Next Contents Previous

5.2. Residual Autocorrelation Function

The sky plots shown above provide visual evidence that the betaI = 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,

Equation 25 (25)

where deltam was defined in equation (23) and the sum is over the Np(tau) distinct pairs with IRAS-predicted separation dij within Delta tau = 100 km s-1 of a given value tau. This definition makes psi(tau) insensitive to the values of sigmaTF and sigmav (because the deltam,i are themselves normalized using their maximum likelihood values for each betaI), but sensitive to the residual correlations that signal a poor fit.

In Figure 17, we plot psi(tau) versus tau for the IRAS plus quadrupole models, with betaI = 0.5, 0.1, and 1.0, as well as the betaI = 0.6, no-quadrupole model. The error bars are described below. The model that fits best according to the likelihood statistic, betaI = 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(meta, cz) are independent, and thus validates the VELMOD likelihood statistic curlyLforw.

Figure 17

Figure 17. VELMOD residual autocorrelation functions, psi(tau), plotted for betaI = 0.5 plus quadrupole (best-fit model), betaI = 0.6 no quadrupole, betaI = 0.1 plus quadrupole, and betaI = 1.0 plus quadrupole. In the betaI = 0.1, several points at large tau 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 betaI = 0.6, no-quadrupole model has noticeable correlations on small and large scales, as does the betaI = 0.1 plus quadrupole model. Indeed, several of the values of psi(tau) for betaI = 0.1 are so large that they are off-scale on the plot. The betaI = 1.0 plus quadrupole model exhibits strong correlations for tau ltapprox 2000 km s-1, although it is well behaved on large scales.

Next Contents Previous