcl} A_{ij}^{aggregate} & = & \left[ A^{L} ,A^{U} \ \right]_{ij}^{FCM_{aggregate}} \\ \\ & = & \left[\frac{\sum _{p\ =\ 1}^{N}\left[ A^{L}\right]_{ij}^{FCM_{p}}}{\sum _{p\ =\ 1}^{N} 1|\left[ A^{L}\right]_{ij}^{FCM_{p}} \neq 0} ,\ \frac{\sum _{p\ =\ 1}^{N}\left[ A^{U}\right]_{ij}^{FCM_{p}}}{\sum _{p\ =\ 1}^{N} 1|\left[ A^{U}\right]_{ij}^{FCM_{p}} \neq 0}\right] \end{array} $$

The median aggregate FCM of a set of IVFN-FCMs is: $$ \begin{array} {lcl} A_{ij}^{aggregate} & = & \left[ \ \mathrm{Median}\left( A_{ij}^{FCM_{1} -L} ,A_{ij}^{FCM_{2} -L} ,...,A_{ij}^{FCM_{N} -L}\right) \right. , \\ & & \ \ \left. \mathrm{Median}\left( A_{ij}^{FCM_{1} -U} ,A_{ij}^{FCM_{2} -U} ,...,A_{ij}^{FCM_{N} -U}\right) \ \right] \end{array} $$

If not including zero-valued edges in aggregation calculations, the median aggregate equation becomes: $$ \begin{array} {lcl} A_{ij}^{aggregate} & = & \left[ \ \mathrm{Median}\left( A_{ij}^{FCM_{1} -L} ,A_{ij}^{FCM_{2} -L} ,...,A_{ij}^{FCM_{N} -L}\right) |A_{ij}^{FCM_{p} -L} \neq A_{ij}^{FCM_{p} -U} \neq 0 \right. , \\ & & \ \ \left. \mathrm{Median}\left( A_{ij}^{FCM_{1} -U} ,A_{ij}^{FCM_{2} -U} ,...,A_{ij}^{FCM_{N} -U}\right) |A_{ij}^{FCM_{p} -L} \neq A_{ij}^{FCM_{p} -U} \neq 0 \ \right] \end{array} $$

TFN-FCMs

The mean aggregate FCM of a set of TFN-FCMs is: $$ \begin{array} {lcl} A_{ij}^{aggregate} & = & \left[ A^{L} ,A^{M} ,A^{U} \ \right]_{ij}^{FCM_{aggregate}} \\ \\ & = & \left[\frac{\sum _{p\ =\ 1}^{N}\left[ A^{L}\right]_{ij}^{FCM_{p}}}{N} ,\frac{\sum _{p\ =\ 1}^{N}\left[ A^{M}\right]_{ij}^{FCM_{p}}}{N} ,\frac{\sum _{p\ =\ 1}^{N}\left[ A^{U}\right]_{ij}^{FCM_{p}}}{N}\right] \end{array} $$

where $\left[ A^{L} , A^{M}, A^{U} \ \right]_{ij}^{FCM_{aggregate}}$ is the TFN weight (with lower bound $A^{L}$, mode $A^{M}$, and upper bound $A^{U}$) of the element in row i and column j of the aggregate, and $\left[ A^{L}\right]_{ij}^{FCM_{p}}$, $\left[ A^{M}\right]_{ij}^{FCM_{p}}$, and $\left[ A^{U}\right]_{ij}^{FCM_{p}}$ are the lower bound, mode, and upper bound of the TFN weight of the element in row i and column j for the p-th FCM adjacency matrix.

If not including zero-valued edges in aggregation calculations, the mean aggregate equation becomes: $$ \begin{array} {lcl} A_{ij}^{aggregate} & = & \left[ A^{L} ,A^{M} ,A^{U} \ \right]_{ij}^{FCM_{aggregate}} \\ \\ & = & \left[\frac{\sum _{p\ =\ 1}^{N}\left[ A^{L}\right]_{ij}^{FCM_{p}}}{\sum _{p\ =\ 1}^{N} 1|\left[ A^{L}\right]_{ij}^{FCM_{p}} \neq 0} ,\frac{\sum _{p\ =\ 1}^{N}\left[ A^{M}\right]_{ij}^{FCM_{p}}}{\sum _{p\ =\ 1}^{N} 1|\left[ A^{M}\right]_{ij}^{FCM_{p}} \neq 0} ,\ \frac{\sum _{p\ =\ 1}^{N}\left[ A^{U}\right]_{ij}^{FCM_{p}}}{\sum _{p\ =\ 1}^{N} 1|\left[ A^{U}\right]_{ij}^{FCM_{p}} \neq 0}\right] \end{array} $$

The median aggregate FCM of a set of TFN-FCMs is:

$$ \begin{array} {lcl} A_{ij}^{aggregate} & = & \left[ \ \mathrm{Median}\left( A_{ij}^{FCM_{1} -L} ,A_{ij}^{FCM_{2} -L} ,...,A_{ij}^{FCM_{N} -L} \right) \right. , \\ & & \ \ \ \mathrm{Median}\left( A_{ij}^{FCM_{1} -M} ,A_{ij}^{FCM_{2} -M} ,...,A_{ij}^{FCM_{N} -M}\right) , \\ & & \ \ \ \left. \mathrm{Median}\left( A_{ij}^{FCM_{1} -U} ,A_{ij}^{FCM_{2} -U} ,...,A_{ij}^{FCM_{N} -U}\right) \right] \end{array} $$

If not including zero-valued edges in aggregation calculations, the median aggregate equation becomes:

$$ \begin{array} {lcl} A_{ij}^{aggregate} & = & \left[ \ \mathrm{Median}\left( A_{ij}^{FCM_{1} -L} ,A_{ij}^{FCM_{2} -L} ,...,A_{ij}^{FCM_{N} -L}\right) |A_{ij}^{FCM_{1} -L} \neq A_{ij}^{FCM_{1} -M} \neq A_{ij}^{FCM_{1} -U} \neq 0 \right. , \\ & & \ \ \ \mathrm{Median}\left( A_{ij}^{FCM_{1} -M} ,A_{ij}^{FCM_{2} -M} ,...,A_{ij}^{FCM_{N} -M}\right) |A_{ij}^{FCM_{1} -L} \neq A_{ij}^{FCM_{1} -M} \neq A_{ij}^{FCM_{1} -U} \neq 0 , \\ & & \ \ \ \left. \mathrm{Median}\left( A_{ij}^{FCM_{1} -U} ,A_{ij}^{FCM_{2} -U} ,...,A_{ij}^{FCM_{N} -U}\right) |A_{ij}^{FCM_{1} -L} \neq A_{ij}^{FCM_{1} -M} \neq A_{ij}^{FCM_{1} -U} \neq 0\right] \end{array} $$

References

Aminpour P, Gray SA, Jetter AJ, Introne JE, Singer A, Arlinghaus R (2020). “Wisdom of Stakeholder Crowds in Complex Social–Ecological Systems.” Nature Sustainability, 3(3), 191–199. ISSN 2398-9629, doi:10.1038/s41893-019-0467-z.

Examples