296 A.F. Shapiro, R. Paul Gorman Insurance: Mathematics and Economics 26 2000 289–307
fuzzy NN can be constructed where the NN possesses fuzzy signals andor has fuzzy weights. Conversely,
FL can use technologies from other fields, like NNs or GAs, to deduce or to tune, from observed data, the
membership functions in fuzzy rules, and may also structure or learn the rules themselves.
4. Functional classes
ANM technologies also can be grouped into func- tional classes. Broadly, these classes include:
• Generalized nonlinear function approximation,
which adaptively constructs a nonlinear relation- ship between observed variables and dependent
variables. Technologies in this class include NNs and generalized regression networks.
• Domain segmentation, which involves finding sub-
domains in the observation space where the rela- tionships between observed variables and dependent
variables are consistent. This is a very crucial step in the development of nonlinear models. This class
includes rule induction and technologies for analyz- ing data and for inducing decision trees from data.
• Generalized knowledge encoding, which uses do-
main knowledge to creatively bias more general adaptive methods. This is an important component
in the evolution of robust nonlinear models. Tech- nologies in this category includes expert systems,
neuro-fuzzy models and case-based reasoning tech- nologies.
• Dimension reduction, which uses variable selection
and aggregation to lower the dimensionality of the problem. This group includes Kohonen networks
and fuzzy clustering Zimmermann, 1991, Section 11.2, which seeks to divide objects into categori-
cally homogeneous subsets called “clusters”.
• Numerical optimization, which allows the numeri-
cal estimation of model parameters based on a sam- ple of data. These optimization technologies are
an indispensable element of an adaptive nonlinear modeling toolkit. Technologies in this class include
gradient descent, simulated annealing Aarts and Van Laarhoven, 1987, and GAs. Gradient descent
iteratively updates the weight vector in the direc- tion of the greatest decrease in the network error
and simulated annealing is a stochastic algorithm that minimizes numerical functions, whose distin-
guishing feature is that it uses a random process to elude local minimums.
5. The team approach
Given this wide range of technologies, it often is ad- vantageous to approach a problem with a team whose
members have diverse backgrounds and experiences. Thus, for example, an ideal team may be composed of
members whose backgrounds include not only math- ematics, economics and statistics, but physics, and
computational and computer science, as well. It is the ability of the team to craft a solution by integrating
complementary advanced technologies that makes this methodology so powerful.
6. Modeling considerations
Modeling considerations include the heuristic na- ture of the approach, data issues, the emphasis on non-
linear relationships, and domain knowledge.
6.1. Heuristic approach There is no canonically optimal approach to the de-
velopment of models for nonlinear problems so solu- tions can vary considerably from problem to problem.
In this sense, the technology is highly heuristic and often ad hoc, and since the field is still in its embry-
onic stage, there is considerable room for improve- ment. Thus, the formulation of a science of adaptive
nonlinear modeling should be considered as a work in progress.
Moreover, while the focus is on ANM technologies, this cannot always be to the exclusion of more tradi-
tional approaches. Regardless of the sophistication, if the signal-to-noise ratios
8
are so poor that reasonable relationships cannot be derived, it may be necessary
to resort to more conventional technologies to attain adequate performance.
8
If it is assumed that a system has a given pattern, µ, and that the signal stabilizes µ, the noise refers to signals from other
patterns, which destabilize µ.
A.F. Shapiro, R. Paul Gorman Insurance: Mathematics and Economics 26 2000 289–307 297
6.2. Data issues There are many issues with data if there is no theo-
retical framework to constrain the solution, since the resolution of the problem depends on and is highly
sensitive to the nature of the sample data. As a con- sequence, considerable resources are devoted to pro-
cessing the data, with an emphasis on missing and corrupted data and the removal of bias from the sam-
ple. Additionally, where multiple sources of data are involved, the consistency of the differential semantics
across these sources have to be verified.
6.3. Emphasis on nonlinear relationships A distinguishing assumption of this approach is that
there are important nonlinearities both between the ob- servables independent variables and the dependent
variable as well as nonlinearities among the observ- ables. The emphasis is on not making unjustified as-
sumptions about the nature of those nonlinearities, and technologies are used that have the capacity, in theory
at least, to extract the appropriate interaction terms adaptively.
6.4. Domain knowledge As mentioned previously, the technologies do not al-
ways achieve their ends because of the signal-to-noise ratios in the sample data. Of necessity, in these in-
stances, the approach is to constrain the solution by introducing expert knowledge into the process. So, it
is not quite a theoretical framework but it is more a heuristic framework to help constrain the solution
space.
7. The model development process
