Pena-Reyes, Carlos A. - Parameters of Fuzzy Inference

The Structure of Concern Project compares many theoretical models from many disciplines to the Adizes PAEI model, arguing that they must all be reflecting the same underlying phenomenon. One concern structure model is described below.

Carlos A. Peña-Reyes works in the domain of fuzzy computer programming, which differs from ordinary programming in important ways. Standard computer programs manipulate precise and explicit information in precise and explicit ways. Fuzzy systems, by contrast, reason with the computerized equivalents of ideas like “maybe”, “sort of”, “kind of” and “almost”. They allow for shades of gray and degrees of possible truth, and when they are well-designed they produce reasoning outcomes similar to human decision-makers, using the same uncertain and incomplete data used by the humans. Fuzzy systems are thus promising models of certain aspects of human cognitions. Designing these systems well is challenging, however.

Fuzzy systems use linguistic variables like “very hot day” to represent very hot days (rather than ‘AvTemp > 29ºC’). Numerical values accompany the linguistic variables, but the fuzzy reasoning is done on the linguistic variables. For greater precision, increasingly specialized vocabularies or quasi-numerical codes can be used instead of natural language terms, but this increased precision hampers the intelligibility of the final decision and the reasoning that produced it.

Fuzzy modeling is also haunted by the curse of dimensionality. Each new variable added to a model needs to be related to the others by rules, exponentially increasing computing requirements with the number of variables. Building a fuzzy system to solve a certain class of problems thus requires developing a good description of relevant variables and rules, with the right balance of precision and readability, capturing the complexity of the problem with a minimally complex model so as to avoid a combinatorial nightmare.

Achieving this balance is a delicate task. It is often done by working with human domain experts and modeling their reasoning, then using the experts’ decisions as a benchmark to evaluate the performance on the model. If the model replicates their decisions, it accurately represents their reasoning. Another tactics is to use artificial neural networks to reveal rules and regularities in classes of input data, and then to model those rules in fuzzy logic. Alternatively, a benchmark can be set, and then genetic algorithms can be used to breed fuzzy rules and select out unfit candidates, using evolutionary programming.

One such evolutionary solution is called Fuzzy CoCo, developed by Carlos Peña-Reyes. It uses a technique called cooperative coevolutionary fuzzy modeling to produce better fuzzy models in shorter periods of time. It also aims to avoid some of the pitfalls of evolutionary programming, such as high computational costs and the tendency to get stuck in local optima (Peña-Reyes , 2004).

My interest here is in Peña-Reyes’ four-type classification of the parameters of fuzzy systems – the elements of fuzzy inference, if you like. According to Peña-Reyes, to build a fuzzy system, one needs to specify logical, structural, connective and operational parameters. This creates a concern structure model, as follows:

P – Operational Parameters: Each linguistic variable and value needs a membership function, defining roughly what does or does not count as a variable of that type. This constitutes the concrete particulars of the knowledge of the system. It links labels with associated numerical values so that computational work can be done.

A – Logical Parameters: These are usually defined by the programmer in advance, and formally or syntactically define the reasoning operations the system will use to transform input into output. Logical parameters do not define the system’s functions, but rather the ‘legal’ tactics that are available for building those functions.

E – Structural Parameters: These define the overall scope, identity and direction of the model, including input-output and state variables, universes of discourse for linguistic representations and the linguistic labels that will be used for the variables. This can be defined in advance but often it emerges in systems that are allowed to improve themselves, dropping old variables and breeding or learning new rules, etc.

I – Connective Parameters: Describes the network of relationships between variables and values in the system by defining rules, weighting them, and determining what conditions are antecedents for the application of a rule, and what the outcomes should be. This describes the model’s actual behavior as a problem-solving system.

1. Peña-Reyes, C. A. (2004). Coevolutionary Fuzzy Modeling. Berlin, Heidelberg: Springer-Verlag.
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