Bilinear modeling

The transformation by log double-centering has received various names among which spectral mapping [13], logarithmic analysis [14], saturated RC association model [15], log-bilinear model [16] and spectral map analysis or SMA for short [17]. 

According to Andersen [12] early applications of LLM are attributed to the Danish sociologist Rasch in 1963 and to Andersen himself. Later on, the approach has been described under many different names, such as spectral map analysis [13,14] in studies of drug specificity, as logarithmic analysis in the French statistical literature [15] and as the saturated RC association model [16]. The term log-bilinear model has been used by Escoufier and Junca [ 17]. In Chapter 31 on the analysis of measurement tables we have described the method under the name of log double-centred principal components analysis. 

In this case, the bilinear model of eq. (37.10) fits the data as well as the parabolic one. A possible reason for the lack of improvement is that the part of the model which accounts for the higher values of log P is not well covered by the data (Fig. 37.1). The parabolic model yields an optimal value for log P of 5.10 while the optimum of the bilinear model is found at 5.18. 

Figures 7.31a-c clearly show that after some critical soy content in dodecane, Pe values decrease with increasing soy, for both sink and sinkless conditions. [This is not due to a neglect of membrane retention, as partly may be the case in Fig. 7.23 permeabilities here have been calculated with Eq. (7.21).] Section 7.6 discusses the Kubinyi bilinear model (Fig. 7.19d) in terms of a three-compartment system water, oil of moderate lipophilicity, and oil of high lipophilicity. Since lipo-some(phospholipid)-water partition coefficients (Chapter 5) are generally higher than alkane-water partition coefficients (Chapter 4) for drug-like molecules, soy lecithin may be assumed to be more lipophilic than dodecane. It appears that the increase in soy concentration in dodecane can be treated by the Kubinyi analysis. In the original analysis [23], two different lipid phases are selected at a fixed ratio (e.g., Fig. 7.20), and different molecules are picked over a range of lipophilicities.

Depending on the data structure, different types of models are possible to be applied for data analysis. Thus, when data are ordered in one direction, linear univariant models can be applied (see (1)), and nonlinear models as well (see (2)). For data ordered in two directions, bilinear models can be applied (see (3)) or nonbilinear models. Finally, for data ordered in three directions, trilinear models can be applied (see (4)) or, failing that, nontrilinear models. 

Fig. 2 PCA decomposition of D data matrix following a bilinear model for a number of components N = 3...

PCA attempts to summarize the relevant information contained in the analyzed data matrix, distinguishing it from noise or error. Mathematically, the original (experimental) data matrix D is decomposed, following a bilinear model, into the product of two orthogonal data matrices, X and YT (see Fig. 2)... 

In this equation, whereas the same loading matrix (YT matrix) is common for the different individual data matrices Dt, k = 1, 2, 3, 4, four different score matrices Xjt, k = 1, 2, 3, 4 are considered to explain the variation in Daug. Since these four D. matrices have equal sizes (same number of rows or samples and of columns or variables) they can also be arranged in a three-way data cube, with the four data matrices in the different slabs of this cube. However, in the frame of the MCR-ALS method and of the general bilinear model in (10), it is preferable to consider them to be arranged in the column-wise augmented data matrix Daug. 

Kubinyi ( ) used distribution coefficients of the same series of compounds with his bilinear model for absorption to obtain an even closer correlation. Equation 5. pis a constant related to the model. 

We ve looked at many other analyses of "simple" processes, but my favorite is a correlation for the absorption of an acid and a base in the same equation, — not only that, but with each one at six different pH s, (Figure 2). The data are from Schurmann Turner ( ) the base is propranolol and the acid, 4-n-hexylphenylacetic. Only a single parameter is required, log D, Equation 6. Kubinyi s bilinear equation gives an even better correlation ( ), Equation 7. (This is a special version of the bilinear model which sets the coefficients of each term equal.)... 

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 ... 

The PCR method (and PLS, partial least squares, discussed in the Section 6.7) assumes that the linear relation (Equation 6.10) between the x- and y-variables is in fact a bilinear model that depends on scores t ... 

FIGURE 11.11 Bilinear models for three-way data, unfolded PCA and unfolded MCR. 

This second-level modeling of the feedback mechanisms leads to nonlinear models for processes, which, under some experimental conditions, may exhibit chaotic behavior. The previous equation is termed bilinear because of the presence of the b [y (/,)] r (I,) term and it is the general formalism for models in biology, ecology, industrial applications, and socioeconomic processes [601]. Bilinear mathematical models are useful to real-world dynamic behavior because of their variable structure. It has been shown that processes described by bilinear models are generally more controllable and offer better performance in control than linear systems. We emphasize that the unstable inherent character of chaotic systems fits exactly within the complete controllability principle discussed for bilinear mathematical models [601] additive control may be used to steer the system to new equilibrium points, and multiplicative control, either to stabilize a chaotic behavior or to enlarge the attainable space. Then, bilinear systems are of extreme importance in the design and use of optimal control for chaotic behaviors. We can now understand the butterfly effect, i.e., the extreme sensitivity of chaotic systems to tiny perturbations described in Chapter 3. 

M. D Zmura and G. Iverson, A Formal Approach to Color Constancy The Recovery of Surface and Light Source Spectral Properties Using Bilinear Models, in C. Dowling, F. Roberts, and P. Theuns, Eds., Recent Progress in Mathematical Psychology, Lawrence Erlbaum, Mahwah, NJ, 99-132 (1998). 

G. Iverson and M. D Zmura, Color Constancy Spectral Recovery Using Trichromatic Bilinear Models, in R. D. Luce, M. D Zmura, D. D. Hoffman, G. Iverson, and K. Romney, Eds., Geometric Representations of Perceptual Phenomena, Lawrence Erlbaum, Mahawah, NJ, 169-185 (1995). 

The delineation of these models led to explosive development in QSAR analysis and related approaches. The Kubinyi bilinear model is a refinement of the parabolic model and, in many cases, it has proved to be superior (21). 

The ascending and descending slopes are equal (== l)and linear. However, a major drawback of this model is that it forces the activity curves to maximize at logP = 0. These studies woe extended by Kubinyi, who developed the elegant and powerful bilinear model, which is superior to the parabolic model and is extensively used in QSAR studies (192). 

The bilinear model has been used to model biological interactions in isolated receptor systems and in adsorption, metabolism, elimina- tion, and toxicity studies, although it has a few limitations. These include the need for at least 15 data points (because of the presence of the additional disposable parameter jS and data points beyond optimum LogP. If the range in values for the dependent variable is limited, unreasonable slopes are obtained. 

It is apparent from QSAR 1.106 and 1.107, that the hydrophobicrequirements of the substrates vary considerably. As expected, renal clearance is enhanced in the case of hydrophilic drugs, whereas nonrenal clearance shows a strong dependency on hydrophobic-ity. Note that QSAR 1.107 is stretching the limits of the bilinear model with only 10 data points The 95% confidence intervals are also large but, nevertheless, the equations serve to emphasize the difference in clearance mechanisms that are clearly linked to hydrophobicity. 

Besides the nonlinear models and, specifically, the parabolic model, other models were proposed for nonlinear dependence of the biological response from hydrophobic interactions. Among them, the most important are the Hansel bilinear models [Kubinyi, 1977 Kubinyi, 1979] such as ... 

Special cases of such bilinear models are the McFarland model [McFarland, 1970], where b2 = 2b and p = 1 and the Higuchi-Davis model [Higuchi and Davis, 1970], where 2 = 1 and P = Vup / Vaq, which is the ratio between the volume of the lipid phase Viip and the volume of the aqueous phase Vaq. 

Kubinyi, H. (1977). Quantitative Structure-Activity Relationships. 7. The Bilinear Model, a New Model for Nonlinear Dependence of Biological Activity on Hydrophobic Character. J.Med. Chem., 20,625-629. 

If we had a theory for this factor, we could calculate quantum relaxation rates using computed classical correlation functions. For the bilinear model (13.11) we get, using (13.29) and (13.30)... 

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