Component, principal

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... 

C10HJ4N2. When pure nicotine is a colourless liquid, b.p. 247 C, but darkens on exposure to air and light. Crude nicotine contains smalt amounts of other alkaloids, but ( —)-nicotine is the principal component. 

Other correlations between NMR and infrared have been studied because the latter technique is less cui ersonje than NMR. Correlations are obtained not just on the two absorption bands but on the whole of the IR spectrum after reduction of the spectrum into its principal components. 

The resolution of capillary columns enables the separation of all principal components of a straight-run gasoline. The most frequently used stationary phases are silicone-based, giving an order of hydrocarbon elution times close to the order of increasing boiling point. 

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. 

Jolliffe I T 1986 Principal Component Analysls New York Springer)... 

The essential degrees of freedom are found by a principal component analysis of the position correlation matrix Cy of the cartesian coordinate displacements Xi with respect to their averages xi), as gathered during a long MD run ... 

The important underlying components of protein motion during a simulation can be extracted by a Principal Component Analysis (PGA). It stands for a diagonalization of the variance-covariance matrix R of the mass-weighted internal displacements during a molecular dynamics simulation. 

The essential slow modes of a protein during a simulation accounting for most of its conformational variability can often be described by only a few principal components. Comparison of PGA with NMA for a 200 ps simulation of bovine pancreatic trypsic inhibitor showed that the variation in the first principal components was twice as high as expected from normal mode analy-si.s ([Hayward et al. 1994]). The so-called essential dynamics analysis method ([Amadei et al. 1993]) is a related method and will not be discussed here. 

Grubmiiller described a method to induce conformational transitions in proteins and derived rate constants for these ([Grubmiiller 1994]). The method employs subsequent modifications of the original potential function based on a principal component analysis of a short MD simulation. It is discussed in more detail in the chapter of Eichinger et al. in this volume. 

Hayward et al. 1994] Hayward, S., Kitao, A., Go, N. Harmonic and anharmonic aspects in the dynamics of BPTI A normal mode analysis and principal component analysis. Prot. Sci. 3 (1994) 936-943 [Head-Gordon and Brooks 1991] Head-Gordon, T., Brooks, C.L. Virtual rigid body dynamics. Biopol. 31 (1991) 77-100... 

Step 2 This ensemble is subjected to a principal component analysis (PCA) [61] by diagonalizing the covariance matrix C G x 7Z, ... 

Steven Hayward, Akio Kitao, and Nobuhiro Go. Harmonic and anharmonic aspects in the dynamics of BPTI A normal mode analysis and principal component analysis. Physica Scripta, 3 936-943, 1994. 

M. A. Balsera, W. Wriggers, Y. Oono, and K. Schulten. Principal component analysis and long time protein dynamics. J. Phys. Chem., 100 2567-2572, 1996. 

We have to apply projection techniques which allow us to plot the hyperspaces onto two- or three-dimensional space. Principal Component Analysis (PCA) is a method that is fit for performing this task it is described in Section 9.4.4. PCA operates with latent variables, which are linear combinations of the original variables. 

The first few principal components store most of the relevant information, the rest being merely the noise. This means that one can use two or three principal components and plot the objects in two or three-dimensional space without losing information. 

It may look weird to treat the Singular Value Decomposition SVD technique as a tool for data transformation, simply because SVD is the same as PCA. However, if we recall how PCR (Principal Component Regression) works, then we are really allowed to handle SVD in the way mentioned above. Indeed, what we do with PCR is, first of all, to transform the initial data matrix X in the way described by Eqs. (10) and (11). 

To gain insight into chemometric methods such as correlation analysis, Multiple Linear Regression Analysis, Principal Component Analysis, Principal Component Regression, and Partial Least Squares regression/Projection to Latent Structures... 

Kohonen network Conceptual clustering Principal Component Analysis (PCA) Decision trees Partial Least Squares (PLS) Multiple Linear Regression (MLR) Counter-propagation networks Back-propagation networks Genetic algorithms (GA)... 

In mathematical terms, PCA transforms a number of correlated variables into a smaller number of uneorrelated variables, the so-called principal components. 

Thus, the principal components are constructed in order of declining importance the first principal component comprise.s as much of the total variation of all variables as possible, the second principal component as much of the remaining variation, and so on. 

The coordinate of an object when projected onto an axis given by a principal component is called its score. Scores arc usually denoted by Tl, T2,. ... Figure 9-7 is a sketch of a score plot the points are the objects in the coordinate system... 

For most data analysis applications the first three to five principal components give the predominant part of the variance. 

PCR is a combination of PCA and MLR, which are described in Sections 9.4.4 and 9.4.3 respectively. First, a principal component analysis is carried out which yields a loading matrix P and a scores matrix T as described in Section 9.4.4. For the ensuing MLR only PCA scores are used for modeling Y The PCA scores are inherently imcorrelated, so they can be employed directly for MLR. A more detailed description of PCR is given in Ref. [5. ... 

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. 

A crucial decision in PLS is the choice of the number of principal components used for the regression. A good approach to solve this problem is the application of cross-validation (see Section 4.4). 

As described above, PCA can be used for similarity detection The score plot of two principal components can be used to indicate which objects are similar. 

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. 

Figure 9-9. Left first two principal components. Right plot of PCI against PC2. ...

The previously mentioned data set with a total of 115 compounds has already been studied by other statistical methods such as Principal Component Analysis (PCA), Linear Discriminant Analysis, and the Partial Least Squares (PLS) method [39]. Thus, the choice and selection of descriptors has already been accomplished. 

Spectral features and their corresponding molecular descriptors are then applied to mathematical techniques of multivariate data analysis, such as principal component analysis (PCA) for exploratory data analysis or multivariate classification for the development of spectral classifiers [84-87]. Principal component analysis results in a scatter plot that exhibits spectra-structure relationships by clustering similarities in spectral and/or structural features [88, 89]. 

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