Multivariate example

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. 

If the task is multivariate calibration, for example, the proper choice of a pre-processing method will essentially aflFect the quality of the resultant model. For more details about the use of these techniques together with PCA and PLS, readers are advised to consider the fundamental monograph by Erikson et al [8]. 

For example, the objects may be chemical compounds. The individual components of a data vector are called features and may, for example, be molecular descriptors (see Chapter 8) specifying the chemical structure of an object. For statistical data analysis, these objects and features are represented by a matrix X which has a row for each object and a column for each feature. In addition, each object win have one or more properties that are to be investigated, e.g., a biological activity of the structure or a class membership. This property or properties are merged into a matrix Y Thus, the data matrix X contains the independent variables whereas the matrix Ycontains the dependent ones. Figure 9-3 shows a typical multivariate data matrix. 

An analysis of variance can be extended to systems involving more than a single variable. For example, a two-way ANOVA can be used in a collaborative study to determine the importance to an analytical method of both the analyst and the instrumentation used. The treatment of multivariable ANOVA is beyond the scope of this text, but is covered in several of the texts listed as suggested readings at the end of the chapter. 

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. 

Three examples of simple multivariable control problems are shown in Fig. 8-40. The in-line blending system blends pure components A and B to produce a product stream with flow rate w and mass fraction of A, x. Adjusting either inlet flow rate or Wg affects both of the controlled variables andi. For the pH neutrahzation process in Figure 8-40(Z ), liquid level h and the pH of the exit stream are to be controlled by adjusting the acid and base flow rates and w>b. Each of the manipulated variables affects both of the controlled variables. Thus, both the blending system and the pH neutralization process are said to exhibit strong process interacHons. In contrast, the process interactions for the gas-liquid separator in Fig. 8-40(c) are not as strong because one manipulated variable, liquid flow rate L, has only a small and indirec t effect on one controlled variable, pressure P. 

This tutorial looks at how MATLAB commands are used to convert transfer functions into state-space vector matrix representation, and back again. The discrete-time response of a multivariable system is undertaken. Also the controllability and observability of multivariable systems is considered, together with pole placement design techniques for both controllers and observers. The problems in Chapter 8 are used as design examples. 

The for-end loop in examp88.m that employs equation (8.76), while appearing very simple, is in faet very powerful sinee it ean be used to simulate the time response of any size of multivariable system to any number and manner of inputs. If A and B are time-varying, then A(r) and B(r) should be ealeulated eaeh time around the loop. The author has used this teehnique to simulate the time response of a 14 state-variable, 6 input time-varying system. Example 8.10 shows the ease in whieh the eontrollability and observability matriees M and N ean be ealeulated using c t r b and ob s v and their rank eheeked. 

This tutorial uses the MATLAB Control System Toolbox for linear quadratie regulator, linear quadratie estimator (Kalman filter) and linear quadratie Gaussian eontrol system design. The tutorial also employs the Robust Control Toolbox for multivariable robust eontrol system design. Problems in Chapter 9 are used as design examples. 

If the reaction series cannot be correlated with one of these univariate LFER, it may be possible to fit the data to Eq. (7-30). a multivariate LFER. Examples of this approach are given by Ehrenson et al. ... 

It should be noted that in this example the performance of only one variable, the three analysts, is investigated and thus this technique is called a one-way ANOVA. If two variables, e.g. the three analysts with four different titration methods, were to be studied, this would require the use of a two-way ANOVA. Details of suitable texts that provide a solution for this type of problem and methods for multivariate analysis are to be found in the Bibliography, page 156. 

Figure 2 is a multivariate plot of some multivariate data. We have plotted the component concentrations of several samples. Each sample contains a different combination of concentrations of 3 components. For each sample, the concentration of the first component is plotted along the x-axis, the concentration of the second component is plotted along the y-axis, and the concentration of the third component is plotted along the z-axis. The concentration of each component will vary from some minimum value to some maximum value. In this example, we have arbitrarily used zero as the minimum value for each component concentration and unity for the maximum value. In the real world, each component could have a different minimum value and a different maximum value than all of the other components. Also, the minimum value need not be zero and the maximum value need not be unity. 

Analytical results are often represented in a data table, e.g., a table of the fatty acid compositions of a set of olive oils. Such a table is called a two-way multivariate data table. Because some olive oils may originate from the same region and others from a different one, the complete table has to be studied as a whole instead as a collection of individual samples, i.e., the results of each sample are interpreted in the context of the results obtained for the other samples. For example, one may ask for natural groupings of the samples in clusters with a common property, namely a similar fatty acid composition. This is the objective of cluster analysis (Chapter 30), which is one of the techniques of unsupervised pattern recognition. The results of the clustering do not depend on the way the results have been arranged in the table, i.e., the order of the objects (rows) or the order of the fatty acids (columns). In fact, the order of the variables or objects has no particular meaning. 

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. 

Given these tables of multivariate data one might be interested in various relationships. For example, do the two panels have a similar perception of the different olive oils (Tables 35.1 and 35.2) Are the oils more or less similarly scattered in the two multidimensional spaces formed by the Dutch and by the British attributes How are the two sets of sensory attributes related Does the... 

Procrustes analysis is a method for relating two sets of multivariate observations, say X and Y. For example, one may wish to compare the results in Table 35.1 and Table 35.2 in order to find out to what extent the results from both panels agree, e.g., regarding the similarity of certain olive oils and the dissimilarity of others. Procrustes analysis has a strong geometric interpretation. The... 

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