Based on Nonlinear Projection

Methods based on nonlinear projection are distinguished from the linear projection methods that they transform input data by projection on a nonlin- 

Local methods, on the other hand, are characterized by input transformations that are approached using partition methods for cluster seeking. The overall thrust is to analyze input data and identify clusters of the data that have characteristics that are similar based on some criterion. The objective is to develop a description of these clusters so that plant behaviors can be compared and/or data can be interpreted. 

Because of these characteristics, local methods are naturally used for interpretation. In this section, we present some of the fundamental elements of these methods as input analysis techniques. However, this discussion closely ties with the interpretation discussion in Section V. 

The most commonly used family of methods for cluster seeking uses optimization of a squared-error performance criterion in the form 

This index is employed by both the k-means (MacQueen, 1967) and the isodata algorithms (Ball and Hall, 1965), which partition a set of data into k clusters. With the A -means algorithm, the number of clusters are prespecified, while the isodata algorithm uses various heuristics to identify an unconstrained number of clusters. 

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Methods based on nonlinear projection exploit the nonlinear relationship between the inputs by projecting them on a nonlinear hypersurface resulting in latent variables that are nonlinear functions of the inputs, as shown in Figs. 6b and 6c. If the inputs are projected on a localized hypersurface such as a hypersphere or hyperellipse, then the basis functions are local, depicted in Fig. 6c. Otherwise, the basis functions are nonlocal, as shown in Fig. 6b. 

Fig. 6. Input transformation in (a) methods based on linear projection, (b) methods based on nonlinear projection, nonlocal transformation, (c) methods based on nonlinear projection, local transformation, and (d) partition-based methods. (From Bakshi and Utojo, 1998.)...

Techniques for multivariate input analysis reduce the data dimensionality by projecting the variables on a linear or nonlinear hypersurface and then describe the input data with a smaller number of attributes of the hypersurface. Among the most popular methods based on linear projection is principal component analysis (PCA). Those based on nonlinear projection are nonlinear PCA (NLPCA) and clustering methods. 

Methods based on linear projection transform input data by projection on a linear hyperplane. Even though the projection is linear, these methods may result in either a linear or a nonlinear model depending on the nature of the basis functions. With reference to Eq. (6), the input-output model for this class of methods is represented as... 

Input-output analysis methods that project the inputs on a nonlocal hypersurface have also been developed, such as BPNs with multiple hidden layers and regression based on nonlinear principal components. 

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. 

Nonlinear methods based on linear projection also can be used for data interpretation. Since these methods require numeric inputs and outputs, the symbolic class label can be converted into a numeric value for their training. Proposed applications involving numeric to symbolic transformations have a reasonably long history (e.g., Hoskins and Himmel-... 

The estimated ultimate load of pile foundation is an important project and involves a number of challenges from the geotechnical and structural safety viewpoint. The strength reduction method of pile foundation and estimation criterion of ultimate load is studied based on nonlinear finite element and cusp catastrophe theory. The finite element limit analysis of pile is performed using the single reduction factor and two reduction factors of strength reduction method and the criterion of the cusp catastrophe curve method, and has been shown to be a reliable and objective method for estimating the ultimate load of pile. 

Fig. 5 Determination of LOD based on the projected standard deviation at zero concentration (So). The LOD is calculated as the concentration that corresponds to a response three times the value of Sq. Although the relationship between concentration and standard deviation shown here is linear, nonlinear relationships between these two variables are likely.

To describe the X-ray imaging system the projection of 3D object points onto the 2D image plane, and nonlinear distortions inherent in the image detector system have to, be modelled. A parametric camera model based on a simple pinhole model to describe the projection in combination with a polynomal model of the nonlinear distortions is used to describe the X-ray imaging system. The parameters of the model are estimated using a two step approach. First the distortion parameters for fixed source and detector positions are calculated without any knowledge of the projection parameters. In a second step, the projection parameters are calculated for each image taken with the same source and detector positions but with different sample positions. 

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. 

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