Non-linear transformation

The transformation from a set of Cartesian coordinates to a set of internal coordinates, wluch may for example be distances, angles and torsional angles, is an example of a non-linear transformation. The internal coordinates are connected with the Cartesian coordinates by means of square root and trigonometric functions, not simple linear combinations. A non-linear transformation will affect the convergence properties. This may be illustrate by considering a minimization of a Morse type function (eq. (2.5)) with D = a = ] and x = AR. 

We will consider two other variables obtained by a non-linear transformation y = e and z = e. The minimum energy is at x = 0, corresponding to y = z = 1. Consider now an NR optimization starting at x = -0.5, corresponding to y = 1.6487 and z = 0.6065. Table 14.1 shows that the NR procedure in the x-variable requires four iterations before X is less than 10"". In the y-variable the optimization only requires one step to reach the y = 1 minimum exactly The optimization in the z-variable takes six iterations before the value is within 10"" of the minimum. 

Non-linear PCA can be obtained in many different ways. Some methods make use of higher order terms of the data (e.g. squares, cross-products), non-linear transformations (e.g. logarithms), metrics that differ from the usual Euclidean one (e.g. city-block distance) or specialized applications of neural networks [50]. The objective of these methods is to increase the amount of variance in the data that is explained by the first two or three components of the analysis. We only provide a brief outline of the various approaches, with the exception of neural networks for which the reader is referred to Chapter 44. 

In an early application to butadiene [16], and later to the ground and excited states of benzene [17], Berry analyzed MO-based wavefunctions using valence bond concepts, simply by considering the overlaps with nonorthogonal VB structures. Somewhat closer than this to a CASVB type of approach, are the procedures employed by Linnett and coworkers, in which small Cl wavefunctions were transformed (exactly) to nonorthogonal representations [18-20]. The main limitation in their case was on the size of systems that may be treated (the authors considered no more than four-electron systems), both because this non-linear transformation must exist, and because it must be possible to obtain it with reasonable effort. 

The co-ordinate transformation from the J9 to the Qr (or qr) is a non-linear transformation which we may write in the form... 

E >Ncu(iiilly dll piugiitiiis i d.li iild.Lc llic fiuuliiiiiciitill piupcilics, (lit We will consider two other variables obtained by a non-linear transformation y = e ... 

First, we define the necessary and sufficient conditions for exact nonlinear lumping. If we start with the system of equations (4.1) then the new lumped variables are defined by an fl-dimensional non-linear transformation c = h(c) and the new n-dimensional equation system becomes... 

Finding canonical forms for each of the A s, is an easier task than for any general form A. A technique has been developed at Princeton [189-191] for finding such canonical forms using an algebraic method in nonlinear perturbation theory. For the application of this technique the leading operator Aq must be chosen to be in canonical form. The basic approach is then to find a non-linear transformation C such that the resultant operator. 

These examples show that non-linear transformations may strongly affer.t the... 

Last, the mathemical formalism behind the REPSWA method is discussed. REPSWA is based on the non-linear transformation,... 

The original 1931 CIE Y, x, y system of colour measurement is not visually uniform (Fig. 3.4a). Constant hue and chroma are distorted and equal visual distances increase several-fold fi om purple-red to green. Improved spacing has been accomplished by both linear and non-linear transformations of Y, x, y (Bems 2000). Near uniform colour spaces of practical importance are the Hunter and the CIELUV and CIELAB spaces. In the Hunter (1958) Z, a, b colour space the lightness co-ordinate L is the square root of the tristimulus value T, and a, and b are the red/green and yellow/blue opponent co-ordinates. The 1976... 

PPR uses a projection technique similar to PCR and PLS which creates a smaller space into which the original response data are projected. However, these new vectors are found through an iterative technique since each newly formed vector is transformed using a non-linear transformation function, ... 

The basic feedforward neural network performs a non-linear transformation of the input data in order to approximate the output data. This net is composed of many simple, locally interacting, computational elements (nodes/neurons), where each node works as a simple processor. The schematic diagram of a single neuron is shown in Fig 1. The input to each i-th neuron consists of a A-dimensional vector X and a single bias (threshold) bj. Each of the input signals Xj is weighted by the appropiate weight Wij, where] = 1- N. 

In this type of enhancement, the input and output data values follow a non-linear transformation. 

Wang et al proposed a multivariate dominant factor based non-linearized PLS model for LIBS measurements. In constructing such a multivariate model, non-linear transformation of multi-characteristic line intensities according to the physical mechanisms of a laser-induced plasma spectrum were made, combined with a linear-correlation-based PLS method, to model the non-linear self-absorption and inter-element interference effects. Moreover, a secondary PLS was applied, utilizing information from the whole spectrum to correct the model results further. The proposed method showed a significant improvement when compared with a conventional PLS model. Even compared with the already improved baseline dominant-factor-based PLS model, the PLS model based on the multivariate dominant factor yielded the same calibration quality while decreasing the RMSEP. 

In the present paper we have presented a new Monte Carlo method where only the extreme simulation results are stored and analyzed. In the numerical examples the exceedance probability was chosen to be 0.001, and the condition for discarding sampled points were chosen accordingly. For even smaller exceedance probabilities this condition can easily be adjusted so that we always have sufficient data in the area of the contour. Using this new method it becomes very easy to obtain numerically as well as statistically stable results even for very low exceedance probabilities without having to store and analyze millions of simulation results. Contrary to the transformation method the Monte Carlo method avoids bias resulting from non-linear transformations. As seen from the examples this bias can be substantial. 

Another approach to handling non-linear relationships is an appropriate non-linear transformation of the features, preferably based on pre-knowledge about the data. This method for instance has been successfully applied to model mass spectra-structure relationships. Instead of the original spectral peak heights new features were calculated by applying spectroscopic interpretation rules. The resulting spectral features are often more simply (and linearly) correlated with chemical structures than the original peak data. 

Shanks, D. Non-linear transformations of divergent and slowly convergent sequences and an example from hydrodynamics. Phys. Rev. 76, 876-876 (1949)... 

---