Dimensionality of the data

When we examine the plots in Figure 56 we see that the PRESS decreases each time we add another factor to the basis space. When all of the factors are included, the PRESS drops all the way to zero. Thus, these fits cannot provide us with any information about the dimensionality of the data. The problem is that we are trying to use the same data for both the training and validation data. We lose the ability to assess the optimum rank for the basis space because we do not have independent validation samples that contain independent noise. So, the more factors we add, the better the calibration is able to model the particular noise in these samples. When we use all of the factors, we are able to model the noise completely. Thus, when we predict the concentrations for... 

When we examine the plots we see that the PRESS decreases each time we add another factor to the basis space. When all of the factors are included, the PRESS drops all the way to zero. Thus, these fits cannot provide us with any information about the dimensionality of the data. The problem is that we are... 

Visualizing Data, the reader may have guessed from previous sections that graphical display contributes much toward understanding the data and the statistical analysis. This notion is correct, and graphics become more important as the dimensionality of the data rises, especially to three and more dimensions. Bear in mind that ... 

A question that often arises in multivariate data analysis is how many meaningful eigenvectors should be retained, especially when the objective is to reduce the dimensionality of the data. It is assumed that, initially, eigenvectors contribute only structural information, which is also referred to as systematic information. 

Filters are designed to remove unwanted information, but do not address the fact that processes involve few events monitored by many measurements. Many chemical processes are well instrumented and are capable of producing many process measurements. However, there are far fewer independent physical phenomena occurring than there are measured variables. This means that many of the process variables must be highly correlated because they are reflections of a limited number of physical events. Eliminating this redundancy in the measured variables decreases the contribution of noise and reduces the dimensionality of the data. Model robustness and predictive performance also require that the dimensionality of the data be reduced. 

Subsequently 36 strains of aerobic endospore-forming bacteria, consisting of six Bacillus species and one Brevibacillus species could be discriminated using cluster analysis of ESMS spectra acquired in the positive ion mode (m/z 200-2000).57 The analysis was carried out on harvested, washed bacterial cells suspended in aqueous acidic acetonitrile. The cell suspensions were infused directly into the ionization chamber of the mass spectrometer (LCT, Micromass) using a syringe pump. Replicates of the experiment were performed over a period of six months to randomize variations in the measurements due to possible confounding factors such as instrumental drift. Principal components analysis (PCA) was used to reduce the dimensionality of the data, fol-... 

Pranckeviciene et al.11 have assessed the NMR spectra of pathogenic fungi and of human biofluids, finding the spectral signature that comprises a set of attributes that serve to uniquely identify and characterize the sample. This use of GAs effectively reduces the dimensionality of the data, and it can speed up later processing as well as make it more reliable. 

One of the keys to multivariate analysis is the ability to reduce the dimensionality of the data so that it can be displayed in two, three or four (time-dependent) dimensional displays. The primary tool for achieving this, principal component analysis (PCA) [158], is the cornerstone of chemometrics as it accomplishes several things ... 

It reduces the dimensionality of the data so that it can be displayed or manipulated further. 

Factor The result of a transformation of a data matrix where the goal is to reduce the dimensionality of the data set. Estimating factors is necessary to construct principal component regression and partial least-squares models, as discussed in Section 5.3.2. (See also Principal Component.)... 

Eigenvectors reduce the dimensionality of the data matrix when the rank of the covariance matrix is E < V, so that V — E eigenvalues vanish, or when some eigenvectors are not significant, the use of some classification methods with the scores on the first eigenvectors, instead of the original variables, can avoid singular matrices or/and noticeably speed up data analysis. 

The PCA process reduces the dimensionality of the data set. For example, consider the spectra of five mixtures of solutions of the chemical warfare agent ethyl-A A-dimethylphosphoroamidocyanidate (GA) in water shown on the left side in Fig. 5-8. The spectra of the pure components are shown... 

PCA is simply a method for reducing the dimensionality of the data set and for removing dependent data (8). Although each of the five mixture spectra in Fig. 5-8 contain almost 4,000 data points, each can be expressed as a sum of two spectra (loadings or pure) containing 4,000 data points each. Thus the dimensionality is reduced from 5 x 4,000, or 20,000 data points, to 2 x 4,000 + 10, or 8,010 values (the value of 10 is for the two coefficients... 

Figure 9.4. Basic types of 2D chemical structure data. The amount of information and the complexity of searching increases with the dimensionality of the data.

The most useful result of multivariate analysis procedures is the reduction in apparent dimensionality of the data. From an initial collection of several hundred mass peaks, the data are reduced to only a few factors, each of which is by definition a linear combination of the original mass peak intensities. By plotting these linear combinations in the form of spectra, significant information about the chemical components underlying the factors can be obtained. Often this requires rotation of the factors in order to optimize the chemical component patterns. 

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