Principal components calculations

Pattern recognition studies on complex data from capillary gas chromatographic analyses were conducted with a series of microcomputer programs based on principal components (SIMCA-3B). Principal components sample score plots provide a means to assess sample similarity. The behavior of analytes in samples can be evaluated from variable loading plots derived from principal components calculations. A complex data set was derived from isomer specific polychlorinated biphenyl (PCBS) analyses of samples from laboratory and field studies. 

By using SIMCA-3B program, FPLOT.EXE, one can plot numerous variables derived from the principal components calculations. Because a printer in the character mode is used with this program to plot variables, the plots are restricted to two-dimensional presentations. 

The Aroclor samples listed in Table I were modeled by principal components to Illustrate how the result from principal components calculations can be used in... 

Table II. Interpretations of 8 principal components calculated from 90 variables based on molecular connectivity indices for 19,972 industrial chemicals. 

As a result of the principal component calculation, the U matrix has a number of columns equal to tlie minimum of the number of samples or variables. Knowing tliat only some of the columns in U contain the relevant information, a subset is selected. Choosing the relevant number of PCs to include in the model is one of the most important steps in the PCR process because it is the key to the stabilization of the inverse. Ordinarily the columns in U are chosen sequentially, from highest to lowest percent variance described. 

If the variables are correlated, the occurring problem of multicollinearity may be circumvented by performing a principal components calculation with the variables x. This will create independent ( orthogonal ) variables and one can continue the regression analysis using the scores (see Section 5.4) instead of the original x values. This method is known as principal components regression. 

Figure 14,2.1. Dendrogram obtained from the Euclidian distance of the first three principal components calculated from results of conventional coal analysis.

These spectra (Fig. 16) are either spectra of specific compounds or of aggregate matrices (residual organics dissolved, colloids, suspended solids). The first group of spectra, very reproducible, is the deterministic part of the model. It includes the compounds that may be found in the type of sample to be examined. The second one, being of experimental or mathematical nature (difference of spectra, for example), can be considered as the stochastic part of the model. Moreover, some of these spectra can be actually related to principal components calculated from the residual matrix. The selection is done between different spectra, which allows taking into account the effect of the main interferences. 

More often, the SIMCA method is used. This finds separate principal component models for each class. By using SIMCA, the object variable number ratio is less critical and the model is constructed around the projected, rather than the original, data. The basic steps of principal component calculations as needed for SIMCA have been outlined in the chapter on projection methods with the NIPALS algorithm (Example 5.1). 

Particle size of flour can be measured by using only the first principal component calculated from its NIR spectrum (29). A plot of Malvern versus NIR flour particle size is shown in Figure 8.1.5. 

Let us autoscale the data, load it into the graphical user interface and selecting calc. Principal component calculation yield the results shown in Table 22.1. 

In addition to being able to predict the energy costs of the heat exchanger network and utilities directly from the material and energy balance, it would be useful to be able to calculate the capital cost, if this is possible. The principal components that contribute to the capital cost of the heat exchanger network are... 

The procedure is as follows first, the principal components for X and Yare calculated separately (cf. Section 9.4.4). The scores of the matrix X are then used for a regression model to predict the scores of Y, which can then be used to predict Y. 

Initially, the first two principal components were calculated. This yielded the principal components which are given in Figure 9-9 (left) and plotted in Figure 9-9 (right). The score plot shows which mineral water samples have similar mineral concentrations and which are quite different. For e3oimple, the mineral waters 6 and 7 are similar whUe 4 and 7 are rather dissimilar. 

Principal component analysis (PCA) takes the m-coordinate vectors q associated with the conformation sample and calculates the square m X m matrix, reflecting the relationships between the coordinates. This matrix, also known as the covariance matrix C, is defined as... 

A distance geometry calculation consists of two major parts. In the first, the distances are checked for consistency, using a set of inequalities that distances have to satisfy (this part is called bound smoothing ) in the second, distances are chosen randomly within these bounds, and the so-called metric matrix (Mij) is calculated. Embedding then converts this matrix to three-dimensional coordinates, using methods akin to principal component analysis [40]. 

As another illustration of the use of Equation, we calculate the enthalpy of combustion of methane, the principal component of natural gas. We begin with the balanced chemical equation ... 

The concept of property space, which was coined to quanhtahvely describe the phenomena in social sciences [11, 12], has found many appUcahons in computational chemistry to characterize chemical space, i.e. the range in structure and properhes covered by a large collechon of different compounds [13]. The usual methods to approach a quantitahve descriphon of chemical space is first to calculate a number of molecular descriptors for each compound and then to use multivariate analyses such as principal component analysis (PCA) to build a multidimensional hyperspace where each compound is characterized by a single set of coordinates. 

To further analyze the relationships within descriptor space we performed a principle component analysis of the whole data matrix. Descriptors have been normalized before the analysis to have a mean of 0 and standard deviation of 1. The first two principal components explain 78% of variance within the data. The resultant loadings, which characterize contributions of the original descriptors to these principal components, are shown on Fig. 5.8. On the plot we can see that PSA, Hhed and Uhba are indeed closely grouped together. Calculated octanol-water partition coefficient CLOGP is located in the opposite corner of the property space. This analysis also demonstrates that CLOGP and PSA are the two parameters with... 

The aim of factor analysis is to calculate a rotation matrix R which rotates the abstract factors (V) (principal components) into interpretable factors. The various algorithms for factor analysis differ in the criterion to calculate the rotation matrix R. Two classes of rotation methods can be distinguished (i) rotation procedures based on general criteria which are not specific for the domain of the data and (ii) rotation procedures which use specific properties of the factors (e.g. non-negativity). 

In Section 34.2 we explained that factor analysis consists of a rotation of the principal components of the data matrix under certain constraints. When the objects in the data matrix are ordered, i.e. the compounds are present in certain row-windows, then the rotation matrix can be calculated in a straightforward way. For non-ordered spectra with three or less components, solution bands for the pure factors are obtained by curve resolution, which starts with looking for the purest spectra (i.e. rows) in the data matrix. In this section we discuss the VARDIA [27,28] technique which yields clusters of pure variables (columns), for a certain pure factor. 

PLS has been introduced in the chemometrics literature as an algorithm with the claim that it finds simultaneously important and related components of X and of Y. Hence the alternative explanation of the acronym PLS Projection to Latent Structure. The PLS factors can loosely be seen as modified principal components. The deviation from the PCA factors is needed to improve the correlation at the cost of some decrease in the variance of the factors. The PLS algorithm effectively mixes two PCA computations, one for X and one for Y, using the NIPALS algorithm. It is assumed that X and Y have been column-centred as usual. The basic NIPALS algorithm can best be demonstrated as an easy way to calculate the singular vectors of a matrix, viz. via the simple iterative sequence (see Section 31.4.1) ... 

Instead of separately calculating the principal components for each data set, the two iterative sequences are interspersed in the PLS-NIPALS algorithm (see Fig. 

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... 

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