Feature vectors continuous-valued

Subheading 2.3. describes the last class of finite feature vectors, namely, those with continuous-valued components, where the components (i.e., features) are usually obtained from computed or experimentally measured properties. An often-overlooked aspect of continuous feature vectors is the inherent nonorthogonality of the basis of the feature space. The consequences of this are discussed in Subheading 2.3.2, Similarity measures derived from continuous... 

Property-Based Continuous-Valued Feature Vectors... 

As shown by Maggiora et al. [99], the denominators of field-based similarity functions correspond to different types of averages, a feature that also applies to similarity functions associated with vectors with continuous valued components. This enabled the authors to show that the different similarity functions are ordered in an identical manner to that given in Equation 15.4.10 and discussed in Section 15.4.4.1 for set-based similarity functions. 

Thus, it is clear from the above discussion that there is an underlying consistency to the FP- and vector-based similarity coefficients. Moreover, for the case of binary FPs and binary feature vectors, the two approaches yield identical results (vide supra). However, for integer-weighted FPs (see Sect. 1.2.1.1) such as arise in cases where the number of occurrences of substmctural features is considered, methods for treating vectors with continuous, real-valued components are no longer appropriate and multiset procedures provide a better, more consistent approach for dealing with such FPs [10,41]. 

The temperature dependence of Tm obtained in these early experiments by both X-ray (open circles) and neutron diffraction (solid circles) is shown in fig. 6. It is clear that in the temperature range above 20 K the wave vector Vm determined by X-rays has preferred, commensurable values, whereas in the lower resolution neutron data there is a continuous variation of the wave vector with temperature. Other noteworthy features of the X-ray data include the appearance of an inflection point near tm = at around 70 K, thermal hysteresis below 50 K, and coexistence among phases with differing wave vectors. At the lowest temperatures, there is a first-order transition between two commensurable wave vectors, namely, = c and tm = gc, and there is an indication of a lock-in transformation at c. The inset in fig. 6 shows the variation of Tm during several cycles of the temperature between 25 K and 13K. The data suggest a clustering of the wave vectors around tjn = jfC and Tm =... 

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