Linear and sigmoidal fuzzy cognitive maps: An analysis of fixed points

Knight, Christopher J. K.; Lloyd, David J. B.; Penn, Alexandra S.

doi:10.1016/j.asoc.2013.10.030

Cited by 65 publications

(36 citation statements)

References 24 publications

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“…In the cases of dimensions related with the sigmoid parameters the search space must be carefully defined in order to avoid situations on which the system only produces a single decision class. Knight et al [9] proved that if λ > 0 is small enough then there is a unique fixed-point attractor. On the contrary, if λ > 0 is large enough, then there can be multiple fixed-points, where many of such equilibrium points may be linearly stable (see next Theorem 1).…”

Section: Definition 4 Let Us Assume Thatmentioning

confidence: 98%

“…It suggests that the theorem cannot be used in pattern recognition scenarios, but in control scenarios where the system requires to be consistent to external perturbations. On the other hand, Knight et al [9] introduced a theorem where the upper boundλ (M) for "small enough" values of λ is estimated (see Theorem 2). Therefore, if 0 ≤ λ <λ (M) then the system will produce the same decision class regardless the input pattern.…”

Section: Definition 4 Let Us Assume Thatmentioning

confidence: 99%

“…For example, Boutalis et al [7] and Kottas et al [8] investigated the existence and uniqueness of equilibrium values of neurons in FCM equipped with sigmoid transfer functions, using the contraction mapping theorem. Knight et al [9] proposed a slightly different theoretical result related with the inclination of the sigmoid function. However, Nápoles et al [10] numerically verified that these theoretical results cannot be directly used in solving pattern classification problems since a FCM-based classifier with a single fixed point-attractor will produce the same decision class for all input patterns.…”

Section: Introductionmentioning

confidence: 98%

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Learning and Convergence of Fuzzy Cognitive Maps Used in Pattern Recognition

Nápoles

Papageorgiou

Bello

et al. 2016

Neural Process Lett

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In recent years fuzzy cognitive maps (FCM) have become an active research field due to their capability for modeling complex systems. These recurrent neural models propagate an activation vector over the causal network until the map converges to a fixed-point or a maximal number of cycles is reached. The first scenario suggests that the FCM converged, whereas the second one implies that cyclic or chaotic patterns may be produced. The nonstable configurations are mostly related with the weight matrix that defines the causal relations among concepts. Such weights could be provided by experts or automatically computed from historical data by using a learning algorithm. Nevertheless, from the best of our knowledge, population-based algorithms for FCM-based systems do not include the map convergence into their learning scheme and thus, non-stable configurations could be produced. In this research we introduce a population-based learning algorithm with convergence features for FCMbased systems used in pattern classification. This proposal is based on a heuristic procedure, called Stability based on Sigmoid Functions, which allows improving the convergence of sigmoid FCM used in pattern classification. Numerical simulations using six FCM-based classifiers have shown that the proposed learning algorithm is capable of computing accurate parameters with improved convergence features.

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Section: Definition 4 Let Us Assume Thatmentioning

confidence: 98%

Section: Definition 4 Let Us Assume Thatmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

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Learning and Convergence of Fuzzy Cognitive Maps Used in Pattern Recognition

Nápoles

Papageorgiou

Bello

et al. 2016

Neural Process Lett

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“…It has the purpose of system evaluation according to FCM inference. The third part [16][17][18] is the state transformation of FCM, where the transforming rule and the transforming process of steady state are focused on. They are used to analyze and control system evolution.…”

Section: Introductionmentioning

confidence: 99%

Research on Steady States of Fuzzy Cognitive Map and its Application in Three-Rivers Ecosystem

Peng

Chen

2016

Sustainability

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Fuzzy Cognitive Map (FCM) offers many advantages such intuitive knowledge representation and fast numerical reasoning ability, etc. It suits modeling and decision-making of dynamic systems. With the aims to effectively help to analyze and control system sustainable evolution, the paper defines the steady states of fixed point and limited cycle of a FCM modeling system. Accordingly, the rules of steady states of the FCM model and the factors influencing the steady states are presented and proved. The Three-Rivers represents a system including population, ecological environment, social development and their relationships. Based on the relationships, the Three-Rivers ecosystem is modeled by FCM and the Three-Rivers ecosystemsustainable evolutionis analyzed bythe rules of the steady states of FCM.

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“…This function s(x) is called an activation function. This is the main idea behind Fuzzy Cognitive Maps (FCM); see, e.g., [2,3,7,8,11,12,16,17,18,19,20,22,24,25,26,27,28,29]. The FCM model is used when experts provide estimates only for some of the properties.…”

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confidence: 99%

Why Fuzzy Cognitive Maps Are Efficient

Kreinovich¹,

Stylios²

2015

INT J COMPUT COMMUN

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In many practical situations, the relation between the experts' degrees of confidence in different related statements is well described by Fuzzy Cognitive Maps (FCM). This empirical success is somewhat puzzling, since from the mathematical viewpoint, each FCM relation corresponds to a simplified one-neuron neural network, and it is well known that to adequately describe relations, we need multiple neurons. In this paper, we show that the empirical success of FCM can be explained if we take into account that human's subjective opinions follow Miller's seven plus minus two law.

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Linear and sigmoidal fuzzy cognitive maps: An analysis of fixed points

Cited by 65 publications

References 24 publications

Learning and Convergence of Fuzzy Cognitive Maps Used in Pattern Recognition

Learning and Convergence of Fuzzy Cognitive Maps Used in Pattern Recognition

Research on Steady States of Fuzzy Cognitive Map and its Application in Three-Rivers Ecosystem

Why Fuzzy Cognitive Maps Are Efficient

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