Distance statistical

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. 

Some sort of estimation of the statistical distance to the overall model (see Section 16.5.4) should be reported for each compound to provide an estimate of how much an intra- or extrapolation in multivariate descriptor space the prediction actually is. 

As soon as observations are considered as samples of random variables, we must redefine the concepts of distance and projection. Let us consider in three-dimensional space a vector y of one observation of three random variables Yj, Y2, and Y3 with its density of probability function fy. The statistical distance c of the vector. p to another point y can be defined by the non-negative scalar c2, which has already been met a few times, e.g., in equations (5.2.1) and (5.3.7), and such that... 

If y1 Y2, and Y3 are normally distributed, the constant probability surfaces are ellipsoids centered at y (Figure 5.12) and the statistical projection y of y will be defined as the point where the plane is tangent to the innermost probability ellipsoid. Points on the same ellipsoid are by definition at the same statistical distance from y. If Sy is the covariance matrix of the vector y, the statistical distance c between y and y is given by... 

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... 

Note that any p-dimensional vector p and p x p positive-definite matrix X, defines a statistical distance ... 

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... 

The DrugScore function created by Gohlke, Hendlich, and Klebe is based on roughly the same formalism, albeit with several differences in the derivation leading to different potential forms. Most notably, the statistical distance distributions p j(r)/p ) for the individual atom pairs ij are divided by a common reference state that is simply the average of the distance distributions of all atom pairs p(r) =X), X]/P K )/imaxjinax5 where the product in the denominator yields the total number of pair functions. Furthermore, no... 

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

The statistical distance discrimination schemes described are simple to implement. Relating increasing distance with lower likelihoods, they have an intuitive appeal. They can use a large number of correlated variables to choose between many possible source populations. Disjointedness and overlap of sets can be accommodated. Unlike other diagnosis methods, additional source populations can easily be incorporated into the discrimination scheme without retraining the whole diagnosis system. 

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

A temperature dependent X—X+N study (100, 135, 170, and 205 K) on naphthalene [66] addresses the problem of thermal de-convolution, that is, the efficiency of the pseudoatom model to decouple density deformations due to chemical bonding from those due to nuclear motion. The authors analyze the self-consistency of multipole populations, extracted from different temperature XRD data, in terms of statistical distances d ) in the parameter space of the same refinement model ... 

Energy transfer experiments with vitamin K-dependent factors. Equimolar mixtures of PC-PS were labelled with either PBPC or PBPA, at the same concentration, namely 3.8 %. The transfer efficiency values (Et) are shown in Table II. Since the two probes have identical spectroscopic features ( ), the comparison of Et values for the membranes having similar amounts of label, allows a direct comparison of the lipid environment of the factors. The higher Et value indicates a lower statistical distance and/or a greater number of labelled phospholipids in the neighborhood of the protein. 

The first subsection presents the electrostatics-based method of building sodium clusters and further the MTA-based method of optimization. Following sections furnishes a geometrical analysis on the best isomer with specific size through deformation parameter and statistical distance analyses. Further, the MESP-guided method is validated by comparing the resulting structures with those reported in the literature. 

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