Bivariate, term

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... 

The bivariate-log-normal analysis of data collected by Guinn and co-workers appears to be the only comprehensive statistical treatment of firearm residue detection by NAA (11). Suspects handswabs were interpreted in terms of accumulated firing test data and handblanks collected from individuals of different occupational backgrounds. A somewhat more empirical interpretation of the same data is also reported (12). Additional data from smaller scale collection of handblanks have been published recently (13,14). 

Equation (42) cannot be used if NO concentrations approach their equilibrium values, since the net production rate then depends on the concentration of NO, thereby bringing bivariate probability-density functions into equation (40). Also, if reactions involving nitrogen in fuel molecules are important, then much more involved considerations of chemical kinetics are needed. Processes of soot production similarly introduce complicated chemical kinetics. However, it may be possible to characterize these complex processes in terms of a small number of rate processes, with rates dependent on concentrations of major species and temperature, in such a way that a function w (Z) can be identified for soot production. Rates of soot-particle production in turbulent diffusion flames would then readily be calculable, but in regions where soot-particle growth or burnup is important as well, it would appear that at least a bivariate probability-density function should be considered in attempting to calculate the net rate of change of soot concentration. 

Let us now consider a closed system which is at least bivariant. Then, apart from the case of indifferent states, we can, from Duhem s theorem (c/. chap. XIII, 6 and 7), describe all equilibrium states of the system in terms of two variables, T and p. We have... 

The variable x in the preceding formulas denotes a quantity that varies. In our context, it signifies a reference value. If the variable by chance may take any one of a specified set of values, we use the term variate (i.e, a random variable). In this section, we consider distributions of single variates (i.e., univariate distributions). In a later section, we also discuss the joint distribution of two or more variates bivariate or multivariate distributions). 

The representation of this equation for anything greater than two variates is difficult to visualize, but the bivariate form (m = 2) serves to illustrate the general case. The exponential term in Equation (26) is of the form x Ax and is known as a quadratic form of a matrix product (Appendix A). Although the mathematical details associated with the quadratic form are not important for us here, one important property is that they have a well known geometric interpretation. All quadratic forms that occur in chemometrics and statistical data analysis expand to produce a quadratic smface that is a closed ellipse. Just as the univariate normal distribution appears bell-shaped, so the bivariate normal distribution is elliptical. 

It is interesting to highlight here that the Jacobian matrix A is exactly the matrix employed in DQMOM for the calculation of the source terms, see for example Eq. (7.118) for a generic univariate case and Eq. (7.128) for a specific bivariate case. 

Note that, while the spread parameter crj is the same for all terms in the summation, the conditional parameter 0-2,0, can depend on a. Also, the functional form used for 6 1 need not be the same as that used for ( -g- 1 could use beta EQMOM, while 2 uses Gaussian EQMOM). Although the form in Eq. (3.134) is not as general as that in Eq. (3.124), we shall see that it leads to a direct method for moment inversion that is very similar to the one used in the CQMOM. The bivariate moments found from Eq. (3.134) have the form... 

Most daughter distribution functions can be easily extended to bivariate problems. Let us consider two examples. In the first example particles with two components A and B are described. The particulate system is defined in terms of the size of these particles dp and the composition of the particles 0, expressed for example as the mass fraction of component A in the particle. When a particle breaks we can assume for example that the amount of component A is partitioned among the daughters proportionally to the mass of the fragments. Under these hypotheses, and the additional assumption of binary breakage following the beta distribution, the resulting bivariate distribution is... 

As introduced in the previous section, class and sectional methods are based on a discretization of the internal coordinate so that the GPBE becomes a set of macroscopic balances in state space. Indeed, the fineness of the discretization will be dictated by the accuracy needed in the approximation of the integrals and derivative terms appearing in the GPBE. As has already been anticipated, the methods differ according to the number of internal coordinates used in the description and depend on the nature of the internal coordinates. Therefore, in what follows, we will discuss separately the univariate, bivariate, and multivariate PBE, and the use of these methods for the solution of the KE. 

Here we have used the statistical symmetry between the second and third directions in velocity phase space to express the granular temperature in terms of the bivariate moments mj k- The NDF n is the equilibrium (Maxwellian) distribution with the conservation properties m QQ = mo,o, tnl = mi,o, wJq i = Physically, these equalities result... 

Numerous relationships exist among the structural characteristics, physicochemical properties, and/or biological qualities of classes of related compounds. Simple examples include bivariate correlations between physicochemical properties such as aqueous solubility and octanol-water partition coefficients (Jtow) and correlations between equilibrium constants of related sets of compounds. Perhaps the best-known attribute relationships to chemists are the correlations between reaction rate constants and equilibrium constants for related reactions commonly known as linear free-energy relationships or LFERs. The LFER concept also leads to the broader concepts of property-activity and structure-activity relationships (PARs and SARs), which seek to predict the environmental fate of related compounds or their bioactivity (bioaccumulation, biodegradation, toxicity) based on correlations with physicochemical properties or structural features of the compounds. Table 1 summarizes the types of attribute relationships that have been used in chemical fate studies and defines some important terms used in these relationships. 

The important feature of these two equations is that the new positions and the new velocities both depend upon an integral over the random force, R(t) (the final terms in Equations (7.124) and (7.125). As both of these integrals depend upon R,(f) they are correlated. Specifically, they obey a bivariate Gaussian distribution. Such a distribution provides the probability that a particle located at X at time t with velocity Vj and experiencing a force fj will be at at time t + 6t with velocity In practice, this means that the distribution for the second variable depends upon the value selected for the first variable. It can be difficult to properly sample from such distributions, but van Gunsteren and Berendsen showed that the equations can be reformulated in terms of sampling from two independent Gaussian functions. 

Marginal probability. A term used to describe the probability of a given event occurring without reference to (that is to say, unconditional upon) the occurrence of other events, in a context in which, however, certain events are being considered together. So called because in a table of bivariate (or joint) probabilities such marginal probabilities may be calculated as the marginal totals (row or column as the case may be). 

CDG [18] show that no bivariate Markov model of the term structure can generate incomplete bond markets. Furthermore, they show that at least a three-dimensional model is needed to generate incomplete bond markets. 

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