Principal components graphical representation

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. 

High loading of PCf indicates that this vector is aligned close to the original data values that is, the transformation to the feature space defined by the new principal components matches the important (perhaps the most important) trend in the raw data. Conversely, low loading means that the PC does not match a significant trend in the data. Typically, 2-3 PCs can characterize most experimental datasets. This then allows a 2-D or a 3-D graphical representation of the results, as shown in Fig. 10.6. As powerful as it is, the PCA fails in cases where the individual sensors in the... 

The most important method for exploratory analysis of multivariate data is reduction of the dimensionality and graphical representation of the data. The mainly applied technique is the projection of the data points onto a suitable plane, spanned by the first two principal component vectors. This type of projection preserves (in mathematical terms) a maximum of information on the data structure. This method, which is essentially a rotation of the coordinate system, is also referred to as eigenvector-projection or Karhunen-Loeve- projection (ref. 8). 

Fig. 3.6 Graphical representation of the loadings of principal components for data set A.

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. 

Fig. 3. Typical gas chromatographic profiles obtained after on-fibre derivatization of carbonyl compounds with pentafluorophenylhydrazine (A) from In vitro sampling of Sansivieria trifasciata flowers and (B) from a standard aqueous solution (1,58 mM of each aliphatic aldehyde, C3-C11). (C) Graphical representation of Sansevieria trifasciata flower scent composition change during the day. Principal component analysis of the compositional data permitted to discern a coordinate system with 87% of the information.

While the mathematical expressions for the radial functions are the more complicated the larger the principal quantum number is, the graphical representation of basic features can easily be sketched as in the nonrelativistic case. By convention, the large components Pj(r) are chosen to start with positive values from the origin of the r coordinate. 

The Mohr circle representation (Fig. 9.6c) is a graphical method of relating stress components in different sets of axes. When the axes in the material rotate by an angle B, the diameter of the circle rotates by an angle 2 B. If the material yields, the circle has radius k, the constant in the Tresca yield criterion. The axes of the Mohr diagram are the tensile and shear stress components. Thus, in the left-hand circle, representing the stresses at A in Fig. 9.6b, the ends of the horizontal diameter are the principal stresses. The principal axes are parallel and perpendicular to the notch-free surface. There is a tensile principal stress Ik parallel to the surface, and a zero stress perpendicular to the surface. The points at the ends of the vertical diameter represent the stress components in the a)3 axes, rotated by 45° from the principal axes. In the a/3 axes, the shear stresses have a maximum value k, and there are equal biaxial tensile stresses of magnitude = k (the coordinate of the centre of the circle). 

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