Principal Component Analysis explained variance

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. 

Musumarra et al. [44] also identified miconazole and other drugs by principal components analysis of standardized thin-layer chromatographic data in four eluent systems and of retention indexes on SE 30. The principal component analysis of standardized R values in four eluents systems ethylacetate-methanol-30% ammonia (85 10 15), cyclohexane-toluene-diethylamine (65 25 10), ethylacetate-chloroform (50 50), and acetone with plates dipped in potassium hydroxide solution, and of gas chromatographic retention indexes in SE 30 for 277 compounds provided a two principal components model that explains 82% of the total variance. The scores plot allowed identification of unknowns or restriction of the range of inquiry to very few candidates. Comparison of these candidates with those selected from another principal components model derived from thin-layer chromatographic data only allowed identification of the drug in all the examined cases. 

Figure 4.12 Principal component analysis of the major elements in Coumiac limestones. 91 percent of the variance is explained by the first two components. The data can be explained by the combination of three chemical end-members calcitic (CaO and C02), detrital (Si02 and A1203), and organic (organic C and Fe203). Because of the closure condition these three end-members translate into only two significant components.

Figure 4.13 Principal component analysis of the mean isotopic data for oceanic islands (courtesy of Vincent Salters). In the top left corner, the plane of the first two components (the Mantle Plane of Zindler et al, 1982) explains 93 percent of the variance. Component 1 is dominated by lead isotopes, component 2 by Sr and Nd isotopes. Other components are plotted for reference. In the top right corner, the Mantle Plane is viewed sideways along the direction of the second component, so the distance of each point to the plane can be easily seen. In the bottom left corner, it is viewed along the axis of the first component. The bottom right corner shows how little variance is left with components 3 and 4.

The method of PLS bears some relation to principal component analysis instead of Lnding the hyperplanes of maximum variance, it Lnds a linear model describing some predicted variables in terms of other observable variables. It is used to Lnd the fundamental relations between two matrices (X andY), that is, a latent variable approach to modeling the covariance structures in these two spaces. A PLS model will try to Lnd the multidimensional direction irMIspace that explains the maximum multidimensional variance direction in flrfspace. 

Spectral data are highly redundant (many vibrational modes of the same molecules) and sparse (large spectral segments with no informative features). Hence, before a full-scale chemometric treatment of the data is undertaken, it is very instructive to understand the structure and variance in recorded spectra. Hence, eigenvector-based analyses of spectra are common and a primary technique is principal components analysis (PC A). PC A is a linear transformation of the data into a new coordinate system (axes) such that the largest variance lies on the first axis and decreases thereafter for each successive axis. PCA can also be considered to be a view of the data set with an aim to explain all deviations from an average spectral property. Data are typically mean centered prior to the transformation and the mean spectrum is used a base comparator. The transformation to a new coordinate set is performed via matrix multiplication as... 

Principal Components Analysis (PCA) Compression by explained variance... 

The wa in equation (6) are the PLS loading weights. They are explained in the theory in references 53 - 62. Equation (7) shows how X is decomposed bilinearly (as in principal component analysis) with its own residual Epls A. T is the matrix with the score vectors as columns, P is the matrix having the PLS loadings as columns. Also the vectors of P and wa can be used to construct scatter plots. These can reveal the data structure of the variable space and relations between variables or groups of variables. Since PLS mainly looks for sources of variance, it is a very good dirty data technique. Random noise will not be decomposed into scores and loadings, and will be stored in the residual matrices (E and F), which contain only non-explained variance . 

Principal component analysis (PCA) and multivariate curve resolution-alternating least squares (MCR-ALS) were applied to the augmented columnwise data matrix D1"1", as shown in Figure 11.16. In both cases, a linear mixture model was assumed to explain the observed data variance using a reduced number of contamination sources. The bilinear data matrix decomposition used in both cases can be written by Equation 11.19 ... 

As an example, Tables 13.14 and 13.15 show the results of applying principal components analysis to the 10 volatile compounds (methanol, 1-propanol, isobutanol, 2-and 3-methyl-1-butanol, 1-hexanol, cw-3-hexen-l-ol, hexanoic acid, octanoic acid, decanoic acid and ethyl octanoate) analyzed in 16 varietal wines (Pozo-Bay6n et al. 2001), obtained with the STATISTICA program Factor Analysis procedure in Multivariate Exploratory Techniques module, and using Principal Components as Extraction method). The results include the factor loadings matrix for the two first principal components selected q = 2), which explains 70.1% of the total variance (Table 13.14). The first principal component is strongly correlated with d.y-3-hexen-l-ol (-0.888), 1-hexanol (-0.885), 1-propanol (0.870), and... 

The structural variance of the dataset was analyzed with principal component analysis (PCA) [9] performed on the complete set of ALMOND descriptors calculated for the compounds which comprised the training and test sets. The first two components explained 35% of the stmctural variance of the dataset. Figure 9.1 shows that no structural outliers are present in the dataset and that the training and test sets share similar chemical space. 

Based on the Principal Component Analysis, the LIN index (leaching index) and the VIN index (volatility index) were defined in terms of the first and second PCs, respectively, explaining 92.7% of the total variance [Gramatica and Di Guardo, 2002]. PCs were calculated on a data set of 135 pesticides, described by vapour pressure (Vp), Henry s law constant (H), water solubility (Sw), and octanol/water (Kqw) and organic carbon (Kqc) partition coefficients. The LIN and VIN indices are defined as the following ... 

One way to try to alleviate the problem of correlated descriptors is to perform a principal components analysis (see Section 9.13). Those principal components which explain (say) 90% of the variance may be retained for the subsequent calculations Alternatively, those principal components for which the associated eigenvalue exceeds unity may be chosen, or the principal components may be selected using more complex approaches based on cross-validation (see Section 12.12.3). It may be important to scale the descriptors (e.g. using autoscaling) prior to calculating the principal components. However, unless each principal component is largely associated with any particular descriptor it can be difficult to interpret the physical meaning of any subsequent results. ... 

The application of principal components analysis to the TLC behaviour of a large number of basic drugs including amodiaquine has been studied (38). A two-component model explains 77% of the total variance in four... 

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