Linear classification defined

Two groups of objects can be separated by a decision surface (defined by a discriminant variable). Methods using a decision plane and thus a linear discriminant variable (corresponding to a linear latent variable as described in Section 2.6) are LDA, PLS, and LR (Section 5.2.3). Only if linear classification methods have an insufficient prediction performance, nonlinear methods should be applied, such as classification trees (CART, Section 5.4), SVMs (Section 5.6), or ANNs (Section 5.5). 

For particles of regular geometric shape, values of and ip can be calculated readily from the equivalent sphere definition and by Eq. (6.20), as listed in Table 6.2 for a number of geometric solids. However, the catalyst most commonly used in ammonia synthesis is a crushed and classified material which consists of a mixture of irregular particles having linear dimensions defined by the sieves used in the classification. The surface area of individual catalyst particles is very irregular and complex, and does not permit direct measurement or calculation. Consequently, a nominal sphericity factor of 0.65, as reported in the literature for... 

The geometrical measurements previously extracted help the making decision system to decide for example whether the defect is linear or not. This defect discrimination into two categories is considered as a first attempt for defect classification. To this end, we define a linearity ratio (Ri) Rl =Length / width. If Rl is equal or near to "1", the defect is volumic, otherwise it is a linear defect. 

FIGURE 5.4 Linear discriminant scores dj for group j by the Bayesian classification rule based on (Equation 5.2). mj, mean vector of all objects in group j Sp1, inverse of the pooled covariance matrix (Equation 5.3) x, object vector (to be classified) defined by m variables Pj, prior probability of group j. 

The dimension of a space equals the number of elements in a basis, which is defined as a set of elements such that every element in the space is equal to a unique linear combination of them. Therefore, P steady-state mechanisms can be chosen in terms of which all others can be uniquely expressed. This gives us a unique way to symbolize each steady-state mechanism and its overall reaction, but it does not provide a classification system for them which is valid from a chemical viewpoint, because the choice of a basis is arbitrary and is not dictated, in general, by any consideration of chemistry. A classification system for mechanisms is our next topic. 

In discussions of structure we must limit ourselves to the case of linear polymers, for with the branched polymers so far studied their complexity has defied analysis. Here, however, as I have discussed elsewhere [11], it is possible to draw up a rational classification of the types of structure as defined by the arrangements of hydrogen bonds taken as the main element in intra- and inter-chain linking. 

This classification technique uses a space that is defined by a unique set of vectors called linear discriminants or LDs. Like the PCs obtained from PCA analysis, LDs are linear combinations of the original M-variables in the X-data that are completely independent of (or orthogonal to) one another. However, the criterion for determining LDs is quite different than the criterion for determining PCs each LD is sequentially determined such... 

From the set E of equations the construction of the system matrices follow directly. The approach outlined before, making use of the classification strategy allows the general reduction of the initial balances into a set of equations smaller in size than that suggested by Vaclavek. The reduced set of balance equations given by eq. (15), or (18) define now the following weighted least squares problem for the reconciliation of the measurement errors. In the linear case... 

Classification methods are designed to deal with well-defined classes. However, many diseases are characterized by an essentially continuous transition from healthy to the disease state. These types of problems are most appropriately dealt with by a regression approach. For two-class problems, classification by LDA and linear regression are equivalent. If the number of samples/class for the two classes is the same, then the two approaches are identical. The regression approach has the additional advantage that robust versions, which can detect the presence of outliers, can be readily implemented. 

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