EQMOM

The value of cr is determined by fixing one additional moment (a total of 21V + 1 moments, i.e. an odd number of moments). In order to distinguish between moment methods using Eq. (3.81) and those using Eq. (3.82), we will refer to the former as the quadrature moment of moments (QMOM) and the latter as the extended quadrature method of moments (EQMOM) (Yuan et al, 2012). The principal advantage of using the EQMOM instead of the QMOM is that with one additional moment it is possible to reconstruct a smooth, nonnegative NDF that exactly reproduces the first 21V + 1 moments. However, there are several... 

A convenient choice for univariate EQMOM is to define 6a (x, y) in terms of the weight function w(t) for a known family of orthogonal polynomials (Gautschi, 2004). For example, on the interval [0, oo) the associated Laguerre polynomials have the weight function... 

In this section, we discuss three types of univariate EQMOM that are based on weight functions with infinite, semi-infinite, and finite support (Chalons et al, 2010 Yuan et al, 2012). Example numerical algorithms are provided in Section A.4 of Appendix A. 

As an example of univariate EQMOM, we will consider an NDE with (-co, -1-00) and define kernel density functions using Gaussian distributions (Chalons et al, 2010) ... 

This set of five moments is used for bi-Gaussian EQMOM with N = 2 (Chalons et at., 2010). 

Univariate EQMOM can be applied to an NDE defined on a finite interval (Yuan et at., 2012). Eor example, if the NDE is nonzero only on the interval [0,1], then we can define the kernel density functions using a beta distribution ... 

While EQMOM allows us to capture an additional moment, the use of kernel density functions can lead to a closure problem when evaluating integrals such as Eq. (3.9) ... 

In principle, the EQMOM introduced in Section 3.3.2 can be generalized to include multiple internal coordinates. However, depending on the assumed form of the kernel density functions, it may be necessary to use a multivariate nonlinear-equation solver to find the parameters (i.e. similar to the brute-force QMOM discussed in Section 3.2.1). An interesting alternative is to extend the CQMOM algorithm described in Section 3.2.3. Here we consider examples using both methods. 

In the literature (Chalons et al, 2010), only a bivariate EQMOM with four abscissas represented by weighted Gaussian distributions with a diagonal covariance matrix has been considered. However, it is likely that brute-force QMOM algorithms can be developed for other distribution functions. Using the multi-Gaussian representation as an example, the approximate NDF can be written as... 

In order to be consistent with the univariate EQMOM discussed in Section 3.3.2, the diagonal components of H are determined from the pure moments of order 2n (e.g. in 2D phase space m(2n,0) and m 0,2n) are used to find (Th and 0-22, respectively). The ofF-diagonal components of H can then be determined by using the cross moments of total order 2n, but of order n in the i and j components. For example, in 2D phase space, the moment m(n, n) is used to find 0-12, and in 3D phase space the moment m n, 0, n) is used to And 0-13. The reader can verify that none of these moments is contained in the optimal moment set and, hence, they are available for use in the extended optimal moment set. The multivariate moments m(k) computed from Eq. (3.124) can be written as... 

The application of multivariate EQMOM to evaluate integrals is very similar to that of univariate EQMOM. For the multi-Gaussian EQMOM, we have... 

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... 

The reconstructed distribution function may be continuous (EQMOM) or discrete (QMOM), but we will assume that it is always realizable (i.e. nonnegative). For the case in which / is a set of weighted delta functions, the computation of the moments and is trivial. With EQMOM the integrals are evaluated using... 

For simplicity, we employ the QMOM to illustrate the numerical methods. In Section 8.3 examples using the EQMOM are provided. 

Both the QMOM and the EQMOM can be written in this discrete form by using the quadrature nodes. 

In Section 8.4, an alternative definition using die EQMOM directly widi the moments is employed to determine the weights and abscissas. In both formulations, it is always possible to spUt die set of abscissas into the positive and negative velocities. 

As was done in Section 8.4, it is also possible to get the weights and abscissas directly from the EQMOM quadrature. 

In this section, we discuss realizable schemes for Eq. (8.61) that are based on the KBFVM presented in Section 8.2 and Appendix B. In order to implement these methods, we must reconstruct the NDE from the moments. For this purpose, we will employ gamma EQMOM (Eq. (3.84)) with N = 2 nodes, with which the reconstructed moments are given by... 

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