Linear projection methods

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

Four pairs of structures with identical descriptors merge at a distance of zero. From the chemist s point of view clustering appears more satisfying than the linear projection method PCA (with only 47.6% of the total variance preserved by the first two PCA scores). A number of different clustering algorithms have been applied to the 20 standard amino acids by Willet (1987). 

An alternative method would be to use complete spectra and hope that discrepancies between the linear calibration model and the real world data are leveraged out by the large pool of spectral information. For example, the PLS method is said to be suitable for such a type of P-matrix analysis. However, deterioration of measured spectra is more likely to be attributed to systematic physical effects than to uncorrelated random noise. Despite excellent results obtained using full spectra PLS calibrations and predictions compared to other linear projection methods, prediction errors can stiQ be significantly higher than for... 

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],... 

Although the result of the example mentioned above is rather satisfactory, the PCA projection method, as a linear projection method, cannot be used for the problems having significant nonlinearity. The following section will consider an example of such kind. 

Methods based on linear projection exploit the linear relationship among inputs by projecting them on a linear hyperplane before applying the basis function (see Fig. 6a). Thus, the inputs are transformed in combination as a linear weighted sum to form the latent variables. Univariate input analysis is a special case of this category where the single variable is projected on itself. 

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... 

PPR is a linear projection-based method with nonlinear basis functions and can be described with the same three-layer network representation as a BPN (see Fig. 16). Originally proposed by Friedman and Stuetzle (1981), it is a nonlinear multivariate statistical technique suitable for analyzing high-dimensional data, Again, the general input-output relationship is again given by Eq. (22). In PPR, the basis functions 9m can adapt their shape to provide the best fit to the available data. 

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. 

With regard to linear projection based methods, the latent variables or scores determined by linear multivariate statistical methods such as... 

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-... 

Here xik is an estimated value of a variable at a given point in time. Given that the estimate is calculated based on a model of variability, i.e., PCA, then Qi can reflect error relative to principal components for known data. A given pattern of data, x, can be classified based on a threshold value of Qi determined from analyzing the variability of the known data patterns. In this way, the -statistic will detect changes that violate the model used to estimate x. The 0-statistic threshold for methods based on linear projection such as PCA and PLS for Gaussian distributed data can be determined from the eigenvalues of the components not included in the model (Jack-son, 1992). 

Bakshi, B. R., and Utojo, U., Unification of neural and statistical methods that combine inputs by linear projection, Comput Chem. Eng. 22(12), 1859-1878 (1998). 

Variational electrostatic projection method. In some instances, the calculation of PMF profiles in multiple dimensions for complex chemical reactions might not be feasible using full periodic simulation with explicit waters and ions even with the linear-scaling QM/MM-Ewald method [67], To remedy this, we have developed a variational electrostatic projection (VEP) method [75] to use as a generalized solvent boundary potential in QM/MM simulations with stochastic boundaries. The method is similar in spirit to that of Roux and co-workers [76-78], which has been recently... 

Gill, P. E. W. Murray M. A. Saunders J. A. Tomlin and M. H. Wright. On Projected Newton Barrier Methods for Linear Programming and an Equivalence to Karmarkar s Projective Method. Math Program 36 183-191 (1986). 

Rosen, J. B. (1961). The gradient projection method for non-linear programming, Part II, Non-linear constraints. J. Soc. Indus. Appl. Math., 9, 414-32. 

The described projection method with scores and loadings holds for all linear methods, such as PCA, LDA, and PLS. These methods are capable to compress many variables to a few ones and allow an insight into the data structure by two-dimensional scatter plots. Additional score plots (and corresponding loading plots) provide views from different, often orthogonal, directions. 

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