Class distances

Various partitions, resulted from the different combinations of clustering parameters. The estimation of the number of classes and the selection of optimum clustering is based on separability criteria such as the one defined by the ratio of the minimum between clusters distance to the maximum of the average within-class distances. In that case the higher the criterion value the more separable the clustering. By plotting the criterion value vs. the number of classes and/or the algorithm parameters, the partitions which maximise the criterion value is identified and the number of classes is estimated. 

In Figure 6 models are fitted separately to each class and the distances for each object to the two classes are plotted. The two classes are well separated and the objects corresponding to the soil samples are located close to the dashed line, indicating equal class distance. 

Star Apparent Magnitude Absolute Magnitude Spectral Class Distance (light-year)... 

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. 

The second is to determine the Mahalanobis distance to the centroid of any given group, a form of class distance. There will be a separate distance to die centre of each group defined, for class A, by... 

Class distance plot horizontal axis = class A, vertical axis = class B... 

The method can be extended to any number of classes. It is easy to calculate the Mahalanobis distance to several classes, and determine which is the most appropriate classification of an unknown sample, simply by finding the smallest class distance. More discriminant functions are required, however, and the computation can be rather complicated. 

The class distance can be calculated as the geometric distance from die PC models see die illustration in Figure 4.35. The unknown is much closer to die plane formed than the line, and so is tentatively assigned to diis class. A more elaborate approach is often employed in which each group is bounded by a region of space, which represents 95 % confidence that a particular object belongs to a class. Hence geometric class distances can be converted to statistical probabilities. 

Class distance of unknown object (represented by an asterisk) to two classes in Figure 4.34... 

For each of the 13 objects in the training set /, calculate the distance from the PC model of each class c by determine dic = yJ Ej=l(cxij — ci, )2, where 7=8 and corresponds to the measurements, and the superscript c indicates a model of for class c. For these objects produce a class distance plot. 

Extend the class distance plot to include the two samples in the test set using the method of steps 6 and 7 to determine the distance from the PC models. Are they predicted correctly ... 

Calculate the centroids of class A (excluding the outlier) and class B. Calculate the Euclidean distance of the 58 samples to both these centroids. Produce a class distance plot of distance to centroid of class A against class B, indicating the classes using different symbols, and comment. 

Determine the variance-covariance matrix for the 11 elements and each of the classes (so there should be two matrices of dimensions 11 x 11) remove the outlier first. Hence calculate the Mahalanobis distance to each of the class centroids. What is die reason for using Mahalanobis distance rather than Euclidean distance Produce a class distance plot for diis new measure, and comment. 

Calculate die %CC using die class distances in question 8, using the lowest distance to indicate correct classification. 

In accordance with our sampling design, the lag class distance interval was chosen... 

Input-domain based category Equivalence classes Probability of execution of an equivalence class Distance between test cases in an equivalence class or correctness probability... 

Figure 10.13 is a bar-chart representation of the Class Projection output for factor 2 from a SIMCA, in which the modulus of the class distinctions is 10 or greater for all spectra except numbers 105, 115 and 125. The values for these spectra are about 7, 4 and 1 respectively, suggesting much less distinction (particularly for spectrum number 125), as might be expected from spectra recorded along a boundary region. A class distance plot of for the two main categories is shown in Figure 10.14. The transition from stroma to tumour along a row of spectra is clearly evident in the regions of spectra numbers 105, 115 and 125 the other crossovers occur at the end of each row of spectra in the grid-map, z.e., at spectra numbers 111 and 121.

Figure 10.14 Class Distance plot output from a SIMCA analysis of the pre-processed absorbance spectra numbered 101-130, see text for details.

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