Nonlinear projections

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

Nonlinear principal component Nonlinear projection, nonlocal Adaptive shape [a, ], minimum input prediction error... 

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. 

Fig. 10.1. Analysis of die fitness landscape for a basic amino acid dope (30% Arg, 30% Lys, 40% His), (a) Nonlinear projection of die seven-dimensional solution space onto two dimensions by a self-organizing map (SOM) [14]. The seven dimensions are (Tl, Cl, Al, T2, C2, A2, C3), encoding fractions of nucleoddes for each NN(G/C) codon position. The mean squared error between die computed and desired amino acid concentrations are indicated by shades of grey here and by color in die copy of diis figure on die CD diat accompanies diis book.

PCA is not the only projection method that can be used. Various types of nonlinear projections have been employed, e.g., Sammon mapping and nonlinear PCA [80], and several software packages can be used to graphically visualize library distributions and aid compound selection [81]. 

While PCA is a linear projection method, there also exist nonlinear projection methods, e.g. multidimensional scaling [Mardia et al. 1979] and nonlinear PCA [Dong McAvoy 1996], A good overview of nonlinear multivariate analysis tools is given by [Gift 1990],... 

Fig. 1.2. Linear classification is easier in higher dimensional spaces. In 2D on the left it is impossible to find a linear subspace (a straight line) to divide grey and black dots. After a nonlinear projection into the higher-dimensional 3D space, it is easy to find such a linear subspace (a plane).

Kohonen networks or self-organizing maps (SOMs) are obtained by a complex, highly nonlinear mathematical approach for dimensionality reduction [102]. After training those networks produce a 2D-map with regions containing similar molecules. As a result of the complex mathematical formalism, models produced by nonlinear projection approaches are often more accurate maintaining the local environment of a molecule, while straightforward interpretation is less obvious. 

Static and dynamic scattering techniques are spectroscopic characterisation methods in the sense of Sect. 2.2. These techniques evaluate the functional dependency of measurement signals on a spectral parameter, i.e. on time, space, or classically on wavelength or frequency. The major advantage of spectroscopic methods is the reduced sample preparation (no fractionation), but they involve the inversion problem. That is, the spectrum is a—most frequently incomplete and discrete— nonlinear projection of the size distribution. Beside the scattering techniques, there are further spectroscopic methods which are based on the extinction of radiation or on any other response of the particle system to an external field. This section describes optical, acoustic, and electroacoustic methods that have gained relevance for the characterisation of colloidal suspensions. 

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