Multivariate procedures

Overfitting is the commonest problem in multivariate statistical procedures when the number of variables is greater than objects (samples) one can fit an elephant with enough variables. Tabachnick and Fidell (1983) have suggested minimum requirements for some multivariate procedures to avoid the overfitting or underfitting that can occur in a somewhat unpredictable manner, regardless of the multivariate procedure chosen. 

The main goal of this section is to provide a summary of several of the most widely used multivariate procedures in food authentication out of the vast array currently available. These are included in well-known computer packages such as BMDP, IMSL, MATLAB, NAG, SAS, SPSS and STATISTIC A. The first three subsections describe unsupervised procedures, also called exploratory data analysis, that can reveal hidden patterns in complex data by reducing data to more interpretable information, to emphasize the natural grouping in the data and show which variables most strongly influence these patterns. The fourth and fifth subsections are focused on the supervised procedures of discriminant analysis and regression. The former produces good information when applied under the strictness of certain tests, whereas the latter is mainly used when the objective is calibration. 

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Normally, one does not have hue values of the elements of the slope mah ix M for comparison. It is always possible, however, to obtain y, the vector of predicted y values at each of the known Xi from any of the slope vectors m obtained by the multivariate procedure... 

W.W. Cooley and P.P. Lohnes, Multivariate Procedures for the Behavioral Sciences. Wiley, New York, 1962. 

The chemometric procedures that are currently applied in empirical investigations have been notably improved in recent years with the assistance of computer science. Researchers have passed from initial application of univariate analyses to extensive use of multivariate procedures in less than one decade. This qualitative step has been possible because, first, the new sophisticated analytical instruments are now able to analyse dozens of chemical compounds in hundreds of samples daily, and, second, due to personal computers that can work with a great diversity of software packages. 

The analyst should check the Shepard diagram that represents a step line so-called D-hat values. If all reproduced distances fall onto the step-line, then the rank ordering of distances (or similarities) would be perfectly reproduced by the dimensional model, while deviations from the step-line mean lack of fit. The interpretation of the dimensions usually represents the final step of this multivariate procedure. As in factor analysis, the final orientation of axes in the plane (or space) is mostly the result of a subjective decision by the researcher since the distances between objects remain invariable regardless of the type of the rotation. However, it must be remembered that MDS and FA are different methods. FA requires that the underlying data be distributed as multivariate normal, whereas MDS does not impose such a restriction. MDS often yields more interpretable solutions than FA because the latter tends to extract more factors. MDS can be applied to any kind of distances or similarities (those described in cluster analysis), whereas FA requires firstly the computation of the correlation matrix. Figure 7.3 shows the results of applying MDS to the samples described in the CA and FA sections (7.3.1 and 7.3.2). 

Like many other statistical methods for the evaluation of biomonitoring data, the above-depicted example of a trend analysis considers only a single variable. Although the multivariate procedures consider several measured variables at the same time, their results are often only limited meaningful. Cluster analyses can reveal structures in a given data set principal component analyses concentrate the information contents of many variables in a set of a few latent variables, which are difficult to interpret correctly. 

After selecting a stationary phase and the mobile-phase components, several isocratic experiments are required to build a retention model. A computer-assisted multivariate procedure is often used to find the best combination of the working parameters.Separation selectivity is often monitored by maximum resolvable components in the shortest time at separation of multicomponent samples. 

The analogous procedure for a multivariate problem is to obtain many experimental equations like Eqs. (3-55) and to extract the best slopes from them by regression. Optimal solution for n unknowns requires that the slope vector be obtained from p equations, where p is larger than n, preferably much larger. When there are more than the minimum number of equations from which the slope vector is to be extracted, we say that the equation set is an overdetermined set. Clearly, n equations can be selected from among the p available equations, but this is precisely what we do not wish to do because we must subjectively discard some of the experimental data that may have been gained at considerable expense in time and money. 

Evidence of the appHcation of computers and expert systems to instmmental data interpretation is found in the new discipline of chemometrics (qv) where the relationship between data and information sought is explored as a problem of mathematics and statistics (7—10). One of the most useful insights provided by chemometrics is the realization that a cluster of measurements of quantities only remotely related to the actual information sought can be used in combination to determine the information desired by inference. Thus, for example, a combination of viscosity, boiling point, and specific gravity data can be used to a characterize the chemical composition of a mixture of solvents (11). The complexity of such a procedure is accommodated by performing a multivariate data analysis. 

Nonlinear Programming The most general case for optimization occurs when both the objective function and constraints are nonlinear, a case referred to as nonlinear programming. While the idea behind the search methods used for unconstrained multivariable problems are applicable, the presence of constraints complicates the solution procedure. 

Mancozeb is a dithiocarbamate pesticide with a very low solubility in organic and inorganic solvent. In this work we have developed a solvent free, accurate and fast photoacoustic FTIR-based methodology for Mancozeb determination in commercial fungicides. The proposed procedure was based on the direct measurement of the solid samples in the middle infrared region using a photoacoustic detector. A multivariate calibration approach based on the use of partial least squares (PLS) was employed to determine the pesticide content in commercially available formulations. 

Many techniques can be used to solve multivariable optimizations. Unfortunately, there is no single best method that applies to every type ofresponse surface. Therefore, I will give a number of different procedures, with the advantages and disadvantages of each one. The reader will then have to decide which one(s) he wishes to use. 

Multivariate analytical images may be processed additionally by chemo-metrical procedures, e.g., by exploratory data analysis, regression, classifica-tion> and principal component analysis (Geladi et al. [1992b]). 

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. 

Instrument standardization, v - a procedure for standardizing the response of multiple instruments such that a common multivariate model is applicable for measurements conducted across these instruments, the standardization being accomplished via adjustment of the spectrophotometer hardware or via mathematical treatment of one or a series of collected spectra. 

Multivariate curve resolution methods (MCR [17]) describe a family of chemometric procedures used to identify and solve the contributions existing in a data set. These procedures have been traditionally applied for the resolution of multiple chemical components in mixtures investigated by spectroscopic analysis techniques [18]. 

The multivariate method MCR-ALS has been used to analyse data in order to identify the main sources of organic pollution affecting the Ebro River delta. Subsequently, an interpolation procedure has been also applied to obtain distribution maps from the punctual resolved data (corresponding to the score values obtained from MCR-ALS). 

Sayre, E. V., Brookhaven Procedures for Statistical Analyses of Multivariate Archaeometric Data, BNL, Brookhaven National Laboratory, Upton, 1975. 

Waller, N. G.,. Meehl, P. E. (1998). Multivariate taxometric procedures. Thousand Oaks, CA Sage. 

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