This function calculates the largest possible lambda for sigmoid and tanh squashing functions that guarantees convergence of an FCM simulation. Lambda is estimated such that 'squashed' simulation values are contained within the near-linear region of the sigmoid or tanh functions.
Note: The lambda value estimated will vary for different FCMs.
This algorithm is intended for conventional fCMs. If an Interval Value
Fuzzy Number (IVFN) FCM or a Triangular Fuzzy Number (TFN) FCM is provided,
estimate_fcm_lambda() will create a conventional FCM by averaging
the edge weights (upper and lower) for IVFN-FCMs and upper, mode, and lower
for TFN-FCMs) for use in calculating lambda,.
Usage
estimate_fcm_lambda(
adj_matrix = data.frame(),
squashing = c("sigmoid", "tanh")
)Details
This algorithm was first explored by Kottas et al. (2010), and expanded upon by Harmati et al. (2018), and Koutsellis et al. (2022)
The function applies an algorithm that can be used to optimize lambda which comparing the lambda calculated based on the Frobenius-norm (\(\lambda^{'}\)) and S-norm (\(\lambda^{*}\)) of an adjacency matrix and selects the minimum.
$$ \lambda _{s} < \min\left( \lambda _{s}^{'} ,\lambda _{s}^{*}\right) $$ $$ \lambda _{h} < \min\left( \lambda _{h}^{'} ,\lambda _{h}^{*}\right) $$
where \(\lambda _{s}\) is the lambda calculation when using the sigmoid squashing function and \(\lambda _{h}\) is the lambda calculation when using tanh.
The equations for \(\lambda ^{'}\) were developed by Harmati et al. 2018 (https://doi.org/10.1109/FUZZ-IEEE.2018.8491447) and are given below.
$$ \lambda _{s} < \lambda _{s}^{'} =\frac{4}{\| \mathbf{W} \| _{F}} $$
$$ \lambda _{h} < \lambda _{h}^{'} =\frac{1}{\| \mathbf{W} \| _{F}} $$
where \(\| \mathbf{W} \| _{F} \ =\ \sqrt{\sum\limits _{i=1}^{n}\sum\limits _{j=1}^{n}\left( w_{ij}^{2}\right)}\)
is the Frobenius norm of the adj. matrix (or norm(x, type = "F")).
The equations for \(\lambda ^{*}\) were developed by Koutsellis et al. 2022 (https://doi.org/10.1007/s12351-022-00717-x). Unlike for \(\lambda ^{'}\), the calculations for \(\lambda ^{*}\) follow different steps based on whether calculating \(\lambda\) for the sigmoid or tanh squashing function.
For sigmoid: $$ |x_{i}^{k} \| _{\max} =\max\left(\left| 0.211\cdot \sum _{i=1}^{p} w_{ij}^{+} +0.789\cdot \sum _{i=1}^{p} w_{ij}^{-}\right| ,\left| 0.211\cdot \sum _{i=1}^{p} w_{ij}^{-} +0.789\cdot \sum _{i=1}^{p} w_{ij}^{+}\right| \right) $$ $$ \| \mathbf{W} \| _{s} =\underset{i}{\max}\left( |x_{i}^{k} \| _{\max}\right) $$ $$ \lambda _{s} < \lambda _{s}^{*} =\frac{1.317}{\| \mathbf{W} \| _{s}} $$
For tanh: $$ \lambda _{h} < \lambda _{h}^{*} =\frac{1}{\| \mathbf{W} \| _{\infty }} $$ $$ \| \mathbf{W} \| _{\infty } =\underset{i}{\max}\sum _{j=1}^{n}| w_{ij}| $$
Finally, the maximum lambda the ensures convergence of the simulation for the input adjacency matrix is the minimum of \(\lambda ^{'}\) and \(\lambda ^{*}\).
Note: This is only algorithm included at present, but the code for
estimate_fcm_lambda is organized to streamline the addition of
new algorithms in the future.
References
Kottas TL, Boutalis YS, Christodoulou MA (2010). “Fuzzy cognitive networks: Adaptive network estimation and control paradigms.” In Kacprzyk J, Glykas M (eds.), Fuzzy Cognitive Maps, volume 247, 89–134. Springer Berlin Heidelberg, Berlin, Heidelberg. ISBN 978-3-642-03219-6 978-3-642-03220-2, doi:10.1007/978-3-642-03220-2_5, Series Title: Studies in Fuzziness and Soft Computing,
Harmati IA, Koczy LT (2018). “On the Existence and Uniqueness of Fixed Points of Fuzzy Set Valued Sigmoid Fuzzy Cognitive Maps.” In 2018 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), 1–7. ISBN 9781509060207, doi:10.1109/FUZZ-IEEE.2018.8491447,
Koutsellis T, Xexakis G, Koasidis K, Nikas A, Doukas H (2022). “Parameter analysis for sigmoid and hyperbolic transfer functions of fuzzy cognitive maps.” Operational Research, 22(5), 5733–5763. ISSN 1109-2858, 1866-1505, doi:10.1007/s12351-022-00717-x,
Examples
estimate_fcm_lambda(sample_fcms$simple_fcms$conventional_fcms[[1]], squashing = "sigmoid")
#> [1] 2.225602
estimate_fcm_lambda(sample_fcms$simple_fcms$ivfn_fcms[[1]], squashing = "tanh")
#> [1] 0.9186304
estimate_fcm_lambda(sample_fcms$simple_fcms$tfn_fcms[[1]], squashing = "sigmoid")
#> [1] 2.384574