Actuarial Theory for Dependent Risks: Measures, Orders, and Models

Perhaps more troubling is the complete lack of motivation for studying FC. Ostensibly, classical controllers are insufficiently flexible to accommodate certain nonlinear input/output relationships. Fuzzy controllers are allegedly superior in this regard; however, the authors do not even attempt to explicate this extremely vital detail. Succeeding chapters are likely of far greater interest here. Applications run the gamut from somewhat straightforward to complex. Different feedback mechanisms are demonstrated in the mix. Again, there is minimal justification regarding the reasons for using FC vis a vis classical control. Two examples will hopefully elucidate. In Chapter 14 a helicopter is controlled using fuzzy gain scheduling. A recently popular technique for controller design in nonlinear plants, gain scheduling is not an intrinsically fuzzy method. It requires specifying a set of state derivatives, and the system is then represented in matrix form. The solution of a Riccati algebra equation is required. Rather than using a traditional numerical method, the authors use a Takagi–Sugeno type fuzzy model. The solution to the specific problem is given. There is no analogous solution shown for a model using “classical” gain scheduling. A second example is the data mining process application given in Chapter 17. Most of the section is spent introducing general data mining needs. The final five pages cover rule extraction, genetic algorithms (not a fuzzy technique), and dependency analysis. Rule extraction is accomplished by using the data to derive a set of rules relating fuzzy sets corresponding to each of two linguistic variables. Here a genetic algorithm is used to select the “best” set of rules to relate the variables. As the author states, “dependency analysis is a heuristic search technique for finding significant relationships among attributes in large datasets. ” Unfortunately, no corresponding example is offered. One need not tax the imagination to find probabilistic techniques that accomplish such a mission that are not heuristic. The author does note that fuzzy logic modeling is probability-based, and also refers to clustering in passing. Fuzzy clustering has been widely used, even though it uses no fuzzy mathematics. Overall, this section does little to remove the “ad hoc” stigma often attributed to fuzzy theoretic methodology. Philosophical issues are not discussed here. This is not terribly troubling, given the book’s applied engineering flavor. However, it becomes an issue when considering FST in data mining applications. There is no dearth of statistical and engineering literature on this topic, including an entire volume of IIE Transactions. Whether FST offers advantages in data mining relative to existing techniques such as neural networks and generalized additive models is beyond the scope of this review, and also is unresolved in the text. It remains a topic worthy of future study. For a reader desiring a collection of advanced industrial FST applications, this text is adequate. The examples are fresh and fairly broad. Few readers outside this group will find this book useful. This is too often an indictment of the narrow foci of most academics and industrial researchers. Texts such as this one require much arduous effort, an effort that is clearly a labor of love given the limited likely readership. Regardless of the previous turf battles between probabilists and statisticians (to which the reviewer has been no stranger), the reader is encouraged to investigate fuzzy methodology. Some familiarity is necessary given the greater frequency of fuzzy theoretic methodology present in previously nonfuzzy territory (such as Technometrics). The aforementioned texts are recommended here, as well those by Ross, Booker, and Parkinson (2002) and Singpurwalla and Booker (2004). These works are strict complementary expositions and probably offer the most balanced view available.