Geometrical Description of PCA

The projection is made so that the first principal component vector, describes the direction through the swarm of data points which shows the largest variation with respect to the distribution of the points in the space. If this vector is anchored at the average point, we can make perpendicular prjections of the data points on 

The next step in the analysis is to determine whether there is systematic variation which was not accounted for by the first component and which could be described by a second component. The second component has a direction perpendicular to the first component and defines the direction through the swarm of points which describes the second next largest variation of the distribution of the data points. This constitutes a projection of the swarm of points to the plane spanned by the two first principal components. As the principal component vectors are orthogonal, they will portray different and independent principal properties. 

This process is continued until all systematic variation has been exhausted and picked up by principal components and the variation which is left is nothing but 

The variation of the principal properties in a set of compounds is quantified by the score values. This variation can be displayed by plotting the scores of different components against each other. Such score plots are very useful for selecting test compounds for experimental studies. Strategies for the selection of test system based upon principal properties are discussed in Sect. 4.6. 

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