First principal component

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. 

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 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. 

Fig. 34.4. The two first principal components of the data matrix of the spectra given in Fig. 34.2.

Fig. 34.34. The three first principal components obtained by a local PCA (a) zero component region, (b) up-slope selective region, (c) down-slope selective region (d) three-component region. The spectra included in the local PCA are indicated in the score plot and in the chromatogram.

Fio. 11. Simple illustration of PCA. is the first principal component and < 2 is the second principal component. 

Mathematically, X is decomposed into a model for the first principal component ... 

A PCA was performed on all relevant properties of the soil. The first principal component explained about 22% of the observed variation, while the second accounted for 17% of the observed variation. On the related scatter plot, the UREA treatment lies in upper right quadrant and the SLUDGE treatment is in the upper left quadrant. The SUNFLOWER and CONTROL treatments were found in lower left and right quadrants respectively (Figure 1). 

Because the first principal component is always situated in direction of the largest variation of the data, the intensity of ypc should be increased compared with the original intensities from which it was calculated, namely theoretically according to the -dimensional Pythagoras principle... 

The PCA can be interpreted geometrically by rotation of the m-dimensional coordinate system of the original variables into a new coordinate system of principal components. The new axes are stretched in such a way that the first principal component pi is extended in direction of the maximum variance of the data, p2 orthogonal to pi in direction of the remaining maximum variance etc. In Fig. 8.15 a schematic example is presented that shows the reduction of the three dimensions of the original data into two principal components. 

Figure 14-3 (a) The representation of two columns of a matrix in row space. The vector sum of the two column vectors is the first principal component (PCI), (b) A close-up view of Figure 14-3a, illustrating the line segments, direction angles, and projection of Columns 1 and 2 onto the first principal component. 

Figure 27-3 First principal component from concentration spectra.

I would expect PLS to outperform PCR, and the loading of the first principal component to be mostly located around the lower wavelength peak for PLS. (Paul Chabot)... 

When applied to electronic nose data the presence of various sources of correlated disturbances has to be considered. As an example, sample temperature fluctuations induce correlated disturbances, which may be described by principal components of highest order. When these disturbances are important the first principal component has to be eliminated in order to emphasize the relevant data properties. A set of algorithms called Minor Component Analysis (MCA) was introduced to take into account these phenomena mainly in image analysis [17]. 

PCA is based only on the variances among spectra. No content information is used to generate the preliminary factors. In a series of mixtures of water and methanol (shown in Fig. 6.3), for instance, the first Principal Component (see Fig. 6.7) shows the positive and negative lobes representing the shifting of water in a positive direction and methanol in a negative direction. This is based solely on the change in... 

Fig. 6.7. First principal component of water-methanol mixture in Fig. 6.4.

The direction in a variable space that best preserves the relative distances between the objects is a latent variable which has maximum variance of the scores (these are the projected data values on the latent variable). This direction is called by definition the first principal component (PCI). It is defined by a loading vector... 

FIGURE 3.1 Scatter plot of demo data from Table 3.1. The first principal component (PCI) is defined by a loading vectorp — [0.839, 0.544], The scores are the orthogonal projections of the data on the loading vector. 

PCA score 1 (PCI, first principal component) is the linear latent variable with the maximum possible variance. The direction of PC2 is orthogonal to the direction of PCI and again has maximum possible variance of the scores. Subsequent PCs follow this mle. 

As in many such problems, some form of pretreatment of the data is warranted. In all applications discussed here, the analytical data either have been untreated or have been normalized to relative concentration of each peak in the sample. Quality Assurance. Principal components analysis can be used to detect large sample differences that may be due to instrument error, noise, etc. This is illustrated by using samples 17-20 in Appendix I (Figure 6). These samples are replicate assays of a 1 1 1 1 mixture of the standard Aroclors. Fitting these data for the four samples to a 2-component model and plotting the two first principal components (Theta 1 and Theta 2 [scores] in... 

The first principal component values (Theta 1) for each sample were determined and these values were correlated with the total PCB concentration (Figure 14) recorded for each sample in a separate computer data base that contained other environmental data such as hydrology and sediment texture. The results indicated that certain samples deviated by factors of about two. Upon examining the sample records, the recorded dilution values... 

Earlier it was mentioned, and demonstrated using the Fisher Iris example (Section 12.2.5), that the PCA scores (T) can be used to assess relationships between samples in a data set. Similarly, the PCA loadings (P) can be used to assess relationships between variables in a data set. For PCA, the first score vector and the first loading vector make up the first principal component (PC), which represents the most dominant source of variability in the original x data. Subsequent pairs of scores and loadings ([score vector 2, loading vector 2], [score vector 3, loading vector 3]...) correspond to the next most dominant sources of variability. 

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