Bilinear regression models

The most relevant bilinear regression models obtained for the binding affinity of the oq-AR subtypes are the following ... 

Some improvement in the statistics of these bilinear regression models was achieved with respect to previous linear equations. Moreover, the possibility to generate in a fast way a large pool of QSAR models able to describe a biological... 

The bilinear regression models then express the many input variables X = (xi, X2,..., xa ) in terms of a few factors T = (ti, t2,..., t/i), and these factors are also used for modeling y. This is... 

The coefficients of the regression are all highly significant (p < 0.0001) and the fit of the model to the observed data is shown in Fig. 37.1. Using non-linear regression we obtain the bilinear Hansch model for the bactericidal activities of the 17 phenol analogs (Table 37.2) ... 

The PLSR is a bilinear regression method that extracts a small number of factors, ta, a = 1,2,..., A that are linear combinations of the KX variables, and use these factors as regressors for y. In addition, the X variables themselves are also modeled by these regression factors. Therefore outliers with abnormal spectra x in the calibration set or in future objects can be detected. 

The whole idea behind spectral alignment is to render the large NMR data sets bilinear and thus suitable for subsequent multivariate chemometric exploratory or regression models such as PCA, PLS, and classification ones such as PLS-DA and ECVA. 

Note that the lipophilicity parameter log P is defined as a decimal logarithm. The parabolic equation is only non-linear in the variable log P, but is linear in the coefficients. Hence, it can be solved by multiple linear regression (see Section 10.8). The bilinear equation, however, is non-linear in both the variable P and the coefficients, and can only be solved by means of non-linear regression techniques (see Chapter 11). It is approximately linear with a positive slope (/ ,) for small values of log P, while it is also approximately linear with a negative slope b + b for large values of log P. The term bilinear is used in this context to indicate that the QSAR model can be resolved into two linear relations for small and for large values of P, respectively. This definition differs from the one which has been introduced in the context of principal components analysis in Chapter 17. 

In many chemical studies, the measured properties of the system can be regarded as the linear sum of the fundamental effects or factors in that system. The most common example is multivariate calibration. In environmental studies, this approach, frequently called receptor modeling, was first applied in air quality studies. The aim of PCA with multiple linear regression analysis (PCA-MLRA), as of all bilinear models, is to solve the factor analysis problem stated below ... 

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... 

Eqs. 93 and 94 may be considered as extensions of eqs. 90—92. In contrast to these equations, the bilinear model is generally applicable to the quantitative description of a wide variety of nonlinear lipophilicity-activity relationships. In addition to the parameters that are calculated by linear regression analysis, it contains a nonlinear parameter p, which must be estimated by a stepwise iteration procedure [440, 441]. It should be noted that, due to this nonlinear term, the confidence intervals of a, b, and c refer to the linear regression using the best estimate of the nonlinear term. The additional parameter P is considered in the calculation of the standard deviation s and the F value via the number of degrees of freedom (compare chapter 5.1). The term a in eq. 93 is the slope of the left linear part of the lipophilicity-activity relationship, the value (a — b) corresponds to the negative slope on the right side. 

However, the parabolic model is still valuable for structure-activity analyses. It is the simpler model, easier to calculate, and most often a sufficient approximation of the true structure-activity relationship. The calculation of bilinear equations is relatively time-consuming, as compared to the parabolic model strange results may be obtained in ill-conditioned data sets. On the other hand, in many cases the, bilinear model gives a better description of the data, especially if additional physicochemical parameters are included in the regression equation. The lipophilicity optimum of symmetrical curves is precisely described by both, the parabolic model (optimum log P = — b/2a) and the bilinear model (optimum log P = — log P). In the case of unsymmetrical curves the site of the lipophilicity optimum is described much better by the bilinear model (optimum log P = log a — log p — log (b — a) eq. 93) than by the parabolic model. 

FIGURE 9.2 ThePLSl regression method is shown for modeling one y variable from ZX variables in objects i — 1,2,...,/. The figure shows that the variability not picked up by the A first bilinear factors remain in the residual matrices E and f. Each factor ta represents a linear combination of the X variables. Each latent variable also has a loading vector pa, which show how it related to the individual X variables, and another loading vector qa that relates it to the y variable. 

The underlying notion in bilinear modeling is that something causes the systematic variabilities in the X data. But we may not correctly know what it is there may be surprises in the data due to unexpected interferents, chemical interactions, nonlinear responses, etc. An approximate model of the subspace spanned by these phenomena in X is created. This X model is used for stabilizing the calibration modeling. The PLS regression primarily models the most dominant and most y-relevant of these X phenomena. Thus neither the manifest measured variables nor our causal assumptions about physical laws are taken for granted. Instead we tentatively look for systematic patterns in the data, and if they seem reasonable, we use them in the final calibration model. 

Compared to, say, SMLR the bilinear PLSR calibration model is usually easier to understand The SMLR (and PLSR) regression coefficient vector b is a contrast between all the different phenomena affecting the NIR spectra, so it can be very confusing. The bilinear PLSR allows us to study these NIR phenomena more or less individually, by graphical inspection of the A-dimensional model subspace. 

The terms a, b, and c are calculated using multiple regression analysis while the term requires an iterative procedure for its calculation. The bilinear model has been successfully used for modeling toxicity and bioconcentration in fish. 

PCR method is a two-step process, in which the projection stage is separated and independent from the regression one. As discussed in Section 3.3, this can lead to the drawback that the components that are extracted in the decomposition step, based only on the information about the X-matrix, can be poorly predictive for the Y-block. Starting from these considerations, another method was proposed, PLS regression [2,18,19], in which information in Y is actively used also for the definition of the latent variable space. Indeed, PLS looks for components which compromise between explaining the variation in the X-block and predicting the responses in Y. This corresponds to a bilinear model, which can be summarized mathematically as ... 

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