Multilinear modeling

Thus, multilinear models were introduced, and then a wide series of tools, such as nonlinear models, including artificial neural networks, fuzzy logic, Bayesian models, and expert systems. A number of reviews deal with the different techniques [4-6]. Mathematical techniques have also been used to keep into account the high number (up to several thousands) of chemical descriptors and fragments that can be used for modeling purposes, with the problem of increase in noise and lack of statistical robustness. Also in this case, linear and nonlinear methods have been used, such as principal component analysis (PCA) and genetic algorithms (GA) [6]. 

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. 

Most conventional filters involve computing local multilinear models, but in certain areas, such as process analysis, there can be spikes (or outliers) in the data which are unlikely to be part of a continuous process. An alternative method involves using... 

The parallel factor analysis (PARAFAC) model [18-20] is based on a multilinear model, and is one of several decomposition methods for a multidimensional data set. A major advantage of this model is that data can be uniquely decomposed into individual contributions. Because of this, the PARAFAC model has been widely applied to 3D and also higher dimensional data in the field of chemometrics. It is known that fluorescence data is one example that corresponds well with the PARAFAC model [21]. 

This is also discussed elsewhere [Smilde 1997], V-PLS can be generalized for higher-order arrays X and Y and in all cases a PARAFAC-like multilinear model is assumed for all the multi-way arrays involved [Bro 1996a],... 

When data can be assumed to be approximately multilinear there is little if any benefit in matricizing the data before analysis. Even though the two-way models describe more variation per definition, the increased modeling power does not necessarily provide more predictive models in terms of modeling either the independent or the dependent variables. Even when the data do not approximately follow a multilinear model (e.g. sensory data), the multilinear models can be preferred if the possible bias in having too simple an X-model is counteracted by the smaller amount of overfit. 

The centering of the Tuckerl model used by MacGregor ensures that the main nonlinear dynamics of the data are removed. Hence, it can be considered as a linearization step, which is a common tool in engineering, and this is a very sensible preprocessing step when linear models are used subsequently. The centering of the Tuckerl model used by Wold is performed over two modes simultaneously, which might introduce extra multilinear components (see Chapter 9). Hence, multilinear models may have problems with Wold s arrangement of the data. 

Retention factors are measured with an approximate constant relative error and this translates to a constant absolute error after the logarithmic transformation [Atkinson 1985], Hence, after these transformations, it is reasonable to apply multilinear models assuming a constant error. 

Multiplicative multilinear models can be used for modeling ANOVA data. Such multilinear models can be interesting, for example, in situations where traditional ANOVA interactions are not possible to estimate. In these situations GEMANOVA can be a feasible alternative especially if a comparably simple model can be developed. As opposed to traditional ANOVA models, GEMANOVA models suffer from less developed hypothesis testing and often the modeling is based on a more exploratory approach than in ANOVA. 

Leurgans SE, Ross RT, Multilinear models applications in spectroscopy, Statistical Science, 1992,7, 289-319. 

Ross RT, Feurgans SE, Component resolution using multilinear models, Methods in Enzymology 1995, 246, 679-700. 

In many circumstances, the spectroscopic properties of a biological specimen are a composite of the properties of several different chromo-phores within the specimen, and the investigator would like to know the properties of the individual chromophores. Physical separation of the chromophores is often difficult, and it may alter important properties of the system being studied. Thus, one looks to mathematical methods for separation of the contributions of the different chromophores. This chapter focuses on one class of mathematical methods, the application of multilinear models. A major advantage of this class of methods is that component resolution may often be achieved with no prior information about the properties of the components. 

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... 

The simplest multilinear models are bilinear models, which can be written 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. 

From the perspective of multilinear modeling, we note that one can use a model in which some components for some ways are described by a parametric model, while other components or other ways continue to be described by a general multilinear model. Such a parametric submodel has fewer parameters and may thus be more parsimonious and more accurate than a general multilinear model. When a parametric model is used for all components of at least one way, use of the resulting multilinear submodel is equivalent to global analysis. 

Whenever one is fitting a model to data, it is helpful to present information about the individual residuals, r = y - fi , in a way that allows the user to get a visual sense of the quality of the fit. When a single experimental variable is used, it is common to plot the individual residuals (divided by the estimated standard deviations if they are available) against the value of that variable. When there are two or more independent variables, as with multilinear models, a similar graph can be made for each... 

Multilinear models are nonlinear models. In our applications, a model with several hundred parameters is fit to a data set of several thousand observations. Fitting a nonlinear model having several hundred parame-... 

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. 

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