Scatter within-class

Fisher suggested to transform the multivariate observations x to another coordinate system that enhances the separation of the samples belonging to each class tt [74]. Fisher s discriminant analysis (FDA) is optimal in terms of maximizing the separation among the set of classes. Suppose that there is a set of n = ni + U2 + + rig) m-dimensional (number of process variables) samples xi, , x belonging to classes tt, i = 1, , g. The total scatter of data points (St) consists of two types of scatter, within-class scatter Sw and hetween-class scatter Sb- The objective of the transformation proposed by Fisher is to maximize S while minimizing Sw Fisher s approach does not require that the populations have Normal distributions, but it implicitly assumes that the population covariance matrices are equal, because a pooled estimate of the common covariance matrix (S ) is used (Eq. 3.45). 

The first FDA vector wi that maximizes the scatter between classes (Sb) while minimizing the scatter within classes Sw) is obtained as... 

In the general case, for several classes, the sums of squares representing between-class and within-class scattering are then extended to matrices B and W ... 

W - pooled within-class scatter matrix, see Eq. 5-33, under the assumption that the covariance matrices of the classes are equal (in the statistical sense) ... 

Prior to the actual classification, the FLDC performs a linear mapping to a lower dimensional subspace optimised for class separability, based on the between-class scatter and the within-class scatter of the training set. In classification, each sample is assigned to the class giving the highest log-likelihood using a linear classifier. 

A classification technique based on Fisher linear discrimination (FED) was applied to features extracted from the phenomenological model just described. To illustrate how this method works, Fig 3 presents a scatter plot of two features (F2 versus FI) for several fictional events, each a member from one of three distinct classes (A, B and C). While neither feature alone provides ample separation between all three classes, a combination of these features can be found which clearly separates all classes. FED finds the line in this feature space that maximizes between-class scatter and minimizes within-class scatter upon projection. The mean and standard deviation of each class projected onto this Fisher line can be used to define a probability distribution function (PDF). When projecting a new event onto the Fisher line, the PDF assigns a probability that it belongs to a particular class. Feature saliency and PDF stability are critical in assessing the confidence level associated with a prediction, and are thoroughly examined in Dills dissertation. ... 

Fig 3. Example data consisting of three classes (A, B C) with two descriptors (FI F2) illustrates the Fisher linear discrimination technique. When the data is projected onto the Fisher line, between-class scatter (dashed arrows) is maximized and within-class scatter (solid arrows) is minimized. Gaussian curves for each class, defined by the mean and standard deviation upon projection onto the Fisher line, are also provided. 

The generalization of the within-class scatter matrix Sw for g classes is... 

Equation 3.54 can be rewritten by adding —Xj + Xj to each term and rearranging the sums so that the total scatter is the sum of the within-class scatter and the between-class scatter as [63] ... 

Between-class scatter matrix Within-class scatter matrix Total scatter matrix Scores matrix (n x a)... 

The within classes scatter matrix, together with its element is given by... 

Detailed analysis of t-RNA molecules (similar results have been obtained with other classes of RNA and with DNA) has shown that although A, G, C, U (or T) are the major nucleotides a variety of other nucleotides are present. Some such as inosine (I) and pseudo-uridylic acid (IP) may be scattered within the molecules whereas others are limited to specific sites. Isopentenylaminopurine (IPA) appears to be located at a specific position near to the anticodon in certain t-RNAs (Fig. 5.15) and may play a part in the control of t-RNA activity through affecting the conformation of the anticodon loop or even the whole secondary structure of the molecule. Much remains to be learned about the potentially important functions of these uncommon nucleotides and considerable research effort is presently being directed to this end (see p. 298). 

Geometrical and flexibility data pertaining to the same polymers are also given in Table 1, namely the persistence length and the average chain-to-chain interaxial distance D. The first five polymers in Table 1 have D values smaller than 6 A, unlike all the following polymers (i.e., no. 6 to 19 in Table 1, Class II). This is a consequence of the relatively bulky substituents carried by Class II polymer chains. For some of the polymers in Table 1 the C0o and P literature values are widely scattered or unavailable. In those cases lower-limit values of P from experimentally determined geometrical parameters, are predicted from our model by suitable interpolation and reported within parentheses. 

The implementation of the scattering approach and of some simphfied electronic structure models for describing the transport behavior of short poly(dG)-poly(dC) DNA wires [14] have been recently independently proposed within two main classes of models. One involves dephasing [123-125] and the other involves the hybridization of the r-stack [122]. 

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