Principle component analysis model

On the other hand, techniques like Principle Component Analysis (PCA) or Partial Least Squares Regression (PLS) (see Section 9.4.6) are used for transforming the descriptor set into smaller sets with higher information density. The disadvantage of such methods is that the transformed descriptors may not be directly related to single physical effects or structural features, and the derived models are thus less interpretable. 

Chemometric evaluation methods can be applied to the signal from a single sensor by feeding the whole data set into an evaluation program [133,135]. Both principle component analysis (PCA) and partial least square (PLS) models were used to evaluate the data. These are chemometric methods that may be used for extracting information from a multivariate data set (e.g., from sensor arrays) [135]. The PCA analysis shows that the MISiC-FET sensor differentiates very well between different lambda values in both lean gas mixtures (excess air) and rich gas mixtures (excess fuel). The MISiC-FET sensor is seen to behave as a linear lambda sensor [133]. It... 

In 1999, Hsiao and Siebert applied the principal components analysis model to 11 different properties of 17 organic acids. Results from this study enabled them to construct four fundamental scales (the first four principal components after Varimax rotation). The points of each of the 17 organic acids on each of these four scales represented their relationships. These were referred to as "principle properties" and could then be used to construct multiple regression models relating the MIC of an acid to its principal properties of a set of organisms (Nakai and Siebert, 2004). 

However, no book on experimental design of this scope can be considered exhaustive. In particular, discussion of mathematical and statistical analysis has been kept brief Designs for factor studies at more than two levels are not discussed. We do not describe robust regression methods, nor the analysis of correlations in responses (for example, principle components analysis), nor the use of partial least squares. Our discussion of variability and of the Taguchi approach will perhaps be considered insufficiently detailed in a few years. We have confined ourselves to linear (polynomial) models for the most part, but much interest is starting to be expressed in highly non-linear systems and their analysis by means of artificial neural networks. The importance of these topics for pharmaceutical development still remains to be fully assessed. 

The appropriate tool for the construction of models is factor analysis and principle component analysis C189D. An introduction to these statistical methods is beyond the scope of this book and therefore only a brief discussion about modelling of clusters is given here. 

To test the performance of wavelet packet principle component analysis in dynamic process monitoring with noise, a 2-in-2-out three order dynamic system with noise is employed. The mathematical models are ... 

For safety and good product quality of process plant, it is important to monitor process dynamic operation and to detect upsets, abnormalities, malfunctions, or other unusual events as early as possible. Since first principle models of complicated chemical processes are in many circumstances difficult to develop, data based approaches have been widely used for process monitoring. Among them, the principle component analysis (PCA) extracts a number of independent components ftom highly correlated process data, has been applied successfully for process monitoring. 

To classify samples with various pathogens, SIMCA (21) was employed to develop models for milk sample classification regarding NIR spectra. SIMCA develops models for each class based on factor analysis, that is, principal components that describe the variations of the spectral data. Once each class has its own model new samples are classified to one or another class according to their spectra. Samples from the calibration set were used to develop SIMCA models for class 1 and class 2, respectively. The obtained models were evaluated with samples from the test set, and different models were compared. SIMCA identified variations that were quite different from the inherent variance of the training set. First, the training set data matrix was decomposed by principle component analysis (PCA) and the optimum number of factors was determined. Mahalanobis distance calculations were applied to the score matrix, for the primary set of factors, to compare unknowns to the training set. If an unknown sample was not a member of the groups, it was rejected. A spectrum was classified as a member of a respective class if its Mahalanobis distance was less than 3 standard deviations from the cluster s centroid. 

If gas selectivity cannot be achieved by improving the sensor setup itself, it is possible to use several nonselective sensors and predict the concentration by model based, such as multilinear regression (MLR), principle component analysis (PCA), principle component regression (PCR), partial least squares (PLS), and multivariate adaptive regression splines (MARS), or data-based algorithms, such as cluster analysis (CA) and artificial neural networks (ANN) (for details see Reference 10) (Figure 22.5). For common applications of pattern recognition and multi component analysis of gas mixtures, arrays of sensors are usually chosen... 

Besides the straightforward fingerprints studies, XANES can also be applied to quantitative speciation. This is because XAS is a local probe teehnique, which implies no long-range order in the sample is required. Therefore, if the absorbing atoms are present in the sample at two different sites, the XANES spectrum obtained from this material can be represented by the weighted addition of the spectra of suitable reference samples. For many systems, XANES analysis based on linear combinations of known spectra from model compounds is sufficient to estimate ratios of different species. More sophisticated linear algorithms, such as principle component analysis and factor analysis, can also be applied to XANES spectra. ... 

As a multivariate model is created, multiple planes are created within the data that best explain the variation and best fit the data. The first step in model creation consists of creating the first and second principle components. In the multidimensional matrix of data, a line is fit to the space that best defines the data and then another line is fit to the data that provides the second most detail. These two lines are called the first principle and second principle components, respectively, and a plane is then fit to these two lines which is called the principle component plane. Each principle component plane that is created can then be analyzed, and is known as Principle Component Analysis (PCA). 

For many applications, quantitative band shape analysis is difficult to apply. Bands may be numerous or may overlap, the optical transmission properties of the film or host matrix may distort features, and features may be indistinct. If one can prepare samples of known properties and collect the FTIR spectra, then it is possible to produce a calibration matrix that can be used to assist in predicting these properties in unknown samples. Statistical, chemometric techniques, such as PLS (partial least-squares) and PCR (principle components of regression), may be applied to this matrix. Chemometric methods permit much larger segments of the spectra to be comprehended in developing an analysis model than is usually the case for simple band shape analyses. 

Principal component analysis is ideally suited for the analysis of bilinear data matrices produced by hyphenated chromatographic-spectroscopic techniques. The principle component models are easy to construct, even when large or complicated data sets are analyzed. The basis vectors so produced provide the fundamental starting point for subsequent computations. Additionally, PCA is well suited for determining the number of chromatographic and spectroscopically unique components in bilinear data matrices. For this task, it offers superior sensitivity because it makes use of all available data points in a data matrix. 

From the redundant species analysis it is clear that all reactions which consume H2O2 and O3 are redundant and can be removed automatically from the mechanism. In order to identify other redundant reactions the techniques of rate sensitivity analysis coupled with a principal component analysis of the resulting matrix can be used. The principal component analysis of the rate sensitivity matrix containing only the remaining important and necessary species will reveal the important reactions leading to reduced mechanisms applicable at various ambient temperatures. In principle it may be possible to produce a reduced scheme which models non-isothermal behaviour from analysis carried out on an isothermal model. An isothermal system is easier to model since thermodynamic and heat-transfer properties can be excluded from the calculations. However,... 

NIR spectroscopy became much more useful when the principle of multiple-wavelength spectroscopy was combined with the deconvolution methods of factor and principal component analysis. In typical applications, partial least squares regression is used to model the relation between composition and the NIR spectra of an appropriately chosen series of calibration samples, and an optimal model is ultimately chosen by a procedure of cross-testing. The performance of the optimal model is then evaluated using the normal analytical performance parameters of accuracy, precision, and linearity. Since its inception, NIR spectroscopy has been viewed primarily as a technique of quantitative analysis and has found major use in the determination of water in many pharmaceutical materials. 

The principles behind principal components analysis are most easily explained by means of a geometrical description of the method. From such a description it will then be evident how a principal components (PC) model can be used to simplify the problem of which test compounds should be selected. The data of the Lewis acids in Table 15.1 will be used to give an example of such selections, after the general presentation of the method which follows. 

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