Linear forward projection

Various schemes have been applied to solve the forward problem of ECT for the visualization of multi-phase flow components. Forward solutions based on the sensitivity model (Huang et al., 1989 Xie et al., 1992) are the most widely used due to the simplicity and speed of applying this model. The simplest form of the sensitivity model is the single iteration linear forward projection (LFP) which is explained in Equation (4). 

Proof, (of Proposition 10.6) First we suppose that (S(/(2), V, p) is a linear irreducible unitary Lie group representation. By Proposition 6.14 we know that p is isomorphic to the representation R for some n. By Proposition 10.5 we know that R can be pushed forward to an irreducible projective representation of SO(3). Hence p can be pushed forward to an irreducible projective Lie group representation of SO(3). 

Figures 11 and 12 illustrate the performance of the pR2 compared with several of the currently popular criteria on a specific data set resulting from one of the drug hunting projects at Eli Lilly. This data set has IC50 values for 1289 molecules. There were 2317 descriptors (or covariates) and a multiple linear regression model was used with forward variable selection the linear model was trained on half the data (selected at random) and evaluated on the other (hold-out) half. The root mean squared error of prediction (RMSE) for the test hold-out set is minimized when the model has 21 parameters. Figure 11 shows the model size chosen by several criteria applied to the training set in a forward selection for example, the pR2 chose 22 descriptors, the Bayesian Information Criterion chose 49, Leave One Out cross-validation chose 308, the adjusted R2 chose 435, and the Akaike Information Criterion chose 512 descriptors in the model. Although the pR2 criterion selected considerably fewer descriptors than the other methods, it had the best prediction performance. Also, only pR2 and BIC had better prediction on the test data set than the null model.

Again, in an effort to be forward looking, experiments on parity-violating electron scattering on Pb (a project called PREX) will be done in the next few years. If snccessful, this experiment will produce data directly sensitive to the linear term in the dependence of the asymmetry energy on density (Horowitz and Piekarewicz 2002). 

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