Regression algorithms

Regression Algorithms. The fitting of structural models to X-ray scattering data requires utilization of nonlinear regression techniques. The respective methods and their application are exhausted by Draper and Smith [270], Moreover, the treatment of nonlinear regression in the Numerical Recipes [154] is recommended. 

Linear least squares regression is the most common method of fitting a response that is a function of a single independent variable. Many nonlinear functions may be transformed to simple linear functions, extending the capabilities of the simplest regression algorithm. 

The full partial least squares regression algorithm is given in Appendix 3.B. NONORTHOGONAL t VECTORS VERSION... 

The background for the extension of two-way partial least squares regression to multiway data (fV-PLS) was provided by Bro [1996] and further elaborated on by Smilde [1997] and de Jong [1998], An improved model of X was later introduced which, however, maintains the same predictions as the original model [Bro et ol. 2001], Only the three-way version of A-PLS is considered here. It differs from the N-way (N > 3) partial least squares regression algorithm in that it has a closed-form solution for the situation with only one dependent variable. 

In extending the PLS regression algorithm to three-way data, the only thing needed is to change the bilinear model of X to a trilinear model of X. For example, the first component in a bilinear two-way model... 

The magnitude of /emp(g) will be referred as the empirical error. All regression algorithms, by minimizing the empirical risk /emp(g), produce an estimate, g(x), which is the solution to the functional estimation problem. 

Characterization of the variogram from actual observations permits the estimation of concentrations at points on the site which were not sampled by application of generalized least-squares type statistical regression algorithms. This type of estimation has come to be referred to as "kriging"(2). Thus, once the similarity of observations with distance has been described in terms of the variogram, contamination across the site can be estimated and... 

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