Principal component analysis method

Using a different set of standard substances, i.e. substituting 1-butanol, pentan-2-one, and 1-nitropropane for the rather volatile ethanol, butan-2-one, and nitromethane, McReynolds developed an analogous approach [103]. Altogether, he characterized over 200 liquid stationary phases using a total of 10 probes. A statistical analysis of the McReynolds retention index matrix using the principal component analysis method has shown that only three components are necessary to reproduce the experimental data matrix [262]. The first component is related to the polarity of the liquid phase, the second depends almost solely on the solute, and the third is related to specific interactions with solute hydroxy groups [262]. 

D Verify the calculations of Section 26.1.2 applying the principal-component analysis method to the correlation matrix for elements measured in Whiteface Mountain, New York (Table 26.4). 

In the course of determinations of trace element concentrations in nails of a group of carpenters by cyclic instrumental neutron activation analysis (CINAA) using the PCA (principal component analysis ) method, the silver concentration in nails was determined to be 0.027 0.037 pgg. The mean concentration of gold was 0.010 pgg. Nails can be used as a dietary and environmental exposure monitor " . ... 

The singular value decomposition (SVD) method, and the similar principal component analysis method, are powerful computational tools for parametric sensitivity analysis of the collective effects of a group of model parameters on a group of simulated properties. The SVD method is based on an elegant theorem of linear algebra. The theorem states that one can represent an w X n matrix M by a product of three matrices ... 

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. 

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. 

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... 

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. 

The dimensionality of a data set is the number of variables that are used to describe eac object. For example, a conformation of a cyclohexane ring might be described in terms c the six torsion angles in the ring. However, it is often found that there are significai correlations between these variables. Under such circumstances, a cluster analysis is ofte facilitated by reducing the dimensionality of a data set to eliminate these correlation Principal components analysis (PCA) is a commonly used method for reducing the dimensior ality of a data set. 

Multiple linear regression is strictly a parametric supervised learning technique. A parametric technique is one which assumes that the variables conform to some distribution (often the Gaussian distribution) the properties of the distribution are assumed in the underlying statistical method. A non-parametric technique does not rely upon the assumption of any particular distribution. A supervised learning method is one which uses information about the dependent variable to derive the model. An unsupervised learning method does not. Thus cluster analysis, principal components analysis and factor analysis are all examples of unsupervised learning techniques. 

The field points must then be fitted to predict the activity. There are generally far more field points than known compound activities to be fitted. The least-squares algorithms used in QSAR studies do not function for such an underdetermined system. A partial least squares (PLS) algorithm is used for this type of fitting. This method starts with matrices of field data and activity data. These matrices are then used to derive two new matrices containing a description of the system and the residual noise in the data. Earlier studies used a similar technique, called principal component analysis (PCA). PLS is generally considered to be superior. 

Other chemometrics methods to improve caUbration have been advanced. The method of partial least squares has been usehil in multicomponent cahbration (48—51). In this approach the concentrations are related to latent variables in the block of observed instmment responses. Thus PLS regression can solve the colinearity problem and provide all of the advantages discussed earlier. Principal components analysis coupled with multiple regression, often called Principal Component Regression (PCR), is another cahbration approach that has been compared and contrasted to PLS (52—54). Cahbration problems can also be approached using the Kalman filter as discussed (43). 

A method of resolution that makes a very few a priori assumptions is based on principal components analysis. The various forms of this approach are based on the self-modeling curve resolution developed in 1971 (55). The method requites a data matrix comprised of spectroscopic scans obtained from a two-component system in which the concentrations of the components are varying over the sample set. Such a data matrix could be obtained, for example, from a chromatographic analysis where spectroscopic scans are obtained at several points in time as an overlapped peak elutes from the column. 

In general, two related techniques may be used principal component analysis (PCA) and principal coordinate analysis (PCoorA). Both methods start from the n X m data matrix M, which holds the m coordinates defining n conformations in an m-dimensional space. That is, each matrix element Mg is equal to q, the jth coordinate of the /th conformation. From this starting point PCA and PCoorA follow different routes. 

One of the main attractions of normal mode analysis is that the results are easily visualized. One can sort the modes in tenns of their contributions to the total MSF and concentrate on only those with the largest contributions. Each individual mode can be visualized as a collective motion that is certainly easier to interpret than the welter of information generated by a molecular dynamics trajectory. Figure 4 shows the first two normal modes of human lysozyme analyzed for their dynamic domains and hinge axes, showing how clean the results can sometimes be. However, recent analytical tools for molecular dynamics trajectories, such as the principal component analysis or essential dynamics method [25,62-64], promise also to provide equally clean, and perhaps more realistic, visualizations. That said, molecular dynamics is also limited in that many of the functional motions in biological molecules occur in time scales well beyond what is currently possible to simulate. 

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]. 

We now consider a type of analysis in which the data (which may consist of solvent properties or of solvent effects on rates, equilibria, and spectra) again are expressed as a linear combination of products as in Eq. (8-81), but now the statistical treatment yields estimates of both a, and jc,. This method is called principal component analysis or factor analysis. A key difference between multiple linear regression analysis and principal component analysis (in the chemical setting) is that regression analysis adopts chemical models a priori, whereas in factor analysis the chemical significance of the factors emerges (if desired) as a result of the analysis. We will not explore the statistical procedure, but will cite some results. We have already encountered examples in Section 8.2 on the classification of solvents and in the present section in the form of the Swain et al. treatment leading to Eq. (8-74). 

The mechanism of free-radical addition follows the pattern discussed in Chapter 14 (pp. 894-895). The method of principal component analysis has been used to analyze polar and enthalpic effect in radical addition reactions. A radical is generated by... 

Principal Component Analysis (PCA). Principal component analysis is an extremely important method within the area of chemometrics. By this type of mathematical treatment one finds the main variation in a multidimensional data set by creating new linear combinations of the raw data (e.g. spectral variables) [4]. The method is superior when dealing with highly collinear variables as is the case in most spectroscopic techniques two neighbor wavelengths show almost the same variation. 

The data from sensory evaluation and texture profile analysis of the jellies made with amidated pectin and sunflower pectin were subjected to Principal component analysis (PC) using the statistical software based on Jacobi method (Univac, 1973). The results of PC analysis are shown in figure 7. The plane of two principal components (F1,F2) explain 89,75 % of the variance contained in the original data. The attributes related with textural evaluation are highly correlated with the first principal component (Had.=0.95, Spr.=0.97, Che.=0.98, Gum.=0.95, Coe=0.98, HS=0.82 and SP=-0.93). As it could be expected, spreadability increases along the negative side of the axis unlike other textural parameters. 

Principal component analysis (PCA) is a statistical method having as its main purpose the representation in an economic way the location of the objects in a reduced coordinate system where only p axes instead of n axes corresponding to n variables (p[Pg.94]

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. 

A first introduction to principal components analysis (PCA) has been given in Chapter 17. Here, we present the method from a more general point of view, which encompasses several variants of PCA. Basically, all these variants have in common that they produce linear combinations of the original columns in a measurement table. These linear combinations represent a kind of abstract measurements or factors that are better descriptors for structure or pattern in the data than the original measurements [1]. The former are also referred to as latent variables [2], while the latter are called manifest variables. Often one finds that a few of these abstract measurements account for a large proportion of the variation in the data. In that case one can study structure and pattern in a reduced space which is possibly two- or three-dimensional. 

According to Andersen [12] early applications of LLM are attributed to the Danish sociologist Rasch in 1963 and to Andersen himself. Later on, the approach has been described under many different names, such as spectral map analysis [13,14] in studies of drug specificity, as logarithmic analysis in the French statistical literature [15] and as the saturated RC association model [16]. The term log-bilinear model has been used by Escoufier and Junca [ 17]. In Chapter 31 on the analysis of measurement tables we have described the method under the name of log double-centred principal components analysis. 

Multivariate chemometric techniques have subsequently broadened the arsenal of tools that can be applied in QSAR. These include, among others. Multivariate ANOVA [9], Simplex optimization (Section 26.2.2), cluster analysis (Chapter 30) and various factor analytic methods such as principal components analysis (Chapter 31), discriminant analysis (Section 33.2.2) and canonical correlation analysis (Section 35.3). An advantage of multivariate methods is that they can be applied in... 

G. Dijksterhuis, Principal component analysis of Tl-curves three methods compared. Food Qual. Pref., 5 (1994) 121-127. 

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