Explained variance principal components

TABLE 12. Percentage of the sample variance explained the principal components for torsion angles around (a) N1—C2 (T1-T4) and (b) C2=C3 (T5-T8) bonds... 

Here we use a PCA on the set of our NaCl spectra after area normalization. The principle of a PCA is to reduce the number of spectral variables using an orthogonal transformation and turn them into uncorrelated variables, the principal components, ranked in order of largest possible variance as explain previously. Principal components and individuals (spectra) represented in the space of the two first components are shown in Fig. 17. 

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. 

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... 

In the method of linear discriminant analysis, one therefore seeks a linear function of the variables, D, which maximizes the ratio between both variances. Geometrically, this means that we look for a line through the cloud of points, such that the projections of the points of the two groups are separated as much as possible. The approach is comparable to principal components, where one seeks a line that explains best the variation in the data (see Chapter 17). The principal component line and the discriminant function often more or less coincide (as is the case in Fig. 33.8a) but this is not necessarily so, as shown in Fig. 33.8b. 

Fig. 36.7. Percentage variance of X-content explained by the principal components from spectral data. Individual percentages (bars) are shown as well as cumulative percentages (circles).

To construct the reference model, the interpretation system required routine process data collected over a period of several months. Cross-validation was applied to detect and remove outliers. Only data corresponding to normal process operations (that is, when top-grade product is made) were used in the model development. As stated earlier, the system ultimately involved two analysis approaches, both reduced-order models that capture dominant directions of variability in the data. A PLS analysis using two loadings explained about 60% of the variance in the measurements. A subsequent PCA analysis on the residuals showed that five principal components explain 90% of the residual variability. 

Figure 38 shows the variance explained by the two principal component (PC) model as a percentage of each of the two indices batch number and time. The lower set of bars in Fig. 38a are the explained variances for the first PC, while the upper set of bars reflects the additional contribution of the second PC. The lower line in Fig. 38b is the explained variance over time for the first PC and the upper line is the combination of PC 1 and 2. Figure 38a indicates, for example, that batch numbers 13 and 30 have very small explained variances, while batch numbers 12 and 33 have variances that are captured very well by the reference model after two PCs. It is impossible to conclude from this plot alone, however, that batches 13 and 30 are poorly represented by the reference model. 

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. 

Fig. 5 Main contamination sources identified by PCA for sediments, fish, and suface water in the Ebro River basin, and explained variances for each principal component. Variable identification. Organic compounds in sediments 1, summatory of hexachlorocyclohexanes (HCHs) 2, summa-tory of DDTs (DDTs) 3, hexachlorobenzene (HCB) 4, hexachlorobutadiene (HCBu) 5, summatory of trichlorobenzenes (TCBs) 6, naphthalene 7, fluoranthene 8, benzo(a)pyrene 9, benzo(b) fluoranthene 10, benzo(g,h,i)perylene 11, benzo(k)fluoranthene 12, indene(l,2,3-cd)pyrene. Organic compounds in fish 1, hexachlorobenzene (HCB) 2, summatory of hexachlorocyclohexanes (HCHs) 3, o,p-DDD 4, o,p-DDE 5, o,p-DDT 6, p,p-DDD 7, />,/>DDE 8, />,/>DDT 9, summatory of DDTs (DDTs) 10, summatory of trichlorobenzenes (TCBs) 11, hexachlorobutadiene (HCBu) 12, fish length. Physico-chemical parameters in water 1, alkalinity 2, chlorides 3, cyanides 4, total coliforms 5, conductivity at 20°C 6, biological oxygen demand 7, chemical oxygen demand 8, fluorides 9, suspended matter 10, total ammonium 11, nitrates 12, dissolved oxygen 13, phosphates 14, sulfates 15, water temperature 16, air temperature...

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.

It was mentioned earlier that PCA is a useful method for compressing the information contained in a large number of x variables into a smaller number of orthogonal principal components that explain most of the variance in the x data. This particular compression method was considered to be one of the foundations of chemometrics, because many commonly used chemometric tools are also focused on explaining variance and dealing with colinearity. However, there are other compression methods that operate quite differently than PCA, and these can be useful as both compression methods and preprocessing methods. 

For the styrene-butadiene example, the use of the PCR method to develop a calibration for di-butadiene is summarized in Table 12.6. It should be mentioned that the data were mean-centered before application of the PCR method. Figure 12.12 shows the percentage of explained variance in both x (the spectral data) andy (the c/i-butadiene concentration data) after each principal component. After four principal components, it does not appear that the use of any additional PCs results in a large increase in the explained variance of X or y. If a PCR regression model using four PCs is built and applied to the calibration data, a fit RMSEE of 1.26 is obtained. 

The difference between PLS and PCR is the manner in which the x data are compressed. Unlike the PCR method, where x data compression is done solely on the basis of explained variance in X followed by subsequent regression of the compressed variables (PCs) to y (a simple two-step process), PLS data compression is done such that the most variance in both x and y is explained. Because the compressed variables obtained in PLS are different from those obtained in PCA and PCR, they are not principal components (or PCs) Instead, they are often referred to as latent variables (or LVs). 

The principal component space does not have the redundancy issue discussed above, because the PCs are orthogonal to one another. In addition, because each PC explains the most remaining variance in the x data, it is often the case that fewer PCs than original x variables are needed to capture the relevant information in the x data. This leads to simpler classification models, less susceptibility to overfitting through the use of too many dimensions in the model space, and less noise in the model. 

Figure 4.35. Tihe percent variance explained by each principal component for PCA Example 2.

Loadings Plot (Model and Variable Diagnostic) The loading plot in Figure 4.64 reveals that the first and se< ond loadings have nonrandom features, while the third is random in nature. This suggests a two-principal component model consistent with the percent variance explained, residuals plots, and mSECV PCA results... 

Computer Programs O) Initial factors were extracted using a principal components solution. The number of factors to be kept for rotation to a final solution was selected from a plot of the variance explained by each factor (its eigenvalue) versus its ordinal number. Usually, factors with eigenvalues larger than about 1.0 were kept. Final solutions were obtained using Varimax rotations. 

The principle of PCA consists of finding the directions in space—known as principal components (PCs)—along which the data points are furthest apart. It requires linear combinations of the initial variables that contribute most to making the samples different from each other. PCs are computed iteratively, with the first PC carrying the most information, that is, the most explained variance, and the second PC carrying most of the residual information not taken into account by the previous PC, and so on. This process can go on until as many PCs have been computed as there are potential variables in the data table. At that point, all between-sample variation has been accounted for, and the PCs form a new set of axes having two... 

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