Abstract
First, it is a book not about fuzzy sets per se, it is a book about computing with fuzzy sets. Second, it is not a book about foundations –although foundations are also mentioned: it is a book about analysis, design, and applications that use fuzzy sets.
In those days, we did not have a problem finding the rules: the rules cames “from the horse’s mouth”, i.e., from the experts themselves. Ingenuity was in how to translate these rules into a precise control strategy: how to elicit degrees, how to best interpolate these degrees to get membership functions, what “and”- and ‘’or”-operations (t-norms and t-conorms) to use, what defuzzification procedure to use, later –whether to use type-1 or type-2 fuzzy sets. In all this, interesting and helping results and heuristics were developed, and this joint effect of theoretical development and practical testing led to the boom.
Fuzzy techniques still lead to many successful applications –but nowadays, it is a rare situation when all we have is expert rules. Usually, we also have some models, we also have some recorded controls –all this needs to be taken into account to get a good decision or control system. In processing expert rules, fuzzy is still very efficient, but nowadays, it is rarely enough just to use fuzzy techniques –we usually need to combine them with other control and decision making techniques.
Nowadays, the situation is opposite: we have an abundance of data. Sensors have become cheaper and cheaper, recording is easy, processing is easy –and, as a result, we have so much data that we do not know what to do with it.
The answer is straightforward: when we see a picture –e.g., when someone shows us a picture of an animal –we do not analyze it pixel-by-pixel as naive image processing algorithms do, we immediately divide this image into granules: tail, paws, face, etc. This granularity is the main way of dealing with large amounts of data.
Methods of processing these fuzzy granules are largely the same as in the past, but the main difference is that now granules do not come from simply translating expert’s words, they come from the detailed analysis of all the available information, including expert rules and available numerical data.
This view of fuzzy sets as a particular case of granules is the main focus, the main leitmotif of this book.
Usual textbooks would jump from here to operations on fuzzy sets –but not this book, it first described operations on another important class of granules –a class of intervals. This is a pedagogically good way to introduce operations on fuzzy sets: indeed, intervals can be naturally viewed as a simple particular case of fuzzy sets, and it is always a good idea to first describe operations on a simple particular case and only then move to the general more complex one.
Once we learn operations on intervals, we can extend them to operations on the usual [0, 1]-based fuzzy sets –and this is also a reasonable starting point to extend the usual arithmetic and logical operations to more general aggregation operations (e.g., averaging), and to extend all these operations to more general types of granules –interval-valued and type-2 fuzzy sets, rough fuzzy sets, probabilistic granules, and hybrid granules –that combine fuzzy and probabilistic uncertainty.
The book starts explaining this, in Chapter 9, with fuzzy clustering –the most frequently used way to form fuzzy granules. In Chapter 10, the granular formation is explained on the general level of granular computing. A granule should cover a sufficient number of examples and at the same time be sufficiently specific –i.e., its elements should be different from others. If we, as frequently happens in fuzzy applications, interpret “and” as a product, the need to maximize the degree to which both objectives are satisfied leads to maximizing the product of appropriately defined degrees of coverage and specificity; this idea is known as the Principle of Justified Granularity. This principle is then explained on the examples of fuzzy granules.
Once we have the conclusion in the form of a granule as well, how do we go from there to the actual recommendation? In fuzzy techniques, this is called defuzzification, for general granules, it is called degranulation. Such methods are described in Chapter 11, after which in Chapters 12 and 13, all this knowledge is combined together on the example of fuzzy models.
What we have described so far is a model based on all available knowledge. But knowledge does not remain unchanged: we get new knowledge all the time. How can we use this new knowledge to update our model? This model update is the domain of what is called machine learning, so a reasonable idea is to use machine learning techniques for the desired update. At present, the most effective machine learning techniques are neural networks.
How to combine fuzzy and neural techniques is the subject of Chapter 14. Finally, Chapter 15 provides advice on how to use all this in applications. In most applications, we want to optimize our models; different optimization techniques are recalled in the corresponding Appendix.