Variance, principal component

Chen, R, Ln, Y., Harrington, P.B. (2008) Biomarker profiling and reprodncibiUty stndy of MALDI-MS measurements of Escherichia coli by analysis of variance-principal component analysis. Analytical Chemistry, 80, 1474-1481. 

The important underlying components of protein motion during a simulation can be extracted by a Principal Component Analysis (PGA). It stands for a diagonalization of the variance-covariance matrix R of the mass-weighted internal displacements during a molecular dynamics simulation. 

For most data analysis applications the first three to five principal components give the predominant part of the variance. 

So we have found a pair of axes that we can use as the basis of a new coordinate system. And since each axis spans the maximum possible amount of variance in the data, we can be assured that there are no axes that can serve as a more efficient frame of reference than these two. Each axis is a factor or principal component of the data. Together, they comprise the basis space of this data set. 

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. 

We have seen that PCR and RRR form two extremes, with CCA somewhere in between. RRR emphasizes the fit of Y (criterion ii). Thus, in RRR the X-components t, preferably should correlate highly with the original T-variables. Whether X itself can be reconstructed ( back-fitted ) from such components t, is of no concern in RRR. With standard PCR, i.e. top-down PCR, the emphasis is initially more on the X-side (criterion i) than on the T-side. CCA emphasizes the importance of correlation whether the canonical variates t and u account for much variance in each respective data set is immaterial. Ideally, of course, one would like to have the best of all three worlds, i.e. when the major principal components of X (as in PCR) and the major principal components of Y (as in RRR) happen to be very similar to the major canonical variables (as in CCA). Is there a way to combine these three desiderata — summary of X, summary of Y and a strong link between the two — into a single criterion and to use this as a basis for a compromise method The PLS method attempts to do just that. 

PLS has been introduced in the chemometrics literature as an algorithm with the claim that it finds simultaneously important and related components of X and of Y. Hence the alternative explanation of the acronym PLS Projection to Latent Structure. The PLS factors can loosely be seen as modified principal components. The deviation from the PCA factors is needed to improve the correlation at the cost of some decrease in the variance of the factors. The PLS algorithm effectively mixes two PCA computations, one for X and one for Y, using the NIPALS algorithm. It is assumed that X and Y have been column-centred as usual. The basic NIPALS algorithm can best be demonstrated as an easy way to calculate the singular vectors of a matrix, viz. via the simple iterative sequence (see Section 31.4.1) ... 

We have seen that PLS regression (covariance criterion) forms a compromise between ordinary least squares regression (OLS, correlation criterion) and principal components regression (variance criterion). This has inspired Stone and Brooks [15] to devise a method in such a way that a continuum of models can be generated embracing OLS, PLS and PCR. To this end the PLS covariance criterion, cov(t,y) = s, s. r, is modified into a criterion T = r. (For... 

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).

A difficulty with Hansch analysis is to decide which parameters and functions of parameters to include in the regression equation. This problem of selection of predictor variables has been discussed in Section 10.3.3. Another problem is due to the high correlations between groups of physicochemical parameters. This is the multicollinearity problem which leads to large variances in the coefficients of the regression equations and, hence, to unreliable predictions (see Section 10.5). It can be remedied by means of multivariate techniques such as principal components regression and partial least squares regression, applications of which are discussed below. 

Fig. 37.3. Principal components biplot showing the positions of 6 substituted oxazepine (O) and 6 substituted thiazepine (S) neuroleptics with respect to three physicochemical parameters and two biological activities [41,43]. The data are shown in Table 37.6. The thiazepine analogs are represented by means of filled symbols. The horizontal and vertical components represent 50 and 39%, respectively, of the variance in the data.

Principal component regression Linear projection Fixed shape, linear a, maximum variance of projected inputs (3, minimum output prediction error... 

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. 

However, there is a mathematical method for selecting those variables that best distinguish between formulations—those variables that change most drastically from one formulation to another and that should be the criteria on which one selects constraints. A multivariate statistical technique called principal component analysis (PCA) can effectively be used to answer these questions. PCA utilizes a variance-covariance matrix for the responses involved to determine their interrelationships. It has been applied successfully to this same tablet system by Bohidar et al. [18]. 

The main difference between factor analysis and principal component analysis is the way in which the variances of Eq. (8.20) are handled. Whereas the interest of FA is directed on the common variance var Xij)comm and both the other terms are summarized as unique variance... 

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. 

---