Bilinear model example

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... 

When the underlying model is low-rank bilinear, the data matrices produced are often close to this bilinear model, except for noise and nonlinearities. In such a case, a low-rank bilinear model can be fitted, for example, by curve resolution methods. When certain added conditions such as selective channels, nonnegative parameters etc. are fulfilled, unique models can sometimes be obtained [Gemperline 1999, Manne 1995],... 

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... 

FIGURE 6 Example of MCR for fingerprinting in food analysis. (A) MCR bilinear model related to the multiset structure formed by HPLC-DAD runs of polyphenol standards and samples of three Lambrusco wine varieties. (B) PCA scores and loadings from the matrix of resolved peak areas obtained for aU the compounds resolved by MCR-ALS (the target phenolic compounds and the coeluting unknown compounds). Source. Salvatore et al. [55],... 

This chapter is devoted to the analysis of variable classification and the decomposition of the data reconciliation problem for linear and bilinear plant models, using the so-called matrix projection approach. The use of orthogonal factorizations, more precisely the Q-R factorization, to solve the aforementioned problems is discussed and its range of application is determined. Several illustrative examples are included to show the applicability of such techniques in practical applications. 

Illustration 7.4.5 This example comes from control systems with time delays and is taken from Psarris and Floudas (1990) (see model P3) where bilinear products of the form xl3 y exist in inequality constraints of the form ... 

Illustration 7.4.6 This example is taken from Voudouris and Grossmann (1992), and corresponds to multiple choice structures that arise in discrete design problems of batch processes. The model has bilinear inequality constraints of the form ... 

Remark 1 The resulting optimization model is an MINLP problem. The objective function is linear for this illustrative example (note that it can be nonlinear in the general case) and does not involve any binary variables. Constraints (i), (v), and (vi) are linear in the continuous variables and the binary variables participate separably and linearly in (vi). Constraints (ii), (iii), and (iv) are nonlinear and take the form of bilinear equalities for (ii) and (iii), while (iv) can take any nonlinear form dictated by the reaction rates. If we have first-order reaction, then (iv) has bilinear terms. Trilinear terms will appear for second-order kinetics. Due to this type of nonlinear equality constraints, the feasible domain is nonconvex, and hence the solution of the above formulation will be regarded as a local optimum. 

An example is given for illustration. Consider a spectral data set. The data have been synthesized according to a bilinear three-component part plus a scalar offset of one. Thus, the model is X = + 11 and no noise is added. Using PCA to model the raw... 

With a stack of low-rank bilinear data matrices such as the one above, a three-way array can be built. The underlying model of such an array is not necessarily low-rank trilinear if for example there are retention time shifts from sample to sample. However, the data can be still be fitted by a low-rank trilinear model. If the deviations from the ideal trilinearity (low-rank - one component per analyte) are small, directly interpretable components can be found. If not, components are still found in the same sense as in PCA, where the components together describe the data, although no single component can necessarily be related to a single analyte. 

Figure 3.6 Example of a non-linear log PQ -dependent model (Veith, Call and Brooke, 1983) in comparison to experimental data on fish toxicity (log I/LC50 in mmol/1). The bilinear function provides a good description of the activity data for the compounds with log < log Pow(max.) whereas the model is less well defined for the highly lipophilic substances because their data are extremely variable.

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