Quantization effects on the equilibrium behavior of combined fuzzy cognitive maps

Abstract

Fuzzy cognitive maps (FCMs) allow experts to express their knowledge by drawing weighted causal digraphs. Experts can pool or fuse their knowledge by adding the underlying FCM causal matrices. This naturally extends the ordered-weighted-averaging (OWA) technique to averaging dynamical systems and can create complex dynamical systems from several simpler ones. Edge quantization allows experts to state their knowledge in the simpler terms of causal increase (1), decrease (−1), or absence (0). We model the expert FCMs as a sequence of random fields to study the small-sample effects of quantizing both the causal edges and the fuzzy-set concept nodes. The averaged quantized random matrices exhibit large-sample convergence to the population means of the unquantized matrices in accordance with the Strong Law of Large Numbers. But the small-sample averages can show substantial diversity of equilibrium attractors (fixed points or limit cycles). We use statistical tests—chi-square tests, Spearman's rank coefficient, the Kolmogorov–Smirnov test, and the fuzzy equality of limit cycle histograms—to show that this small-sample equilibrium diversity increases as the node multivalence or fuzzy-set quantization increases. The appendix presents a new probabilistic convergence theorem that shows that edge quantization or thresholding does not affect FCM combination for large expert sample sizes: the sample mean of quantized expert causal edge values converges with probability one to the population mean causal edge values. © 2007 Wiley Periodicals, Inc. Int J Int Syst 22: 181–202, 2007.