Bivariant

Rules of matrix algebra can be appHed to the manipulation and interpretation of data in this type of matrix format. One of the most basic operations that can be performed is to plot the samples in variable-by-variable plots. When the number of variables is as small as two then it is a simple and familiar matter to constmct and analyze the plot. But if the number of variables exceeds two or three, it is obviously impractical to try to interpret the data using simple bivariate plots. Pattern recognition provides computer tools far superior to bivariate plots for understanding the data stmcture in the //-dimensional vector space. 

Figure 6.19 Small organisms found inside a tubercle. Each is about 0.05 in. (0.13 cm) high. The organisms have segmented, fibrous stalks and bivariate heads.

By the variance, or number of degrees of freedom of the system, we mean the number of independent variables which must be arbitrarily fixed before the state of equilibrium is completely determined. According to the number of these, we have avariant, univariant, bivariant, trivariant,. . . systems. Thus, a completely heterogeneous system is univariant, because its equilibrium is completely specified by fixing a single variable— the temperature. But a salt solution requires two variables— temperature and composition—to be fixed before the equilibrium is determined, since the vapour-pressure depends on both. 

It should be emphasized that very little is quantitatively known about how the total pressures of plutonium-bearing species vary with oxygen potential, stoichiometry, and temperature in the bivariant region of oxygen-deficient plutonia between the phase limits at very high temperatures. New (though limited) oxygen potential data have been obtained in our laboratory above the fluorite, diphasic, and sesquioxide phases in the Pu-0 system at 1750, 2050, and 2250 K. 

It should be emphasized that a survey of the vapor pressure measurements of plutonium-bearing species above bivariant Pu02-x(s) revealed that in general these measurements suffer from a lack of knowledge of the composition of the condensed phase. 

H2O ratios) of the carrier gas. This Is consistent with the requirements based on the phase rule for a bivariant system. 

Fig. 10-3 Simulated three-dimensional T-S diagram of the water masses of the world ocean. Apparent elevation is proportional to volume. Elevation of highest peak corresponds to 26.0 x 10 km per bivariate class 0.1°C X 0.01%o. (Reproduced with permission from L. V. Worthington, The water masses of the world ocean some results of a fine-scale census. In B. A. Warren and C. Wunsch (1981). Evaluation of Physical Oceanography," MIT Press, Cambridge, MA.)...

Beck, L.A. 1985 Bivariate analysis of trace elements in bone. Journal of Human Evolution 14 493-502. 

For long linear chains the second condition is supported by the Stockmayer bivariate distribution (8,9) which shows the bivariate distribution of chain length and composition is the product of both distributions, and the compositional distribution is given by the normal distribution whose variance is inversely proportional to chain length. 

Figure 1.31. Bivariable plot of oxygen versus carbon isotopic compositions of carbonates. Solid circle magnesite open circle dolomite open square calcite A oxygen and carbon isotopic compositions of igneous carbonates B oxygen and carbon isotopic compositions of marine carbonates (Shikazono et al., 1995).

The interpretation of biplots is made easier by the construction of axes in it. These axes are used in the same way as in a bivariate Cartesian diagram. Perpendicular projection of the points in the diagrams upon a coordinate axis allows us to determine (or reconstruct) the values in the table. 

Figure 31.4 shows the biplot of the trace elements and wind directions for the case when a = p = 0.5. Since here we have that a + P equals 1, we can reconstruct the values in the columns of the data table X by means of perpendicular projections upon unipolar axes. In Fig. 31.4a we have drawn a unipolar axis through Cl. Perpendicular projection of the four wind directions upon this axis reconstructs the order of the concentrations of Cl at the four wind directions as listed in Table 31.1. Now we have established a way which leads back from the graphic display to the tabulated data. This interpretation of the biplot emphasizes the one-to-one relationship between the data and the plot. Such a relationship is also inherent in the ordinary bivariate (or Cartesian) diagram. 

Fig, 33.4. Nintety-five percent confidence limit for a bivariate distribution as class envelope. 

Beilken et al. [ 12] have applied a number of instrumental measuring methods to assess the mechanical strength of 12 different meat patties. In all, 20 different physical/chemical properties were measured. The products were tasted twice by 12 panellists divided over 4 sessions in which 6 products were evaluated for 9 textural attributes (rubberiness, chewiness, juiciness, etc.). Beilken etal. [12] subjected the two sets of data, viz. the instrumental data and the sensory data, to separate principal component analyses. The relation between the two data sets, mechanical measurements versus sensory attributes, was studied by their intercorrelations. Although useful information can be derived from such bivariate indicators, a truly multivariate regression analysis may give a simpler overall picture of the relation. 

In association with caffeine intake a lower bone mineral content was shown bivariately, but not multivariately, however, there was no relationship between coffee and fracture risk Caffeine intake unrelated to hip or wrist fracture. 

Statistical dimensions number of variables (manifest or latent) taken into account in evaluation. Statistical dimensions define the type of data handling and evaluation, e.g. univariate, bivariate, multivariate... 

The goal of EDA is to reveal structures, peculiarities and relationships in data. So, EDA can be seen as a kind of detective work of the data analyst. As a result, methods of data preprocessing, outlier selection and statistical data analysis can be chosen. EDA is especially suitable for interactive proceeding with computers (Buja et al. [1996]). Although graphical methods cannot substitute statistical methods, they can play an essential role in the recognition of relationships. An informative example has been shown by Anscombe [1973] (see also Danzer et al. [2001], p 99) regarding bivariate relationships. 

The above discussion has assumed that the rank of a coal can be adequately measured by a single parameter, such as the reflectance, the volatile matter yield or the organic carbon content. This assumption is commonly made, but it has for a long time appeared a pretty improbable proposition. The discussion also was restricted to bivariate correlations, that is, plots of a single variable against another. 

A statistical study of the conversion with tetralin of 68 coals (60) must now be regarded as superseded by a later, more comprehensive paper (61), but it did show very clearly that bivariate plots are of little value in interrelating liquefaction behavior with coal properties at least two or three coal properties must be taken into account in seeking to explain the variance of liquefaction behavior, and some of these properties are not related to the rank of the coal. The paper implies strongly that any interrelationships of coal characteristics must necessarily be multivariate. Hence in any study of coal a large sample and data base is essential if worthwhile generalizations are to be made. 

Ua and Ub. Assume further that the second-order perturbation theory applies. This means that Po(AU) can be represented as a bivariate Gaussian. Then, AA, from (2.30), is given by... 

The CFD model described above is adequate for particle clusters with a constant fractal dimension. In most systems with fluid flow, clusters exposed to shear will restructure without changing their mass (or volume). Typically restructuring will reduce the surface area of the cluster and the fractal dimension will grow toward d — 3, corresponding to a sphere. To describe restructuring, the NDF must be extended to (at least) two internal coordinates (Selomulya et al., 2003 Zucca et al., 2006). For example, the joint surface, volume NDF can be denoted by n(s, u x, t) and obeys a bivariate PBE. 

Thus, it would be natural to attempt to extend the QMOM approach to handle a bivariate NDF. Unfortunately, the PD algorithm needed to solve the weights and abscissas given the moments cannot be extended to more than one variable. Other methods for inverting Eq. (125) such as nonlinear equation solvers can be used (Wright et al., 2001 Rosner and Pykkonen, 2002), but in practice are computationally expensive and can suffer from problems due to ill-conditioning. 

The extension of DQMOM to bivariate systems is straightforward and, for the surface, volume NDF, simply adds another microscopic transport equation as follows ... 

Example calculations for a bivariate system can be found in Marchisio and Fox (2006) and Zucca et al. (2006). We should note that for multivariate systems the choice of the moments used to compute the source terms is more problematic than in the univariate case. For example, in the bivariate case a total of 3 M moments must be chosen to determine am, bm and cm. In most applications, acceptable accuracy can be obtained with 3[Pg.283]

In the second step we must add the micromixing terms from the DQMOM model to Eqs. (133)—(135). Fiowever, as we discussed earlier, we need to keep in mind that micromixing conserves the moments of the NDF, and not the weights and abscissas (see Eq. 113). The micromixing model in environment n for the bivariate moments has the form... 

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