Equality methods classification

Clays (qv) are aluminosihcate minerals, some of which have definite chemical compositions. In regard to tar sands, however, clay is only a size classification and is usually deterrnined by a sedimentation method. According to the previous definition of fines, the fines fraction equals the sum of the silt and clay fractions. The clay fraction over a wide range of fines contents is a relatively constant 30% of the fines. 

Sieving Methods and Classification Sieving is probably the most frequently used and abused method of analysis because the equipment, an ytical procedure, and basic concepts are deceptively simple. In sieving, the particles are presented to equal-size apertures that constitute a series of go-no-go gauges. Sieve analysis presents three major difficulties (1) with woven-wire sieves, the weaving process produces three-dimensional apertures with considerable tolerances, particularly for fine-woven mesh (2) the mesh is easily damaged in use (3) the particles must be efficiently presented to the sieve apertures. 

The approach of Fisher (1938) was originally proposed for discriminating two populations (binary classification), and later on extended to the case of more than two groups (Rao 1948). Here we will first describe the case of two groups, and then extend to the more general case. Although this method also leads to linear functions for classification, it does not explicitly require multivariate normal distributions of the groups with equal covariance matrices. However, if these assumptions are not... 

Not all aspects of the theory are dealt with in equal depth. Some are just mentioned, some even omitted. For example, the method of effective (equivalent) operators deserves mentioning [321]. It allows one to take into account the main part of relativistic effects but at the same time to preserve the LS coupling used for classification of the energy spectra of the atoms or ions considered. 

It is not always possible to divide the classes exactly into two groups by this method (see Figure 4.26), but the misclassified samples are far from the centre of both classes, with two class distances that are approximately equal. It would be possible to define a boundary towards the centre of the overall dataset, where classification is deemed to be ambiguous. 

Crystalline compounds exist in a great many crystal forms. The accepted method for the crystallographic classification of crystals is based on the angles between the crystal faces. In this classification system, the types of crystal forms are not related to the relative sizes of the crystal faces, since the relative development of the faces is characteristic of the specific material. The cubic system, for example, is characterized by the fact that the faces of a cubic crystal can be referred to three equal and mutually perpendicular axes. The actual macrocrystal may be a cube, a needle, a plate, or an aggregate of imperfect crystals. 

This supervised classification method, which is the most used, accepts a normal multivariate distribution for the variables in each population ((Ai,..., A ) Xi) ), and calculates the classification functions minimising the possibility of incorrect classification of the observations of the training group (Bayesian type rule). If multivariate normality is accepted and equality of the k covariance matrices ((Ai,..., Xp) NCfti, X)), Linear Discriminant Analysis (LDA) calculates... 

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