Pattern space

Many pattern recognition methods can be explained by an intuitive geometric description of the classification problem. This point of view will be used throughout this book. 

The basic concept is the following An obj ect or an event j is described by a set of d features x. j (i = 1...d). All features of one object form a pattern. For a simpler description of the method one may assume that only two features (measurements) x and are known for 

FIGURE 1. An object is characterized by two measurements x and x j and represented in a 2-dimensional pattern space by a point with the coordinates or by a pattern vector Xj. 

FIGURE 2. Two-dimensional pattern space with two distinct clusters for the classes plus (+) and minus (-). The unknown Co) is classified to one of the two classes. 

In practical applications, a pattern space with much more than two dimensions is necessary. Clustering in a d-dimensional hyperspace is of course not directly visible or imaginable to the scientist. In this case, pattern recognition methods are helpful. 

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Compound Diffraction Pattern Spacing, A Intensity, % Refs... 

Supervised methods rely on some prior training of the system with objects known to belong to the class they define. Such methods can be of the discriminant or modeling types.11 Discriminant methods split the pattern space into as many regions as the classes encompassed by the training set and establish bounds that are shared by the spaces. These methods always classify an unknown sample as a specific class. The most common discriminant methods include discriminant analysis (DA),12 the K-nearest neighbor... 

KNN)13 14 and potential function methods (PFMs).15,16 Modeling methods establish volumes in the pattern space with different bounds for each class. The bounds can be based on correlation coefficients, distances (e.g. the Euclidian distance in the Pattern Recognition by Independent Multicategory Analysis methods [PRIMA]17 or the Mahalanobis distance in the Unequal [UNEQ] method18), the residual variance19,20 or supervised artificial neural networks (e.g. in the Multi-layer Perception21). 

Van Zele and Diener 1990 To investigate the effectiveness of water sprays in reducing hazards from HF releases. Many key variables were studied to enhance HF removal efficiency. Water-to-HF ratio is key. Upflow water sprays are more efficient than downflow sprays. Removal efficiency depends on spray nozzle configuration, nozzle size, spray pattern, spacing, etc. 

The principal aim of performing a cluster analysis is to permit the identification of similar samples according to their measured properties. Hierarchical techniques, as we have seen, achieve this by linking objects according to some formal rule set. The K-means method on the other hand seeks to partition the pattern space containing the objects into an optimal predefined number of... 

Although the layout in Figure 13 correctly classifies the data, by applying two linear discriminating functions to the pattern space, it is unable to learn from a training set and must be fully programmed before use, i.e. it must be manually set-up before being employed. This situation arises because the... 

Figure 18 Scatter plot of absorbance data at three wavelengths. An. A , and A, g, from Table 11. The high degree of colinearity, or correlation, between these data is evidenced by their lying on a plane and not being randomly distributed in the pattern space...

One of the attractions of supramolecular chemistry is the extraordinary potential for synthesis of new materials that can be achieved much more rapidly and more effectively than with conventional covalent means. For supramolecular synthesis to advance, it is obviously important to characterize, classify, and analyze structural patterns, space group frequencies, and symmetry operators [118], However, at the same time we also need to bring together this information with the explicit aim of improving and developing supramolecular synthesis - the deliberate combination of different discrete molecular building blocks within periodic crystalline materials. 

Symmetry of three-dimensional patterns space groups... 

Many tests exist for detecting outliers in univariate data, but most are designed to check for the presence of a single rogue value. Univariate tests for outliers are not designed for multivariate outliers. Consider Figure 1.6, the majority of data exists in the highlighted pattern space with the exception of the two points denoted A and B. Neither of these points may be considered a univariate outlier in terms of variable x or x2, but both are well away from the main cluster of data. It is the combination of the two variables that identifies the presence of these outliers. Outlier detection and treatment is of major concern to analysts, particularly with multivariate data where the presence of outliers may not be immediately obvious from visual inspection of tabulated data. 

CA is similar to PCA and is based on the assumption of a close position of similar samples in multidimensional pattern space. Any similarity between two close samples is calculated as a function of the distance between them and displayed on a dendrogram. 

Necessary assumptions of LDA are the normality of data distributions and the existence of different class centroids, as well as the similarity of variances and covariances among the different groups. Classification problems therefore arise if the variances of groups differ substantially or if the direction of objects in the pattern space is different, as depicted in Figure 5.28. 

Lattice- An open framework of criss-crossed wood or metal strips that form regular, patterned spaces. 

The hypothesis for all pattern recognition methods is that similar objects - similar with regard to a certain property - are close together in the pattern space and form clusters. 

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