Multilinear form

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. 

The variables that are combined hnearly are In / 17T, and In C, Multilinear regression software can be used to find the constants, or only three sets of the data smtably spaced can be used and the constants found by simultaneous solution of three linear equations. For a linearized Eq. (7-26) the variables are logarithms of / C, and Ci,. The logarithmic form of Eq. (7-24) has only two constants, so the data can be plotted and the constants read off the slope and intercept of the best straight line. 

POLYMATH multilinear regression handles the rate equation in linearized form, In r = a - b/T + q ln(C/C0)... 

Use POLYMATH multilinear regression with the equation in the form,... 

Treat the linearized form by POLYMATH multilinear regression,... 

Multilinear Regression Analysis. As an entry to the problem we have selected simple gas phase reactions involving proton or hydride ion transfer which are influenced by only a few effects and for which reactivity data of high accuracy are available. In these situations where a larger set of numerial data are available multilinear regression analysis (MLRA) was applied. Thus, the simplest mathematical form, a linear equation is chosen to describe the relationship between reactivity data and physicochemical factor. The number of parameters (factors) simultaneously applied was always kept to a minimum, and a particular parameter was only included in a MLRA study if a definite indication of its relevance existed. 

Perform experimental tests on this subset of compounds and then use some form of modelling to relate the desired activity to structural data. Note that this modelling does not have to be multilinear modelling as discussed in this section, but could also be PLS (partial least squares) as introduced in Chapter 5. 

Multilinear regression can be used to find the constants k0, E, and n. For constant-temperature (isothermal) data, Eq. (7-167) can be simplified by using the Arrhenius form as... 

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. 

Staying with this example, what if the absorption spectrum of a component varies with some other experimental variable, such as temperature This is a situation in which the application of a multilinear model is problematic. Either of two adjustments in the situation might allow both wavelength and temperature to be used as independent variables in a multilinear model (1) If the dependence of its absorption spectrum on temperature is because a component is shifting between discrete low-temperature and high-temperature forms, then each form can properly be regarded as a separate component whose absorption spectrum is independent of temperature. (2) Alternatively, it may be necessary to restrict the temperatures used to a sufficiently narrow range that the absorption spectrum is essentially constant. 

The following fourth-order example contains non-convexities in the form of multilinear... 

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]. 

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