Hydroinformatics: modelling and information systems for integrated water resources management

Hydroinformatics

Dimitri Solomatine's pages

FUZZY SYSTEMS

Fuzzy logic was initiated in 1965 by Lotfi A. Zadeh, professor in computer science at the University of California in Berkeley. Since then Fuzzy logic has emerged as a powerful technique for the controlling industrial processes, household and entertainment electronics, diagnosis systems and other expert systems. Rapid growth of this technology has actually started from Japan and then spread to the USA and Europe. Most applications of Fuzzy logic are in the are of control.

The motivation for fuzzy logic was expressed by Zadeh (1984) in the following way: "The ability of the human mind to reason in fuzzy terms is actually of a great advantage. Even though a tremendous amount of information is presented to the human senses in a given situation – an amount that would choke a typical computer – somehow the human mind has the ability to discard most of this information and to concentrate only on the information that is task relevant. This ability of the human mind to deal only with the information that is task relevant is connected with its possibility to process fuzzy information. By concentrating only on the task-relevant information, the amount of information the brain has to deal with is reduced to a manageable level.".

Fuzzy logic is basically a multi-valued logic that allows intermediate values to be defined between conventional evaluations like yes/no, true/false, black/white, etc. Notions like rather warm or pretty cold can be formulated mathematically and algorithmically processed. In this way an attempt is made to apply a more human-like way of thinking in the programming of computers ("soft" computing).

Fuzzy logic systems address the imprecision of the input and output variables by defining fuzzy numbers and fuzzy sets that can be expressed in linguistic variables (e.g. small, medium and large).

The present research is aimed at the use of fuzzy rule-based systems (FRBS). Fuzzy rule-based approach to modelling is based on verbally formulated rules overlapped throughout the parameter space. They use numerical interpolation to handle complex non-linear relationships.

Many of existing systems, like Fuzzle and FuzzyCLIPS, need the rules to be formulated by an expert. Our aim was to construct a tool that would generated such rules automatically on the basis of numerical data describing a certain phenomenon, and to apply it to a real problem related to water resources.

Fuzzy rule-based systems

Fuzzy rules are linguistic IF-THEN- constructions that have the general form "IF A THEN B" where A and B are (collections of) propositions containing linguistic variables. A is called the premise and B is the consequence of the rule. In effect, the use of linguistic variables and fuzzy IF-THEN- rules exploits the tolerance for imprecision and uncertainty. In this respect, fuzzy logic mimics the crucial ability of the human mind to summarize data and focus on decision-relevant information.

In a more explicit form, if there are I rules each with K premises in a system, the i^th rule has the following form.

In the above equation a represents the crisp inputs to the rule and A and B are linguistic variables. The operator 1 can be AND or OR or XOR.

Example: If a HIGH flood is expected and the reservoir level is MEDIUM, then water release is HIGH.

Several rules constitute a fuzzy rule-based system.

Another example comes from Kosko (1993). Figures below are adapted from this book and illustrate the notion of a simple fuzzy rule with one input and one output applied to the problem of an air motor speed controller for air conditioning. Rules are given. Let us say the temperature is 22 degrees. This temperature is "right" to a degree of 0.6 and "warm" to a degree of 0.2 and it belongs to all others to a degree of zero. This activates two of the rules shown in Figure 1. The rule responses are combined to give those shown in Figure 2 (thick lines).

Figure 1. Air motor speed controller. Temperature (input) and spedd (output) are fuzzy variables used in the set of rules.

Figure 2. Temperature of 22 deg. "fires" two fuzzy rules. The resulting fuzzy value for air motor speed is "defuzzified" – abscissa of the centroid of area gives the "crisp" value

AFUZ - fuzzy rule-based system tool

A Windows based fuzzy rule-based tool AFUZ was built in the framework of the MSc research of A.J. Abebe (he continues currently as a Ph.D. fellow). It allows modelling an input – output relationship (function of several variables) of any nature. Training of a set of fuzzy rules is performed on the basis of a given set of "examples" of input – output data. Being trained, the resulting system allows for accurate reproduction of output variable, given values of input variables.

The system will shortly be available to run across Internet. See this page later for further details.

Applications

FRBS methodology has been successfully applied to a problem of representing the spatial precipitation pattern at rain gauge stations in Italy using rules generated from historical data. The number of rules has been found to be the key parameter in overcoming problems of over-fitting and generalization arising from uncertainties due to incomplete or non-representative data. For this particular case study, the performance indices have shown its best performance compared to two other possible methods of solution – a traditional normal ratio method and artificial neural network (ANN) solution.

Another area of application is the use of FRBS (along with ANNs) in the problems of reproducing the behaviour of a control system responsible for regulating water levels in a water system.