Statistical distance measure

Another popular method is to express the observed value by a statistical distance measure. Such distances are aU ratios of the following type ... 

Local interpretation methods encompass a wide variety of approaches that resolve decisions about input data relative to annotated data or known features that cluster. By characterizing the cluster or grouping, it is possible to use various measures to determine whether an arbitary pattern of data can be assigned the same label as the annotated grouping. All approaches are statistical, but they vary in terms of measures, which include statistical distance, variance, probability of occurrence, and pattern similarity. 

A vector x of n random variables has been measured m times, the ith measurement resulting in an estimate of the mean value x, and of the covariance matrix St. A best estimate Jt of the pooled ( weighted ) average makes the sum of squared statistical distances to each x minimum. The scalar expression... 

Cluster analysis is simply a method to group entities, for which a number of properties or parameters exist, by similarity [292, 308-313]. Various distance measurements are used, and the analysis is performed in a sequential manner, reducing the number of clusters at each step. Such a procedure has been described for use in drug design and environmental engineering research as a way to group substituents that have the most similarity when various combinations of the electronic, steric, and statistically derived parameters are considered. 

A theoretical analysis of the conformational energies of PDCP is presented. The results indicate that the bond pair P—N—P possesses a considerable conformational freedom, whereas the bond pair N-P-N is relatively rigid. This difference explains the low glass transition temperatures and large end-to-end distances measured for polyphosphazenes. All the calculated magnitudes are extremely sensitive to the energy Eler) that controls the statistical weight of the conformations tg, to, tg , gt, ct, and g t, relative to tt for the bond pair P-N-P. A qualitative explanation for this sensitivity is discussed. 

The following statistical measures are those most commonly found in software packages. First we mention HOTELLING S T2 for the 2-class case which is based on a generalized distance measure, the MAHALANOBIS distance D2, and from which a / -test can be derived ... 

In supervised pattern recognition, a major aim is to define the distance of an object from the centre of a class. There are two principle uses of statistical distances. The first is to obtain a measurement analogous to a score, often called the linear discriminant function, first proposed by the statistician R A Fisher. This differs from the distance above in that it is a single number if there are only two classes. It is analogous to the distance along line 2 in Figure 4.26, but defined by... 

A Raich and A Cinar. Diagnosis of process disturbances by statistical distance and angle measures. Comput. Chem. Engg., 21(6) 661-673, 1997. 

These data can be used to statistically infer the position of chromosomal breaks, and, from that point on, the procedures for map construction are similar to those used in linkage mapping. A map function is used to convert estimates of breakage frequency to additive units of distance measured in centirays (cR), with 1 cR representing a 1% probability that a chromosomal break has occmred between two markers in a single hybrid. The resolution of a radiation hybrid map depends on the size of the chromosomal fragments contained in the hybrids, which in turn is proportional to the amount of irradiation to which the human cell line was exposed. 

In SIMCA, an independent principal component model (cf PCA) is estabhshed for each individual class of the test data set. The evaluation of the assignment of objects to these classes of an estabhshed model is performed by statistically backed distance measures. 

Ultrafast shape recognition (USR) [19] is a recent and unusually rapid descriptor-based shape similarity technique. USR is based on the observation that the shape of a molecule is determined by the relative positions of its atoms. This 3D spatial arrangement of atoms is accurately described by a set of distributions of interatomic distances measured from four strategically located reference points, which are in turn characterized by its first three statistical moments. The shape similarity of two molecules is Anally calculated through an inverse of the sum of least absolute differences in their respective descriptors (full details about this recent technique along with applications can be found in a recent review [20]). 

The neutron diffraction studies confirmed earlier X-ray results about other important structural changes with temperature [2.5] the mean Si-0 and Al-0 tetrahedral distances were found to decrease continuously with temperature, whereas the Li-0 distances increased see Table 2.2. The decrease in the Si-O and Al-0 distances is comparable to the decrease in the Si-0 distance measured in quartz, also listed in Table 2.2. In fact, zero expansion or slight apparent contraction of T-O bond lengths with temperature has been found in high-temperature structural analyses of many silicates. Hazen and Finger compiled data for silicates and other compounds and evaluated them statistically [2.33]. The large positive thermal expansion of the Li-0 bonds in /3-eucryptite is also in line with their findings for electrostatically weak M-0 bonds. 

The statistic U measures how far the zero vector is from the posterior mean vector using distance based on the covariance structure found from the posterior sample. 

Unlike the solid state, the liquid state cannot be characterized by a static description. In a liquid, bonds break and refomi continuously as a fiinction of time. The quantum states in the liquid are similar to those in amorphous solids in the sense that the system is also disordered. The liquid state can be quantified only by considering some ensemble averaging and using statistical measures. For example, consider an elemental liquid. Just as for amorphous solids, one can ask what is the distribution of atoms at a given distance from a reference atom on average, i.e. the radial distribution function or the pair correlation function can also be defined for a liquid. In scattering experiments on liquids, a structure factor is measured. The radial distribution fiinction, g r), is related to the stnicture factor, S q), by... 

Similarity is often used as a general term to encompass either similarity or dissimilarity or both (see Section 6.4.3, on similarity measures, below). The terms "proximity" and distance are used in statistical software packages, but have not gained wide acceptance in the chemical literature. Similarity and dissimilarity can in principle lead to different rankings. 

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