Kernel density function multivariate

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. 

Even when multivariate EQMOM is used with kernel density functions, a Dirac delta function (dualquadrature) representation is employed to close the terms in the GPBE. See Section 3.3.4 for more details. 

To conclude this section, we discuss three technical points that arise when applying the realizable high-order scheme for free transport. The first point is how to choose the value of N given that iV = is used for multivariate EQMOM. To answer this question, we first note that the highest-order transported moment in any one direction is of order 2n (which corresponds to 2n -t-1 pure moments in any one direction). The extra (even-order) moment is used in the EQMOM to determine the spread of the kernel density functions (see Section 3.3.2 for details). When free transport is applied to the moment of order 2n, the flux function involves the moment of order 2n -i- 1. Therefore, in order to exactly predict the flux function for free transport using the half-moment sets, we must have fV > (n-i-1). The obvious choice for N is thus fV = (n -i- 1). ... 

A number of methods allow the estimation of probability densities, (a) A multivariate Gaussian distribution can be assumed the parameters are the class mean and the covariance matrix, (b) The p-dimensional probability density is estimated by the product of the probability densities of the p features, assuming they are independent, (c) The probability density at location x is estimated by a weighted sum of (Gaussian) kernel functions that have their centers at some prototype points of the class (neural network based on radial ba.sis functions, RBF ). (d) The probability density at location x is estimated from the neighboring objects (with known class memberships or known responses) by applying a voting scheme or by interpolation (KNN, Section 5.2). 

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