Multilinear function

Also, note that for the case of linear systems considered in this work, only multihnear terms appear as the non-convex terms in the problem formulation. Detailed description of the convex underestimators for trilinear and multilinear functions in general can be found in [20]. 

Algebraic expressions for terms M and C were derived using Dewar s PMO method (for C in a version similar to the co-technique [57] in order to calculate carbocation stabilization energies). The size factor S is simply a cubic function of the number of carbon atoms [97], The three independent variables of the model were assumed to be linearly related to the experimental Iball indices (vide supra). By multilinear regression analysis (sample size = 26) an equation was derived for calculating Iball indices from the three theoretical parameters. The correlation coefficient for the linear relation between calculated and experimental Iball indices is r = 0.961. 

When spectroscopic intensity is linear in functions of each of k independent variables, a multilinear model can be fit to the k-v/ay array of data. For example, consider absorption measurements made on a specimen containing F components, with wavelength and some environmental variable, such as pH, being the two experimental variables. If the environmental variable alters the relative concentrations of the different components in the specimen without affecting their absorption spectra, then the expected absorbance is described by the bilinear model... 

In Section II we presented the standard general multilinear models, of which the bilinear and the PARAFAC and Tucker2 (T2) trilinear models are most important in spectroscopy. These models contain no information about the specimen except the linear dependence of spectral intensity on functions of each of the independent variables. However, some properties of the specimen are known, and a model that incorporates these known properties is preferred to one that does not. This is particularly true when the model is indeterminate without side conditions. In this section we discuss three settings for the application of knowledge about the specimen identifiable bilinear and T2 submodels, penalized general multilinear models, and submodels in which the dependence of the expected intensity from some components for some ways has a specific mathematical form. 

A class of algorithms which is specialized for multilinear problems is known as alternating least-squares (ALS). Multilinear models are all conditionally linear in a function of each of the three or so independent variables for example, spectral intensity is linear in concentration if the other variables are fixed. Each step of an ALS algorithm fixes the vectors for all but one independent variable, then applies linear regression to select the vectors for the one variable to minimize the error sum of squares. The algorithm cycles among the sets of parameters to be estimated, updating each in turn. Most applications of multilinear models use ALS code. ... 

The attraction of the ALS algorithm for general multilinear models is its use of linear least-squares steps. However, these steps become nonlinear regressions for any way containing a nonlinear parametric model, and most parametric models in spectroscopy will be nonlinear. Thus, the ALS approach is unattractive for most situations in which the dependence of the spectral intensity of any component on any experimental variable is described by a specific mathematical function. 

When a multilinear model is fit to a multiway array, the ways of the array must correspond to independent variables having the property that the observed spectroscopic intensity is separately linear in a function of... 

When the dependence of the spectroscopic intensity from every chro-mophore on at least one experimental variable can be described by a highly specific mathematical function, then the approach known as global analysis is preferred. When this condition is not known to be met, but spectroscopic intensity is separately linear in functions of two or more experimental variables, then the multilinear models described in this chapter are valuable. 

The variation of refractive index (n) as a function of wavelength in the visible region is shown in Figure 9.12. For theoretical comparison, a fit to Cauchy-type formula is made by the multilinear regression method (Ostle 1954, Dayal 1965) using the following formula ... 

Equation (9.69) is the structure function after the idempotences have been eliminated. It is called the multilinear form of the structure function and is a polynomial in which any independent variable figures to the power of 1 only. 

If the primary events, which are represented by the Boolean variables, are independent from one another, they can be replaced by their corresponding probabilities provided the structure function is in its multilinear form (cf. [30]). Thus, the probability of failure or the unavailability of the system is obtained in accordance with the meaning of the probabilities involved. 

The procedure shown by means of the foregoing example holds universally. Any stmcture function may be brought into its multilinear form. The binary variables it contains can then be replaced by the pertinent probabilities. In this way the corresponding reliability parameter of the system described by the structure function is obtained. It was already pointed out that this procedure requires the primary events to be independent from one another. The treatment of dependencies is explained in Sect. 9.6. 

In other words, Eq. (6.18) is satisfied by an appropriate function /(/I) = F( 2 - 11). Obviously, 2 and 2 obey the above relations. The reason for symmetry (6.18) will be explained in Appendix A in terms of a duality transformation well-known in the multilinear algebra literature. In Appendix A one can also find a possible generalization of indices N and Neff. Various examples of Nes and related indices will be given throughout the rest of this chapter. 

Perhaps a superior method of multilinear regression is the so-called p matrix method. Here, we assume that concentration is a function of absorbance, rather than vice versa ... 

Seaton et al. [16] used a multilinear least-squares fitting of the parameters of the assumed PSD function so as to match measured isotherm data. A similar method was employed by Lastoskie et al. [18] in their analysis using the nonlocal density functional theory (NL-DFT). Later, an important contribution toward the numerical deconvolution of the distribution result was made by Olivier et al. [35]. They developed a program based on the regularization method [65], in which no restrictions were imposed on the form of PSD. Moreover, this method was found to be numerically robust. Also, a simpler optimization technique has recently been suggested by Nguyen and Do [66]. 

To depict the material behavior of a part of this kind in optimum terms, the material model must ideally distinguish between elements subjected primarily to tensile loading and those subjected primarily to compressive loading. The IKV has a multilinear elastic material model for 3D volume elements as a subroutine for the ABAQUS software, which allocates the corresponding material behavior (tension or compression) as a function of the distinction described. 

In addition, the potential for using characteristic compression values for part layout is explained taking two examples. It is seen that a material model that makes correct allowance for the load ought to take in not only the level of the load, the loading velocity and the time for which the load acts but also, ideally, the load type (tension or compression). A multilinear elastic material model that is presented can be used to assign the corresponding tensile or compressive characteristic values as a function of the local load type, thus making it possible to optimally exploit the material s potential. 

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