Nonlocal Interpretation Methods

Among nonlocal methods, those based on linear projection are the most widely used for data interpretation. Owing to their limited modeling ability, linear univariate and multivariate methods are used mainly to extract the most relevant features and reduce data dimensionality. Nonlinear methods often are used to directly map the numerical inputs to the symbolic outputs, but require careful attention to avoid arbitrary extrapolation because of their global nature. 

Univariate methods are among the simplest and most commonly used methods that compose a broad family of statistical approaches. Based on 

The most commonly implemented method of limit checking is the absolute value check as referred to by Iserman (1984), 

Although usable in many situations, limit-checking methods have limited applicability. First, these methods are univariate and depend on reducing the time series of data points into a single feature. In cases where the 

Because a hyperplane corresponds to a boundary between pattern classes, such a discriminant function naturally forms a decision rule. The global nature of this approach is apparent in Fig. 19. An infinitely long decision line is drawn based on the given data. Regardless of how closely or distantly related an arbitrary pattern is to the data used to generate the discriminant, the pattern will be classified as either o i or 02. When the arbitrary pattern is far removed from the data used to generate the discriminant, the approach is extremely prone to extrapolation errors. 

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Data interpretation methods can be categorized in terms of whether the input space is separated into different classes by local or nonlocal boundaries. Nonlocal methods include those based on linear and nonlinear projection, such as PLS and BPN. The class boundary determined by these methods is unbounded in at least one direction. Local methods include probabilistic methods based on the probability distribution of the data and various clustering methods when the distribution is not known a priori. 

Work on dimension reduction methods for both input and input-output modeling and for interpretation has produced considerable practical interest, development, and application, so that this family of nonlocal methods is becoming a mainstream set of technologies. This section focuses on dimension reduction as a family of interpretation methods by relating to the descriptions in the input and input-output sections and then showing how these methods are extended to interpretation. 

Alternatively, methods based on nonlocal projection may be used for extracting meaningful latent variables and applying various statistical tests to identify kernels in the latent variable space. Figure 17 shows how projections of data on two hyperplanes can be used as features for interpretations based on kernel-based or local methods. Local methods do not permit arbitrary extrapolation owing to the localized nature of their activation functions. 

The above consideration can be interpreted as deduction of the cyclic cluster model of the infinite crystal when the Hartree-Fock LCAO method (or its semiempirical version with nonlocal exchange) is applied. 

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