Mixture Models Based on Neural Network Averaging

Abstract

A modified version of the single hidden-layer perceptron architecture is proposed for modeling mixtures. A particular flexible mixture model is obtained by implementing the Box-Cox transformation as transfer function. In this case, the network response can be expressed in closed form as a weighted power mean. The quadratic Scheffé K-polynomial and the exponential Wilson equation turn out to be special forms of this general mixture model. Advantages of the proposed network architecture are that binary data sets suffice for "training" and that it is readily extended to incorporate additional mixture components while retaining all previously determined weights.

Revisiting the Classic Activity Coefficient Models

“…This is because they are based on special weighted means that conform to a set of axioms, which imply the indicated consistency requirements. 1,18 General multicomponent expressions for excess Gibbs energies and the corresponding activity coefficients, 19 and in particular, specific expressions for Margules, 20 van Laar, 21 and Redlich−Kister 22−24 have been put forward. Of these, the latter has found much favor and it forms the basis of the CALPHAD (calculation of phase diagrams) method used in thermodynamic simulation software.…”

“…In what follows, the apparent physical properties of the pure components in the multicomponent mixture will be denoted by a i , a ii , or a iii depending on the order of the model or polynomial approximation. Let E(y) = a denote the expected value of the physical property of the mixture of interest; then, the first-order ScheffeḰ -polynomial for a ternary mixture, that is, K 3 (1), is given by a a x a x a x…”

“…A useful empirical excess Gibbs free energy model should satisfy the following requirements: 1 the mathematical form should be sufficiently flexible to adequately represent the underlying information with no unnecessary restrictions; it should be consistent with available physical theory, especially in some limits of simplification; the parameters should be easy to interpret and estimable with common multiple regression techniques; and it should be parameter-parsimonious. Ideally, it should also be possible to predict multicomponent behavior from knowledge of pure component properties and binary mixture behavior alone.…”

“…This is attempted by reconsidering their derivation by following previous mixture model ap- proaches. 1,[9][10][11]14 The mixture models that will be considered here are the ScheffèK-polynomials, 12 Padèrational approximations, 10 and the double-weighted power mean (DWPM) mixture model. 13 Toward this goal, Section 3 introduces a variety of consistent mixture models suitable for correlating the physical properties of multicomponent mixtures.…”

“…The purpose of this communication is to revisit some of the classical activity coefficient models with the objective to explore their relative utility for predicting multicomponent mixture behavior. This is attempted by reconsidering their derivation by following previous mixture model approaches. ,− , The mixture models that will be considered here are the Scheffè K -polynomials, Padè rational approximations, and the double-weighted power mean (DWPM) mixture model …”