Eigenvectors significant

If we can find A, we shall have found an orthogonal set of eigenvectors. It is interesting and significant to note at this point that A is only one of many equally valid orthogonal sets of eigenvectors. 

So, we can discard the third eigenvector and, along with it, that portion of the variance in our spectra that displaced the data out of the plane of the noise-free data. We are in fact, discarding a portion of the noise without significantly distorting the spectra The portion of the noise we discard is called the extracted error or the residuals. Remember that the noise we added also displaced the points to some extent within the plane of the noise-free data. This portion of the noise remains in the data because it is spanned by the eigenvectors that we must retain. The noise that remains is called the imbedded error. The total error is sometimes called the real error. The relationship among the real error (RE), the extracted error (XE), and the imbedded error (IE) is... 

So now we understand that when we use eigenvectors to define an "abstract factor space that spans the data," we aren t changing the data at all, we are simply finding a more convenient coordinate system. We can then exploit the properties of eigenvectors both to remove noise from our data without significantly distorting it, and to compress the dimensionality of our data without compromising the information content. 

Observe that in variant 1 the accuracy in specifying the first eigenvector is of significant importance. For the values... 

Hence, the number of structural eigenvectors is the largest r for which Malinowski s F-ratio is still significant at a predefined level of probability a (say 0.05) ... 

Each mixture spectrum is a linear combination of the nc significant eigenvectors. Equally, the pure spectra are linear combinations of the first nc PCs. A target... 

Having derived a solution for two-component systems, we could try and extend this solution to three-component systems. A PCA of a data set of spectra of three-component mixtures yields three significant eigenvectors and a score matrix with three scores for each spectrum. Therefore, the spectra are located in a three-dimensional space defined by the eigenvectors. For the same reason, explained for the two-component system, by normalization, the ternary spectra are found on a surface with one dimension less than the number of compounds, in this case, a plane. 

The two eigenvectors define a plane in the original variable space. This process can be repeated systematically until the eigenvalue associated with each new eigenvector is of such a small magnitude that it represents the noise associated with the observations more than it does information. In the limit where the number of significant eigenvectors equals the number... 

There are many advantages in selecting only the significant ne eigenvectors and singular values for the representation of Y. In fact, from now on we only use this selection and introduce an appropriate nomenclature. 

Y in equation (5.9) is a good representation of the original matrix Y, but not identical. There is a residual matrix of decreasing significance the more eigenvectors are used to compute Y. ... 

Similar to the structure of U and V that reveals the significance of the eigenvectors, the structure of R allows the identification of the correct rank. 

Recall, the standard deviation of the added noise in Y was lxlO-3. It is reached approximately after the removal of 3 sets of eigenvectors (at t=4). Note that, from a strictly statistical point of view, it is not quite appropriate to use Matlab s std function for the determination of the residual standard deviation since it doesn t properly take into account the gradual reduction in the degrees of freedom in the calculation of R. But it is not our intention to go into the depths of statistics here. For more rigorous statistical procedures to determine the number of significant factors, we refer to the relevant chemometrics literature on this topic. 

Figure 2. Projection of hoursO, with complaints, and , without complaints, of air pollution on the two most significant eigenvectors of the Karhunen-Loeve transformed, seven-dimensional feature space. Reproduced with permission from Ref. 7. Copyright 1984,...

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