Principal components analysis problem

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). 

Principal Component Analysis (PCA) PCA is used to recognize patterns in data and reduce the dimensionality of the problem. Let the matrix A now represent data with the columns of A representing different samples and the rows representing different variables. The covariance matrix is defined as... 

The authors did not attempt to address this issue. Although construct validity is important, it does not guarantee taxometric validity, so both issues must be examined, especially in the case of null finding. For example, Franklin et al. could have performed a principal component analysis and examined loadings of the three indicators on the first unrotated component. As mentioned previously, these loadings can give a sense of indicator validity, and if the INTR failed to load sufficiently, this would indicate a measurement problem. 

Finally, approaches are emerging within the data reconciliation problem, such as Bayesian approaches and robust estimation techniques, as well as strategies that use Principal Component Analysis. They offer viable alternatives to traditional methods and provide new grounds for further improvement. 

In Chapter 11 some recent approaches for dealing with different aspects of the data reconciliation problem are discussed. A more general formulation in terms of a probabilistic framework is first introduced and its application in dealing with gross error is discussed in particular. In addition, robust estimation approaches are considered, in which the estimators are designed so they that are insensitive to outliers. Finally, an alternative strategy that uses Principal Component Analysis is reviewed. 

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... 

In the past few years, PLS, a multiblock, multivariate regression model solved by partial least squares found its application in various fields of chemistry (1-7). This method can be viewed as an extension and generalization of other commonly used multivariate statistical techniques, like regression solved by least squares and principal component analysis. PLS has several advantages over the ordinary least squares solution therefore, it becomes more and more popular in solving regression models in chemical problems. 

The rapid classification of polymeric species is an important problem in the area of analytical chemistry in general and of particular relevance to recycling and waste management. To accomplish classification tasks, a combination of spectral data and principal component analysis (PCA) is often employed. 

Important conclusions about the interrelationship among aromaticity indices drawn from energetic, structural, and magnetic criteria stem from principal component analysis of the problem (89JA7). The scheme of principal components is given by... 

There are many advantages in using this approach to feature selection. First, chance classification is not a serious problem because the bulk of the variance or information content of the feature subset selected is about the classification problem of interest. Second, features that contain discriminatory information about a particular classification problem are usually correlated, which is why feature selection methods using principal component analysis or other variance-based methods are generally preferred. Third, the principal component plot... 

In Chapter 16, Lavine colleagues return to a compound classification problem by using a combination of principal component analysis and a genetic algo-... 

Problems like overlapping and interfering of fluorophores is overcome by the BioView sensor, which offers a comprehensive monitoring of the wide spectral range. Multivariate calibration models (e.g., partially least squares (PLS), principal component analysis (PCA), and neuronal networks) are used to filter information out of the huge data base, to combine different regions in the matrix, and to correlate different bioprocess variables with the courses of fluorescence intensities. 

Various approaches can be taken for constructing the U matrix. With PCR, a principal components analysis is used because PCA is an efficient method for finding linear combinations of variables that describe variation in the row space of R (See Section 4.2.2). With analytical chemistry data, it is usually possible to describe the variation in R using significantly fewer PCs than the number of original variables. This small number of columns effectively eliminates the matrix inversion problem. 

Both component and factor analysis as defined by equations 17 and 18 aim at the identification of the causes of variation in the system. The analyses are performed somewhat differently. For the principal components analysis, the matrix of correlations defined by equation 10 is used. For the factor analysis, the diagonal elements of the correlation matrix that normally would have a value of one are replaced by estimates of the amount of variance that is within the common factor space. This problem of separation of variance and estimation of the matrix elements is discussed by Hopke et al. (4). 

Prior Applications. The first application of this traditional factor analysis method was an attempt by Blifford and Meeker (6) to interpret the elemental composition data obtained by the National Air Sampling Network(NASN) during 1957-61 in 30 U.S. cities. They employed a principal components analysis and Varimax rotation as well as a non-orthogonal rotation. In both cases, they were not able to extract much interpretable information from the data. Since there is a very wide variety of sources of particles in 30 cities and only 13 elements measured, it is not surprising that they were unable to provide much specificity to their factors. One interesting factor that they did identify was a copper factor. They were unable to provide a convincing interpretation. It is likely that this factor represents the copper contamination from the brushes of the high volume air samples that was subsequently found to be a common problem ( 2). 

Unfortunately, there are some technical difficulties associated with the determinant criterion (ref. 28). Minimizing the determinant (3.66) is not a trivial task. In addition, the method obviously does not apply if det[ V(p) ] is zero or nearly zero for all parameter values. This is the case if there exist affine linear relationships among the responses y, y2,. .., yny, as we discussed in Section 1.8.7. To overcome this problem the principal component analysis of the observations is applied before the estimation step. 

As seen from Fig. 5.3, the substrate concentration is most sensitive to the parameters around t = 7 hours. It is therefore advantageous to select more observation points in this region when designing identification experiments (see Section 3.10.2). The sensitivity functions, especially with respect to Ks and Kd, seem to be proportional to each other, and the near—linear dependence of the columns in the Jacobian matrix may lead to ill-conditioned parameter estimation problem. Principal component analysis of the matrix STS is a powerful help in uncovering such parameter dependences. The approach will be discussed in Section 5.8.1. 

Practical identifiability is not the only problem that can be adressed by principal component analysis of the sensitivity matrix. In (refs. 29-30) several examples of model reduction based on this technique are discussed. 

There are apparently many multivariate statistical methods partly overlapping in scope [11]. For most problems occurring in practice, we have found the use of two methods sufficient, as discussed below. The first method is called principal component analysis (PCA) and the second is the partial least-squares projection to latent structures (PLS). A detailed description of the methods is given in Appendix A. In the following, a brief description is presented. 

The mathematical procedure for finding such discriminant functions is to solve the eigenvalue problem of the quotient B i.e. to find the characteristic roots and eigenvectors as known from principal components analysis ... 

The exclusive consideration of common factors seems to be promising, especially for such environmental analytical problems, as is shown by the variance splitting of the investigated data material (Tab. 7-2). Errors in the analytical process and feature-specific variances can be separated from the common reduced solution by means of estimation of the communalities. This shows the advantage of the application of FA, rather than principal components analysis, for such data structures. Because the total variance of the data sets has been investigated by principal components analysis, it is difficult to separate specific factors from common factors. Interpretation with regard to environmental analytical problems is, therefore at the very least rendered more difficult, if not even falsified for those analytical results which are relatively strongly affected by errors. 

---