Fuzzy Logic in Natural Resource Modelling William Silvert Emeritus Research Scientist Bedford Institute of Oceanography Dartmouth, Nova Scotia

There is a Problem with Numbers 4We usually express model results as numerical values, possibly with confidence limits. 4The more rigorously we express these numbers, the more complicated we make things. 4A really good scientific description of model output is unlikely to be adequately assimilated by managers and other client groups. 4We have to get good answers and present them in a way that can be understood and used.

TAC, Version 1 The TAC (Total Allowable Catch) for a fish stock is often presented in this way: TAC = 42,000 tonnes

TAC, Version 2 Of course we know that the TAC should be presented like this: TAC = (42,000 ± 6,000) tonnes

TAC, Version 3 Even better, the TAC should be presented as:

Model Output is More and More Complex! The expression of model results gets more and more complex as it becomes more sophisticated. 4A simple TAC is easy to grasp, but not very informative. It is just one number. 4A TAC with a confidence interval is more informative, but there is an additional number. 4The full Probability Distribution Function is most informative, but probably too complex for most fisheries managers to deal with.

Discrete Model Output 4Our conceptual understanding of model output tends to use discrete categories. 4Although model output is usually numerical, when we look at the numbers we think “That is a very low TAC” or “It will be a good year for the fishing industry”. In other words, we map the numbers into discrete categories. 4We should therefore consider using discrete categories for model output.

Counter-Arguments The main problem with using discrete categories is that it gives rise to artificial discontinuities. Bad Medium Good

The Fuzzy Solution We can avoid these artificial discontinuities by using fuzzy categories, so that we can have a mixture of bad, medium, and good to characterise model output.

Flexibility Fuzzy classification can be more flexible than numerical descriptions. For example, the bimodal Probability Distribution Function shown earlier, is described as 40% bad, 25% medium, and 35% good. An error bar tells us less.

Other Considerations 4Fuzzy Logic is ideally suited for including subjective information, such as expert but non- quantitative judgements about the quality of a natural resource (e.g., the taste of fish). 4Different types of information, possibly contradictory, can be reconciled within the framework of Fuzzy Logic. 4Missing data does not pose as great a problem as it does with more quantitative methods.

Defuzzification 4The use of Fuzzy Logic is not a final commitment — it is possible to convert fuzzy memberships into a simple number through a process known as “defuzzification”. 4Many managers equate Fuzzy Logic with Fuzzy Thinking and resist it vigorously. 4This may be why countries like Japan are so far ahead of North America in applications of Fuzzy Logic, such as steady-shot camcorders!

Defuzzified TAC’s For example, we can convert the fuzzy TAC (40% Bad, 25% Medium, 35% Good) into a single numerical value if we associate Medium with the range of values kt, Bad with lower values under 40 kt, and Good with TACs over 45 kt. We get a defuzzified value from the weighted sum which is a reasonable approximation to the weighted mean TAC obtained from the Probability Distribution Function.

Defuzzification Issues 4Defuzzification is always possible and lets us hide the use of Fuzzy Logic if it doesn’t “sell”. 4Unfortunately, defuzzification also hides useful information, such as the bimodal distribution in this example. 4So defuzzify if you have to, but not if you can avoid it.

Model Validation 4Discrete predictions are easy to test, since they are either right or wrong. 4Continuous prediction is inherently fuzzy. 4How do we test whether a result was predicted? If we overfish and the TAC should have been 38 kt, can we say that the model was correct if the predicted value was (42 ± 6) kt? What about (42 ± 2) kt? How do we evaluate models that make vague predictions against ones with more precision?

Conclusions 4Fuzzy Logic offers an alternative form of model output, and a mechanism for model testing. 4It facilitates discrete classification and allows for a degree of subjective evaluation. 4Numerical output may appear more scientific and objective, but this can be both misleading and impractical. 4Additional advantages include means to merge contradictory data and even to deal with missing information.