On Complexity and Fuzziness

In the previous examples we have dealt with very simple fuzzy cognitive maps that contain up to six factors. We have used some of these fuzzy maps to model ecosystems. Someone could argue that a fuzzy cognitive map is not the best way to model the behaviour of an ecosystem. In fact, more accurate and detailed predictions could be obtained by using differential equations; providing, of course, that the modeller has enough knowledge and information about the system to be able to express its behaviour in differential equations, which is not always true.

The Offer and Demand system, and for that matter any economic system, is not so amenable to differential equations as ecosystems. Economists keep trying to model economic systems using non-fuzzy mathematical methods, but as far as I know they are not very successful (please correct me if you can prove that I am wrong). Usually economic systems are far too complex to be expressed in equations. Nevertheless, classical (non-fuzzy) mathematics are to some extent applicable to economy; after all, economy deals with measurable magnitudes such as capital, rates, time, percentages and the like.

The Bad Weather Driving system could be regarded as more suitable for the use of fuzzy cognitive maps, if nothing else because it is not suitable for non-fuzzy methods. After all, how could one express "risk aversion" in an equation and relate it to other factors? It is not easy. Certainly, one could try and model the Bad Weather Driving system using classical mathematical methods, but the outcome is not likely to be any more accurate than what can be obtained with the fuzzy cognitive map, with the advantage that the fuzzy cognitive map is very easy to draw. And of course, one could argue that the system is far too simple to need any modelling at all. Any discerning person can predict that there are more accidents and more traffic congestion when the weather is bad.

Fuzzy cognitive maps are useful when the system is not suitable to be modelled using traditional mathematical methods because it is not easy to express the factors or the relationships between factors mathematically. For example, in an economic system it is very straightforward to express mathematically factors such as salaries, interest rates, percentages of population, revenues or market shares. On the other hand, it is not so easy to express sloppy factors such as investors' fear, consumer optimism, anxiety, leadership, trust, fashion or unpopularity.

Fuzzy cognitive maps are also useful when there are so many factors and relationships in the system that nobody, no matter how discerning and knowledgeable, can make meaningful predictions about the behaviour of the system. An expert may be able to know or to find out how a given factor of a complex system relates to any other given factor, but he does not know how the whole system behaves. The expert can understand single relationships between factors, but he cannot understand more than a few interactions at once; he does not have a grasp of the whole picture. It is in this situation when the fuzzy cognitive map is helpful. The expert can identify all, or most, factors involved, and all the relationships between them. Therefore he is able to draw the fuzzy cognitive map. Then he can use the map to make predictions that otherwise he could not make.

In the fuzzy coginitive map we assign levels to factors, and intensities of effect to relationships. These levels and intensities are expressed by numerical values, which in the applet range between 0 and 100. The applet then uses these values to operate and carry out a mathematical calculation of the evolution of the system. So, if in the fuzzy cognitive map and the simulator are nothing more than crisp numbers and equations, where is the fuzziness then?

The fuzziness of a cognitive map is, first, in the selection of factors and relationships that form the map. Different modellers may come up with different fuzzy maps for representing the same real-life system. The modellers select the factors and the relationships to put in the map according to their knowledge about the subject, their experience, their perceptions and their prejudicies. There is no single "right" fuzzy cognitive map. A map is right when it is useful to make predictions about the system. Two different maps may make the same accurate predictions, and therefore, in this sense, be both right.

The fuzziness of a cognitive map is also in the assignment of numerical values to the levels of the factors and to the intensities of effect of the relationships. This assignment is entirely subjective. For example, let's suppose that we have to assign a value to the consumer optimism factor in a fuzzy map. We may know that the consumer optimism is high but, how much is "high" consumer optimism? Is it 60 out of 100? Is it 75? Is it 80? It is not clear, hence the fuzziness. We know that "little" is less than "a little" is less than "much" is less than "very much" is less than "a disordinately large amount", but we do not know the absolute differences between them. We can only guess. And probably it does not really matter if we do not guess entirely right. The simulation in a fuzzy cognitive map may produce very similar results no matter whether high consumer optimism is represented by a level of 70 or 85. The same kind of fuzziness is in the assigment of values to the intensities of effect. We all know that the level of happiness has a positive causality relationship to the level of good health (i.e. more happiness, better health) and that self-steem is has a (negative?) causality relationship to violent behaviour. But we do not know the values for sure; we have to estimate and assign values according to our best understanding.

This fuzziness does not mean that anything goes in a fuzzy cognitive map. It does not. In fact, it is unescapable that a fuzzy map that has most of the factors and relationships in place will produce better and more complete simulations that another fuzzy map that is not so well mapped to the real system. Regarding the values for levels of factors and intensities of effect we have to do educated guesses.

In the next tutorial we are going to draw a fuzzy map that is much more open to interpretation and opinion than the previous ones.