Cross multivariate

The eigenvectors extracted from the cross-product matrices or the singular vectors derived from the data matrix play an important role in multivariate data analysis. They account for a maximum of the variance in the data and they can be likened to the principal axes (of inertia) through the patterns of points that represent the rows and columns of the data matrix [10]. These have been called latent variables [9], i.e. variables that are hidden in the data and whose linear combinations account for the manifest variables that have been observed in order to construct the data matrix. The meaning of latent variables is explained in detail in Chapters 31 and 32 on the analysis of measurement tables and contingency tables. 

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... 

Galego and Arroyo [14] described a simultaneous spectrophotometric determination of OTC, hydrocortisone, and nystatin in the pharmaceutical preparations by using ratio spectrum-zero crossing derivate method. The calculation was performed by using multivariate methods such as partial least squares (PLS)-l, PLS-2, and principal component regression (PCR). This method can be used to resolve accurately overlapped absorption spectra of those mixtures. 

Figure 1-1 Development of the concept of the Multivariate Normal Distribution (this one shown having three dimensions) - see text for details. The density of points along a cross-section of the distribution in any direction is also an MND, of lower dimension.

The first two terms on the RHS of equation 70-20 are the variances of X and Y. The third term, the numerator of which is known as the cross-product term, is called the covariance between X and Y. We also note (almost parenthetically) here that multiplying both sides of equation 70-20 by (re - 1) gives the corresponding sums of squares, hence equation 70-20 essentially demonstrates the partitioning of sums of squares for the multivariate case. 

Arrays were introduced in the mid-eighties as a method to counteract the cross-selectivity of gas sensors. Their use has since become a common practice in sensor applications [1], The great advantage of this technique is that once arrays are matched with proper multivariate data analysis, the use of non-selective sensors for practical applications becomes possible. Again in the eighties, Persaud and Dodds argued that such arrays has a very close connection with mammalian olfaction systems. This conjecture opened the way to the advent of electronic noses [2], a popular name for chemical sensor arrays used for qualitative analysis of complex samples. 

AU multivariate calibrations must be based on empirical training and validation data sets obtained in fully realistic situations acoustic chemometrics is no exception. Many models are in addition based on indirect multivariate calibration. All industrial applications must always be evaluated only based on test set validation. Reference [2] deals extensively with the merits of the various validation methods, notably when it is admissible, and when not, to use cross-validation. See also Chapters 3 and 12, which give further background for the test set imperative in light of typical material heterogeneity and the Theory of Sampling . 

PLS falls in the category of multivariate data analysis whereby the X-matrix containing the independent variables is related to the Y-matrix, containing the dependent variables, through a process where the variance in the Y-matrix influences the calculation of the components (latent variables) of the X-block and vice versa. It is important that the number of latent variables is correct so that overfitting of the model is avoided this can be achieved by cross-validation. The relevance of each variable in the PLS-metfiod is judged by the modelling power, which indicates how much the variable participates in the model. A value close to zero indicates an irrelevant variable which may be deleted. 

Luo et al. [83] used an ANN to perform multivariate calibration in the XRF analysis of geological materials and compared its predictive performance with cross-validated PLS. The ANN model yielded the highest accuracy when a nonlinear relationship between the characteristic X-ray line intensity and the concentration existed. As expected, they also found that the prediction accuracy outside the range of the training set was bad. 

Process analytical chemistry (PAC) is a field that has existed for several decades in many industries. It is now gaining renewed popularity as the pharmaceutical industry begins to embrace it as well. PAC encompasses a combination of analytical chemistry, process engineering, process chemistry, and multivariate data analysis. It is a multidisciplinary field that works best when supported by a cross-functional team including members from manufacturing, analytical chemistry, and plant maintenance. 

A relevant question is the plotting of multivariate auto- or cross-correlation functions to determine multivariate relationship to lag. 

The multivariate cross-correlation function was treated in the same way ... 

The auto- and cross-correlation matrices are now reduced to one-dimensional simply called multivariate auto- and cross-correlation functions. 

In the multivariate case, the significant cross-correlation or autocorrelation coefficients for each variable add up to the significant multivariate correlation value. 

Example Multivariate cross-correlation for the computation of transport rates of dissolved metals in river water... 

It has been possible by means of multivariate cross-correlation analysis to include the interaction between the metals which arise as a result of emission and transformation. The assumption is that the metals are transported at the same rate. The multivariate cross-correlation function Rxy(f) expresses a broad maximum at t = 3, i.e. 1.5 h (Fig. 6-20). This means that the mean transport rate of the metals is 3 km h Hydrological data gathered on the same day renders this result plausible ... 

Fig. 6-20. Multivariate cross-correlation function of metal concentrations in a river...

By means of the multivariate cross-correlation function, transport rates in streams can be calculated from natural compound concentrations in water without adding any tracer substances. 

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