5.2. Residual Autocorrelation Function
The sky plots shown above provide visual evidence that the I = 0.5 plus quadrupole fit has small residuals generally, although they are correlated to some degree. In this section, we quantify these correlations with the residual autocorrelation function,
where m was defined in equation (23) and the sum is over the Np() distinct pairs with IRAS-predicted separation dij within = 100 km s-1 of a given value . This definition makes () insensitive to the values of TF and v (because the m,i are themselves normalized using their maximum likelihood values for each I), but sensitive to the residual correlations that signal a poor fit.
In Figure 17, we plot () versus for the IRAS plus quadrupole models, with I = 0.5, 0.1, and 1.0, as well as the I = 0.6, no-quadrupole model. The error bars are described below. The model that fits best according to the likelihood statistic, I = 0.5 plus quadrupole, shows no significant residual correlations on any scale. The correlation function is everywhere consistent with zero, as we would expect if the IRAS velocity field plus the quadrupole is indeed a good fit to the data. Indeed, the absence of residual correlations is the basis for a statement made in Section 2.2.1, namely, that the individual galaxy probabilities P(m, cz) are independent, and thus validates the VELMOD likelihood statistic forw.
Figure 17. VELMOD residual autocorrelation functions, (), plotted for I = 0.5 plus quadrupole (best-fit model), I = 0.6 no quadrupole, I = 0.1 plus quadrupole, and I = 1.0 plus quadrupole. In the I = 0.1, several points at large have residuals that are so large that they fall beyond the plot boundaries.
The other models shown in Figure 17 all exhibit significant residual correlations. The I = 0.6, no-quadrupole model has noticeable correlations on small and large scales, as does the I = 0.1 plus quadrupole model. Indeed, several of the values of () for I = 0.1 are so large that they are off-scale on the plot. The I = 1.0 plus quadrupole model exhibits strong correlations for 2000 km s-1, although it is well behaved on large scales.