Principal component analysis representation

Principal component analysis (PCA) is a statistical method having as its main purpose the representation in an economic way the location of the objects in a reduced coordinate system where only p axes instead of n axes corresponding to n variables (p[Pg.94]

The scope of Principal Component Analysis (PCA) is a consistent portrayal of a data set in a representation space. Mathematically, PCA is a linear transformation that may be described as S=WX. Here X is the original data set, W is the transformation matrix, and S are the data in the representation space. PCA is the simplest and most widely used method of multivariate analysis. Nonetheless, most users are seldom aware of its assumptions and sometimes results are badly interpreted. 

For principal component analysis (PCA), the criterion is maximum variance of the scores, providing an optimal representation of the Euclidean distances between the objects. 

Fig. 3. Coverage of chemistry space by four overlapping sublibraries. (A) Different diversity libraries cover similar chemistry space but show little overlap. This shows three libraries chosen using different dissimilarity measures to act as different representations of the available chemistry space. The compounds from these libraries are presented in this representation by first calculating the intermolecular similarity of each of the compounds to all of the other compounds using fingerprint descriptors and the Tanimoto similarity index. Principal component analysis was then conducted on the similarity matrix to reduce it to a series of principal components that allow the chemistry space to be presented in three dimensions.

Principal components analysis is used to obtain a lower dimensional graphical representation which describes a majority of the variation in a data set. With PCA, a new set of axes arc defined in which to plot the samples. They are constructed so that a maximum amount of variation is described with a minimum number of axes. Because it reduces the dimensions required to visualize the data, PCA is a powerftil method for studying multidimensional data sets. 

Basic Concepts. The goal of factor and components analysis is to simplify the quantitative description of a system by determining the minimum number of new variables necessary to reproduce various attributes of the data. Principal components analysis attempts to maximally reproduce the variance in the system while factor analysis tries to maximally reproduce the matrix of correlations. These procedures reduce the original data matrix from one having m variables necessary to describe the n samples to a matrix with p components or factors (p[Pg.26]

Figure 3.4 Principal components analysis of a 6 x 3 matrix (a) the six samples in the original space of three measured variables (b) the new axes (principal components PCi and PC2) obtained from the SVD of the 6 x 3 matrix (c) representation of the six samples in the space of the principal components. Note how the three original variables are correlated (the higher Xi and X2 are, the higher is. Vj). Note also how by using only the coordinates (scores) of the samples on these two principal components, the relative position of the samples in the initial variable space is captured. This is possible because the original variables are correlated. Principal components regression (PCR) uses the scores on these two new variables (the two principal components) instead of the three originally measured variables.

Principal Components Analysis is a data compression method that reduces a set of data collected on M-variables over N samples to a simpler representation that uses a much fewer number (A M) of compressed variables, called principal components (or PCs). The mathematical model for the PCA method is provided below ... 

Principal component analysis makes it possible to find a set of representations for mixture spectra in which noise and interactions are taken into account without knowing anything about the spectra of the pure components or their concentrations. The basic idea is to find a set of representations that can be linearly combined to reproduce the original mixture spectra. In PCA, Equation (4.3) is rewritten as... 

Figure 5.8b Principal Component Analysis on the four varietal wines. Representation of the loadings of the variables in PCI and PC2 in the by-plot schema of Figure 5.8a...

Because of their fixed length, descriptors are valuable representations of molecules for use in further statistical calculations. The most important methods used to compare chemical descriptors are linear and nonlinear regression, correlation methods, and correlation matrices. Since patterns in data can be hard to find in data of high dimension, where graphical representation is not available, principal component analysis (PCA) is a powerful tool for analyzing data. PCA can be used to identify patterns in data and to express the data in such a way as to highlight their similarities and differences. Similarities or diversities in data sets and their properties data can be identified with the aid of these techniques. 

Corresponding to the dimension d = 2, the poset shown in Fig. 19 can alternatively be visualized by a two-dimensional grid as is shown in Fig. 22. Both visualizations have their advantages. Structures within a Hasse diagram, e.g., successor sets, or sets of objects separated from others by incomparabilities, can be more easily disclosed by a representation like that of Fig. 19. In multivariate statistics reduction of data is typically performed by principal components analysis or by multidimensional scaling. These methods minimize the variance or preserve the distance between objects optimally. When order relations are the essential aspect to be preserved in the data analysis, the optimal result is a visualization of the sediment sites within a two-dimensional grid. 

Multipoint pharmacophore fingerprints have also been used to compare libraries. For example, Pickett et al. [17] have represented libraries by the union of the individual molecular fingerprints and were able to identify regions of multipoint pharmacophore space that were not covered or that were underrepresented. McGregor and Muskal [37,38] developed a similar approach that is based on a low-dimensional pharmacophore space obtained by applying principal components analysis to the three-point pharmacophore representations of the compounds. 

Principal components analysis is a well-established multivariate statistical technique that can be used to identify correlations within large data sets and to reduce the number of dimensions required to display the variation within the data. A new set of axes, principal components (PCs), are constructed, each of which accounts for the maximum variation not accounted for by previous principal components. Thus, a plot of the first two PCs displays the best two-dimensional representation of the total variance within the data. With pyrolysis mass spectra, principal components analysis is used essentially as a data reduction technique prior to performing canonical variates analysis, although information obtained from principal components plots can be used to identify atypical samples or outliers within the data and as a test for reproducibihty. 

---