Nonlinear transformation

The 1976 CIELAB Color Space. Defiaed at the same time as the CIELUV space, the CIELAB space, propedy designated CIE E i , is a nonlinear transformation of the 1931 CIE X, Y, Z space. It also uses the metric lightness coordinate E, together with ... 

A key featui-e of MPC is that a dynamic model of the pi ocess is used to pi-edict futui e values of the contmlled outputs. Thei-e is considei--able flexibihty concei-ning the choice of the dynamic model. Fof example, a physical model based on fifst principles (e.g., mass and energy balances) or an empirical model coiild be selected. Also, the empirical model could be a linear model (e.g., transfer function, step response model, or state space model) or a nonhnear model (e.g., neural net model). However, most industrial applications of MPC have relied on linear empirical models, which may include simple nonlinear transformations of process variables. 

Numerical calculations were carried out in order to test the whole numerical algorithm and accuracy of the rate calculation. The potential system employed is a nonlinearly transformed model of the separable case [31]. That is... 

Numerical calculations were performed after applying the following nonlinear transformation,... 

There are also forms of nonlinear PCR and PLS where the linear PCR or PLS factors are subjected to a nonlinear transformation during singular value decomposition the nonlinear transformation function can be varied with the nonlinearity expected within the data. These forms of PCR/PLS utilize a polynomial inner relation as spline fit functions or neural networks. References for these methods are found in [7], A mathematical description of the nonlinear decomposition steps in PLS is found in [8],... 

The criterion of mean-unbiasedness seems to be occasionally overemphasized. For example, the bias of an MLE may be mentioned in such a way as to suggest that it is an important drawback, without mention of other statistical performance criteria. Particularly for small samples, precision may be a more important consideration than bias, for purposes of an estimate that is likely to be close to the true value. It can happen that an attempt to correct bias results in lowered precision. An insistence that all estimators be UB would conflict with another valuable criterion, namely parameter invariance (Casella and Berger 1990). Consider the estimation of variance. As remarked in Sokal and Rohlf (1995), the familiar sample variance (usually denoted i ) is UB for the population variance (a ). However, the sample standard deviation (s = l is not UB for the corresponding parameter o. That unbiasedness cannot be eliminated for all transformations of a parameter simply results from the fact that the mean of a nonlinearly transformed variable does not generally equal the result of applying the transformation to the mean of the original variable. It seems that it would rarely be reasonable to argue that bias is important in one scale, and unimportant in any other scale. 

Exercise. Apply to (1.1) the nonlinear transformation y = 4>(y) and show that the transformed density P(y, t) obeys a Fokker-Planck equation with coefficients... 

If one applies formally the transformation to the meaningless equation (4.5) one also obtains (4.13). Hence the connection between (4.5) and (4.8) is invariant for nonlinear transformations of the variable y. 

Exercise. The deterministic equation obtained by the naive device of dropping the second term from the Fokker-Planck equation is not invariant for nonlinear transformations of x. 

M. J. Feigenbaum discovered a remarkable regularity in period doublings whereby the ratio of the parameter intervals between successive doublings approaches a universal constant 4.669. See M. J. Feigenbaum, Quantitative universality for a class of nonlinear transformations. J. Stat. Phys. 19,25-52 (1978). 

Feigenbaum, M. J., 1978, Quantitative university for a class of nonlinear transformations. J. Slat. Phys. 19, 25-52. 

Image transformations are operations that alter the value of pixels in an image. Transformation results do not depend on the value of neighboring pixels. These include simple linear transformations such as image rotation, translation, and reflection that may be required for correlation of images acquired using different techniques, as well as nonlinear transformations such as shearing, which is used to skew objects. 

For nonlinear systems the solution of the governing equations must generally be obtained numerically, but such solutions can be obtained without undue difficulty for any desired rate expression with or without axial dispersion. The case of a Langmuir system with linear driving force rate expression and negligible axial dispersion is a special case that is amenable to analytical solution by an elegant nonlinear transformation. 

This "self-scaling idea was further developed by Spedicato 15 who considered formulae which were invariant to a scalar nonlinear transformation of f(x), and this also generalizes other attempts to approximate f(x) using more general classes than quadratic functions 16,17,18,19. ... 

Nonlinear Transformations of Fuzzy Electron Density Fragments... 

Whereas for the general case of five or more nuclei within the fragment no linear homotopy exists that can interconvert two arbitrary sets of nuclear coordinates, nevertheless, there are infinitely many nonlinear transformations which can accomplish this. 

Two simple choices for nonlinear transformations, the DER method, and the WAT method, are described below. In both of these techniques, the main step of the transformation is linear, and nonlinearity is included in a rather transparent way. 

Based on these vectors, a simple, nonlinear transformation is defined that places all the n nuclei to the required locations and also transforms the electronic density so that it follows the nuclear distortion. The transformation for the electron density is not unique and is coordinate-dependent for most dimensions, however, the coordinate dependence is small if the distortion of the nuclear arrangement is small. The transformation based on this approach is expected to provide the best results if the dimension of vectors a( ) is 3 (equivalent to the case of linear homotopy for four nuclei, discussed in section 4.5), or 9 (ten nuclei), or 19 (20 nuclei), and in general, for cases where all coordinate products up to a given overall degree are included as components of the vectors a( ). 

Although the terms in Eq. [1] are linear, a nonlinear transformation of one or more simple descriptors may also be included. For example, a nonlinear term might be volume squared, polarizability divided by volume, an electrostatic potential times a charge, or some other term involving a product, division, exponentiation, or logarithm of the descriptors. In effect, the nonlinear terms can be introduced into a linear regression. 

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