Hidden patterns in combined and adaptive knowledge networks

Abstract

Uncertain causal knowledge is stored in fuzzy cognitive maps (FCMs). FCMs are fuzzy signed digraphs with feedback. The sign (+ or -) of FCM edges indicates causal increase or causal decrease. The fuzzy degree of causality is indicated by a number in [−1, 1]. FCMs learn by modifying their causal connections in sign and magnitude, structurally analogous to the way in which neural networks learn. An appropriate causal learning law for inductively inferring FCMs from time-series data is the differential Hebbian law, which modifies causal connections by correlating time derivatives of FCM node outputs. The differential Hebbian law contrasts with Hebbian output-correlation learning laws of adaptive neural networks. FCM nodes represent variable phenomena or fuzzy sets. An FCM node nonlinearly transforms weighted summed inputs into numerical output, again in analogy to a model neuron. Unlike expert systems, which are feedforward search trees, FCMs are nonlinear dynamical systems. FCM resonant states are limit cycles, or time-varying patterns. An FCM limit cycle or hidden pattern is an FCM inference. Experts construct FCMs by drawing causal pictures or digraphs. The corresponding connection matrices are used for inferencing. By additively combining augmented connection matrices, any number of FCMs can be naturally combined into a single knowledge network. The credibility w i in [0, 1] of the ith expert is included in this learning process by multiplying the ith expert's augmented FCM connection matrix by w i. Combining connection matrices is a simple type of adaptive inference. In general, connection matrices are modified by an unsupervised learning law, such as the differential Hebbian learning law. Under special conditions, differential Hebbian dynamical systems are proved globally stable: they resonate on fixed-point attractors.