The three major issues worrying people today are the economy, the environment and terrorism. Uncertainty pervades all three issues, which is unsettling for us today, and could well remain so for our children long into the future.
A quick search on Google for 'We live in uncertain times' produces harbingers of doom on the economy, as well as religious sites offering how to cope with this uncertainty.
A search for 'Uncertainty and the environment' highlights concerns among the government agencies and funding bodies, and numerous articles and books on the topic.
The uncertainty in the policing and politics of terrorism is important to many government and non-government organisations, and many of them employ armies of mathematicians, statisticians and computer scientists to help them react to events and plan for the future.
The modellers may ask themselves: “If only this uncertainty would go away? Perhaps we can forget about it when modelling environmental changes? Statistics will help but we don’t have enough data. Grrr... this uncertainty gets in the way of a mathematical model...”
I don’t believe there is a religious solution to these major challenges to mankind - some people may take solace from religion when considering these problems but that’s different to solving them. At the same time, there is no mathematical or computational solution to effective modelling of these difficult issues. But there is an approach known as fuzzy logic that may help solve these, and other problems where a lot of uncertainty is embedded within the historical data that are typically used to forecast the future or model a particular situation.
Fuzzy logic sounds like an oxymoron, but that’s what us fuzzy guys were stuck with when Lotfi Zadeh, in 1965, proposed this new way of looking at uncertainty and imprecision, by introducing the simple notion of a fuzzy set. Until that point we were dealing with Boolean logic, which assumed we were in a true/false world that didn’t allow 'greyness' or soft boundaries between objects or classes, whereas a fuzzy set allows us to have objects that belong to a set 'to some degree'.
To try to illustrate this I’ll use the example of a hurricane. A hurricane is categorised by the National Hurricane Centre in the US as Category 4 if the winds are between 131mph and 155mph. This is a wide range and if the wind is 130mph it would be classed as Category 3 despite being only 1mph below the Category 4 range. Obviously, this is a handy reference for defining categories of wind but in a modelling environment we might want to use the notion of Category 4 to help with decision making.
A fuzzy approach would say that given a wind speed of 140mph we have a category 4 wind to degree 0.3, for example, whereas a speed of 150mph might be category 4 to degree 0.9 as opposed to both these speeds being simply just Category 4. So, this notion of degree of membership of a set, underpinned by a vast amount of mathematical theory, allows us to capture, represent and model uncertainty.
In the real world, fuzzy logic pervades control applications such as the automatic transmission system on the Volkswagen Jetta that models driver behaviour. There is this story about the head of Motorola experiencing an incredibly smooth ride on an elevator and discovering that fuzzy logic was behind it, and there are numerous other applications, particularly in consumer goods such as cameras and washing machines.
While fuzzy logic has proved very effective in these types of applications, there has been less success in modelling human decision making. Something called type-2 fuzzy logic is now coming to the fore in real-world, difficult problem solving.
Put simply, type-2 fuzzy logic models a higher order of uncertainty by allowing, for example, that a wind of 140mph is a Category 4 wind to 'about 0.3'. This approach allows us fuzzy modellers to better represent the uncertainty that exists in all measurements in all situations. We are now seeing applications in such areas as modelling sand dune movement or medical decision making and these are producing excellent results.
The challenge then is for mathematicians and statisticians to move on from those methods that ignore the uncertainty in this world and deploy fuzzy logic that celebrates uncertainty.
Fuzzy logic to the rescue!
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