Multivariate space

Any data matrix can be considered in two spaces the column or variable space (here, wavelength space) in which a row (here, spectrum) is a vector in the multidimensional space defined by the column variables (here, wavelengths), and the row space (here, retention time space) in which a column (here, chromatogram) is a vector in the multidimensional space defined by the row variables (here, elution times). This duality of the multivariate spaces has been discussed in more detail in Chapter 29. Depending on the chosen space, the PCs of the data matrix... 

Although INAA and LA-ICP-MS Groups 2 and 3 are easily differentiable from one another, examination of the data in bivariate and multivariate space suggest that they are chemically closer to one another than to the other groups. The chemical similarity of these two groups may indicate derivation from separate production centers within the same region, a topic we return to below. 

An univariate search does not necessarily lead to an optimum for a multivariable space that is, convergence to an optimum is not guaranteed. For this reason a series of experiments were made to map the region close to the identified optimum conditions to test whether a local optimum, at least, had been located. 

This distance is also called the Mahalanobis distance by many practitioners after the famous Indian mathematician, Mahalanobis [4], The distance in multivariate space is analogous to the normalized univariate squared distance of a single point (in units of standard deviations) from the mean ... 

Stakeholders, groups of concerned individuals, and organizations can also define a reference condition. Such a multivariate space would be bound by the limits that stakeholders placed on the acceptable conditions for a particular site. It is also possible to construct such a site by having the stakeholders identify current sites that meet their conditions and then sampling those sites to define an acceptable community structure. In this use of a stakeholder-defined reference condition, the goal is not necessarily to identify causality due to contaminants but to identify those sites that require some form of management in order to meet the goals of the stakeholders. 

Different approaches to estimate interpolation regions in a multivariate space were evaluated by Jaworska [Jaworska, Nikolova-Jeliazkova et al, 2005], based on (a) ranges of the descriptor space (b) distance-based methods, using Euclidean, Manhattan, and Mahalanobis distances. Hotelling T method and leverage values and (c) probability density distribution methods based on parametric and nonparametric approaches. Both ranges and distance-based methods were also evaluated in the principal component space by Principal Component Armlysis. 

The mathematics of what is described above is equivalent to principal component analysis. The ideas of principal component analysis (PCA) go back to Beltrami [1873] and Pearson [1901], They tried to describe the structural part of data sets by lines and planes of best fit in multivariate space. The PCA method was introduced in a rudimentary version by Fisher and Mackenzie [1923], The name principal component analysis was introduced by Hotelling [1933], An early calculation algorithm is given by Muntz [1913], More details can be found in the literature [Jackson 1991, Jolliffe 1986, Stewart 1993],... 

The most commonly used classification techniques are Linear Discriminant Analysis (LDA) and Quadratic Discriminant Analysis (QDA). They define a set of delimiters (according to the number of categories under smdy) in such a way that the multivariate space of the objects is divided into as many subspaces as the number of categories, and that each point of the space belongs to one... 

With real-life measurements, it is indeed very difficult to identify a single direction in a multivariate space that is only correlated to sensor drift. So, for each sensor, an individual multiplicative factor was calculated by estimating the drift slope for a standard gas. 

Factor Analysis. Several choices had to be made in preparing the data for factor analysis as well as in choosing criteria for selecting the number of factors needed to describe the data space (e.g. eigenvalue > 1.0, ratio adjacent eigenvalues > 2.0, etc.) and the number of factor scores to be used as input into the canonical correlation analysis. These choices may have affected subsequent interpretation of the multivariate spaces and evaluation of the chemometric analysis methods. Table II shows the types of spectral data input into factor analyses of the first 13 subfractions. 

All American Pharmaceutical Company analyzes more than 100 herbals with the FT-NIR. The herbals are grouped into several library models. Figure 31.9 shows the reference spectra for herbals in one of the herbal libraries. Mathematical data treatment such as first derivative makes the differences more noticeable to the eye. The mathematical algorithms view the data in multivariate space and are able to resolve very small differences that are not obvious to the human eye. Figure 31.10 shows the same materials in Figure 31.9 now graphed using first-derivative data treatment. Other... 

Chapter 4 retrieves the basic ideas of classical univariate calibration as the standpoint from which the natural and intuitive extension of multiple linear regression (MLR), arises. Unfortunately, this generalization is not suited to many laboratory tasks and, therefore, the problems associated with its use are explained in some detail. Such problems justify the use of other more advanced techniques. The explanation of what the multivariate space looks like and how principal components analysis can tackle it is the next step forward. This constitutes the root of the regression methodology presented in the following chapter. 

The concept of determining the molar absorption coefficient e from the calibration line can be transferred to chemometrics. Although it is not practicable in a multivariate space to draw something like a calibration line in Figure 7.1, a multivariate parameter corresponding to e can be defined. 

In the multivariate space, Beer s law may be re-expressed in the following form by fixing the pathlength to 1 cm. 

The above matrix formulation can be visualized by using vectors in a multivariate space as shown in Figure 7.3. The two continuous line arrows in this figure correspond to the vectors and and the five dots (points) represent five observed spectra. The position of each point, which corresponds to a row vector in the A matrix, is determined by adding the two k vectors multiplied by different concentrations given in the C matrix. The five points are therefore in a plane spanned by the two k vectors. This means that the number of dimensions needed for spanning the space to contain all the points is equal to the number of chemical components. This concept will be used positively later for considering PCA (principal component analysis). 

Figure 7.3 Variation of two-component spectra plotted in the multivariate space and vectors in CIS and PCA. Filled circles indicate two-component spectra, and and k depict vectors in CLS, and PC 1 and PC2 orthogonal vectors in PCA.

Scores indicate new positions of the points (observed spectra) on the coordinate axes corresponding to the loading vectors. It should be noted that the variation of spectral quantity is fully recorded in the scores. The number of scores on each loading vector is exactly the same as the number of points in the multivariate space. This plays an important role in principal component regression (PCR), described in the next section. 

To apply the ILS method to spectroscopic calibration properly, the A matrix corresponding to spectra must have a square or portrait form. To fulfill this requirement, it is necessary to reduce greatly the number of wavenumber points selected from the measured spectra. However, the information required for spectroscopic calibration need not be taken from the measured spectra it can also be obtained from the PCA scores, which have information equivalent to the recorded spectra. The number of scores is equal to that of the points corresponding to the measured spectra in the multivariate space. Therefore, A with a size of A X M can be replaced by a matrix of scores with a reduced size of A x ft, the rank of which never exceeds A. In this manner, the singularity problem inherent in the ILS method can be overcome. Note that no reduction of wavenumber points is needed and full use of the measured spectra is made, as the scores are obtained by Equations (7.16) and (7.17). 

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