Distance from the origin

Asymptotes The hmiting position of the tangent to a curve as the point of contact tends to an infinite distance from the origin is called an asymptote. If the equation of a given curve can be expanded in a Laurent power series such that... 

The second step concerns distance selection and metrization. Bound smoothing only reduces the possible intervals for interatomic distances from the original bounds. However, the embedding algorithm demands a specific distance for every atom pair in the molecule. These distances are chosen randomly within the interval, from either a uniform or an estimated distribution [48,49], to generate a trial distance matrix. Unifonn distance distributions seem to provide better sampling for very sparse data sets [48]. 

It is often helpful to examine the regression errors for each data point in a calibration or validation set with respect to the leverage of each data point or its distance from the origin or from the centroid of the data set. In this context, errors can be considered as the difference between expected and predicted (concentration, or y-block) values for the regression, or, for PCA, PCR, or PLS, errors can instead be considered in terms of the magnitude of the spectral... 

Plot this average value against the distance from the original interface. Calculate the rate of evaporation. 

On the other hand, in accordance with Equation (2.121) the distance from the origin R is... 

A similar formula describes the dependence of the distance from the origin to any point of the outer surface of the spheroid. In both expressions terms proportional to the third and higher order of flattening are discarded. This reference ellipsoid and its field are defined by four constants. The best-known and widely used values are... 

Denoting the distance from the origin to the center of mass by a, we obtain... 

The retention of a solute in TLC and PLC is characterized by the R value defined as the ratio of the distance from the origin to the center of the separated zone and the distance from the origin to the mobile phase front. 

In Euclidean space we define squared distance from the origin of a point x by means of the scalar product of x with itself ... 

This relationship is of importance in multivariate data analysis as it relates distance between endpoints of two vectors to distances and angular distance from the origin of space. A geometrical interpretation is shown in Fig. 29.2. 

The same geometrical considerations can be applied to the dual representation of the column-pattern in row-space S" (Fig. 31.2b). Here u, is the major axis of symmetry of the equiprobability envelope. The projection of theyth column Xy of X upon u, is at a distance from the origin given by ... 

By analogy we obtain for the distances from the origin dj and dy of the columns j, f and for the distance d, between these columns ... 

From the current example we can calculate the distances from the origin of the trace elements, either from the data X, from the loadings L (with P = 1) or from the sums of cross-products C. In the case of Na ... 

The distance from the origin or dj is a measure of the information contained in the corresponding row or column. If the distance is relatively large, in comparison to others, then the corresponding row or column can be seen to contribute more information (inertia, variance) to the result of the analysis. In the case when the distance is zero, then the row or column carries no information (inertia, variance) since all its elements are zero. 

There are two outstanding poles on this biplot. DMSO and dimethylchloride are at a large distance from the origin and from one another. These poles are the most likely candidates for the construction of unipolar axes. As has been explained in the previous section, perpendicular projections of points (representing compounds) upon a unipolar axis (representing a method) leads to a reproduction of the data in Table 31.3. In this case we have to substitute the untransformed value in eq. (31.35) by Zy of eq. (31.42) ... 

In the corresponding column-standardized biplot of Fig. 31.7 we find all representations of the eight chromatographic methods more or less at the same distance from the origin of space. The circle is distorted because of the large difference between the contributions of the first and second latent variables (95 and 4%) and the choice of a = [3 = 0.5 which has been made at the outset. The combined effect is an apparent dilation of the vertical axis. 

The biplot of Fig. 31.9 shows that both the centroids of the compounds and of the methods coincide with the origin (the small cross in the middle of the plot). The first two latent variables account for 83 and 14% of the inertia, respectively. Three percent of the inertia is carried by higher order latent variables. In this biplot we can only make interpretations of the bipolar axes directly in terms of the original data in X. Three prominent poles appear on this biplot DMSO, methylene-dichloride and ethylalcohol. They are called poles because they are at a large distance from the origin and from one another. They are also representative for the three clusters that have been identified already on the column-standardized biplot in Fig. 31.7. 

In the following section on the analysis of contingency tables we will relate the distances of chi-square in terms of contrasts. In the present context we use the word contrast in the sense of difference (see also Section 31.2.4). For example, we will show that the distance of chi-square from the origin 5, can be related to the amount of contrast contained in row i of the data tables, with respect to what can be expected. Similarly, the distance 5 can be associated to the amount of contrast in column j, relative to what can be expected. In a geometrical sense, one will find rows and columns with large contrasts at a relatively large distance from the origin of and S", respectively. The distance of chi-square 5- then represents the amount of contrast between rows i and i with respect to the difference between their expected values. Similarly, the distance of chi-square indicates the amount... 

The two plots can be superimposed into a biplot as shown in Fig. 32.7. Such a biplot reveals the correspondences between the rows and columns of the contingency table. The compound Triazolam is specific for the treatment of sleep disturbances. Anxiety is treated preferentially by both Lorazepam and Diazepam. The latter is also used for treating epilepsy. Clonazepam is specifically used with epilepsy. Note that distances between compounds and disorders are not to be considered. This would be a serious error of interpretation. A positive correspondence between a compound and a disorder is evidenced by relatively large distances from the origin and a common orientation (e.g. sleep disturbance and Triazolam). A negative correspondence is manifest in the case of relatively large distances from the origin and opposite orientations (e.g. sleep disturbance and Diazepam). 

Figure J.l Distance between two particles 1 and 2 and their respective distances from the origin.

The actual movement of a specific deep-well-injected hazardous substance depends on the types of processes that act on the waste and on the ways in which different processes interact. Figure 20.3 shows the expected change in concentration over time of a deep-well-injected organic compound in an observation well at an unspecified distance from the original point of injection. 

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