The Multivariate Approach

Some methods that paitly cope with the above mentioned problem have been proposed in the literature. The subject has been treated in areas like Cheraometrics, Econometrics etc, giving rise for example to the methods Partial Least Squares, PLS, Ridge Regression, RR, and Principal Component Regression, PCR [2]. In this work we have chosen to illustrate the multivariable approach using PCR as our regression tool, mainly because it has a relatively easy interpretation. The basic idea of PCR is described below. 

This chapter constitutes an attempt to demonstrate the utility of multivariate statistics in several stages of the scientific process. As a provocation, it is suggested that the multivariate approach (in experimental design, in data description and in data analysis) will always be more informative and make generalizations more valid than the univariate approach. Finally, the multivariate strategy can be really enjoyable, not the least for its capacity to reveal hidden treasures in data that in a univariate analysis look like a set of random numbers. 

CONTENTS 1. Chemometrics and the Analytical Process. 2. Precision and Accuracy. 3. Evaluation of Precision and Accuracy. Comparison of Two Procedures. 4. Evaluation of Sources of Variation in Data. Analysis of Variance. 5. Calibration. 6. Reliability and Drift. 7. Sensitivity and Limit of Detection. 8. Selectivity and Specificity. 9. Information. 10. Costs. 11. The Time Constant. 12. Signals and Data. 13. Regression Methods. 14. Correlation Methods. 15. Signal Processing. 16. Response Surfaces and Models. 17. Exploration of Response Surfaces. 18. Optimization of Analytical Chemical Methods. 19. Optimization of Chromatographic Methods. 20. The Multivariate Approach. 21. Principal Components and Factor Analysis. 22. Clustering Techniques. 23. Supervised Pattern Recognition. 24. Decisions in the Analytical Laboratory. 

The main characterizing strategies are the multivariate approach to the problem, searching for relevant information, model validation to build models with predictive power, comparison of the results obtained by using different methods, and definition and use of indices capable of measuring the quality of extracted information and the obtained models. 

Albert A, Heusghem C. Relating observed values to reference values The multivariate approach. In Grasbeck R, Alstrom T, eds. Reference values in laboratory medicine. Chichester, England John Wdey, 1981 289-96. 

For these reasons, multivariate approaches seem better. The multivariate approaches, the topic of this chapter, can be further divided into sequential and simultaneous strategies (7-9). In sequential optimization strategies, initially only a few experiments are performed and their results are used to define the next experiment(s) (7, 8,10). In simultaneous approaches, a predefined number of experiments are performed according to a well-defined experimen-... 

Statistical tests make possible the automatic detection of an outlier in both cases (they are defined as outliers in the first case and Q outliers in the second case). With these simple tests it will be possible to detect a fault in a process or to reject a bad product by checking just two plots, instead of as many plots as variables as in the case of the Sheward charts commonly used when the univariate approach is applied. Furthermore, the multivariate approach is much more robust, since it will lead to a lower number of false negatives and false positives, and much more sensitive, since it allows the detection of faults at an earlier stage. Finally, the contribution plots will easily outline which variables are responsible for the sample being an outlier. 

Benigni, R., and Giuliani, A. (1994). Quantitative modeling and biology The multivariate approach. Am. J. Physiol KKfS, PI. 2), R1697-R1704. 

A multivariate approach of analyzing the psychophysiological measures obtained from heart period has been proposed as a method for obtaining cardiac autonomic information (Backs, 1995,1998). The multivariate approach attempts to improve the sensitivity and diagnosticity of heart rate by identifying the neurogenic activity of the sympathetic and parasympathetic nervous systems responsible for the observed heart rate in a task. Principal components analysis (PCA) was used in the present study to extract information about the sympathetic and parasympathetic nervous systems common to RSA, low-frequency HRV, residual heart period, and heart period. Details of how the components were derived are presented in the Method section. 

There are significant advantages in using the multivariate approach ... 

Compared to classical measurement series restricted to a single variable, the multivariate approach yields a considerably greater amount of information for object characterization. This advantage. however, is connected with a loss of direct interpretability. If only two variables were measured, the whole data structure might be still visualized in a diagram, but this approach fails for larger problems. That is why the increase in information must be accompanied by some efficient tool of data reduction that enables interpretation... 

The first approach consists of assuming some multivariate distribution model for the random function P(x), xeA A convenient... 

Multivariate analysis of these different types of measurements (heterogeneous, homogeneous, compositional, ordered) may require special approaches for each of them. For example, compositional tables that are closed with respect to the rows, require a different type of analysis than heterogeneous tables where the columns are defined with different units. The basic approach of principal components... 

The aim of all the foregoing methods of factor analysis is to decompose a data-set into physically meaningful factors, for instance pure spectra from a HPLC-DAD data-set. After those factors have been obtained, quantitation should be possible by calculating the contribution of each factor in the rows of the data matrix. By ITTFA (see Section 34.2.6) for example, one estimates the elution profiles of each individual compound. However, for quantitation the peak areas have to be correlated to the concentration by a calibration step. This is particularly important when using a diode array detector because the response factors (absorptivity) may considerably vary with the compound considered. Some methods of factor analysis require the presence of a pure variable for each factor. In that case quantitation becomes straightforward and does not need a multivariate approach because full selectivity is available. 

Several approaches have been investigated recently to achieve this multivariate calibration transfer. All of these require that a small set of transfer samples is measured on all instruments involved. Usually, this is a small subset of the larger calibration set that has been measured on the parent instrument A. Let Z indicate the set of spectra for the transfer set, X the full set of spectra measured on the parent instrument and a suffix Aor B the instrument on which the spectra were obtained. The oldest approach to the calibration transfer problem is to apply the calibration model, b, developed for the parent instrument A using a large calibration set (X ), to the spectra of the transfer set obtained on each instrument, i.e. and Zg. One then regresses the predictions (=Z b ) obtained for the parent instrument on those for the child instrument yg (=Z b ), giving... 

In recent years there has been much activity to devise methods for multivariate calibration that take non-linearities into account. Artificial neural networks (Chapter 44) are well suited for modelling non-linear behaviour and they have been applied with success in the field of multivariate calibration [47,48]. A drawback of neural net models is that interpretation and visualization of the model is difficult. Several non-linear variants of PCR and PLS regression have been proposed. Conceptually, the simplest approach towards introducing non-linearity in the regression model is to augment the set of predictor variables (jt, X2, ) with their respective squared terms (xf,. ..) and, optionally, their possible cross-product... 

There are four main types of data that frequently occur in sensory analysis pair-wise differences, attribute profiling, time-intensity recordings and preference data. We will discuss in what situations such data arise and how they can be analyzed. Especially the analysis of profiling data and the comparison of such data with chemical information calls for a multivariate approach. Here, we can apply some of the techniques treated before, particularly those of Chapters 35 and 36. 

A table of correlations between the variables from the instrumental set and variables from the sensory set may reveal some strong one-to-one relations. However, with a battery of sensory attributes on the one hand and a set of instrumental variables on the other hand it is better to adopt a multivariate approach, i.e. to look at many variables at the same time taking their intercorrelations into account. An intermediate approach is to develop separate multiple regression models for each sensory attribute as a linear function of the physical/chemical predictor variables. 

Sets of spectroscopic data (IR, MS, NMR, UV-Vis) or other data are often subjected to one of the multivariate methods discussed in this book. One of the issues in this type of calculations is the reduction of the number variables by selecting a set of variables to be included in the data analysis. The opinion is gaining support that a selection of variables prior to the data analysis improves the results. For instance, variables which are little or not correlated to the property to be modeled are disregarded. Another approach is to compress all variables in a few features, e.g. by a principal components analysis (see Section 31.1). This is called... 

Wavelength database libraries of >32000 analytical lines can be used for fast screening of the echellogram. Such databases allow the analyst to choose the best line(s) for minimum interferences, maximum sensitivity and best dynamic range. Further extension of the wavelength range (from 120 to 785 nm) is desirable for alkali metals, Cl, Br, Ga, Ge, In, B, Bi, Pb and Sn, and would allow measurement of several emission lines in a multivariate approach to spectral interpretation [185]. 

A more subjective approach to the multiresponse optimization of conventional experimental designs was outlined by Derringer and Suich (22). This sequential generation technique weights the responses by means of desirability factors to reduce the multivariate problem to a univariate one which could then be solved by iterative optimization techniques. The use of desirability factors permits the formulator to input the range of property values considered acceptable for each response. The optimization procedure then attempts to determine an optimal point within the acceptable limits of all responses. 

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