5. ANALYSIS OF RESIDUALS: DO PREDICTIONS MATCH OBSERVATIONS?

The VELMOD analysis can tell us which velocity field models - which values of _I, _v, w_LG, and quadrupole parameters - are "better" than others. However, as with maximum likelihood approaches generally, by itself it cannot tell us which, if any, of these models is an acceptable fit to the data. This is because we do not have precise, a priori knowledge of the two sources of variance, the velocity noise _v and the TF scatter _TF. Instead, we have treated these quantities as free parameters and determined their values by maximizing likelihood. As a result, a standard ² statistic will be ~ 1 per degree of freedom (dof), even if the fit is poor.

Of course, we can ask whether or not the values of _v and _TF obtained from VELMOD agree with independent estimates. It is reassuring that they do. We find _TF 0.46 mag for both the A82 and MAT samples, within the range estimated by Willick et al. (1996) by methods independent of peculiar velocity models. However, this agreement is of limited significance. TF scatter is very sensitive to non-Gaussian outliers (Section 4.1), and thus to precisely which objects have been excluded. Furthermore, the MAT subsample used here is only about half as large as the MAT subsample used by Willick et al. (1996) to estimate its scatter. The VELMOD result for the velocity noise, _v 125 km s^-1, is remarkably small and appears consistent with recent studies for the value of the velocity field outside of clusters based on independent methods (e.g., Miller, Davis, & White 1996 and Strauss et al. 1997). Indeed, because ~ 90 km s^-1 may be attributed to IRAS velocity prediction errors (Section 3.2), our value of _v suggests a true one-dimensional velocity noise of 90 km s^-1. Still, the small _v is not necessarily diagnostic; for demonstrably poor models (e.g., _I 0.2), we find an even smaller value of _v. Thus, an alternative approach is required for identifying a poor fit.

Let us consider fitting a straight line y = ax + b by least squares to data (x_i, y_i) whose errors are unknown. One obtains a, b, and also the rms scatter about the fit. Because the scatter is derived from the fit, the ² statistic is ~ 1 per dof by construction. However, if the straight line is a bad fit - if, say, the relation between y and x is actually quadratic - then the residuals from the fit will exhibit coherence. Coherent residuals in excess of what is expected from the observed scatter would signify that a model is a poor fit. In this section, we will make such an assessment for the VELMOD residuals. First, we will define a suitable residual and plot it on the sky. We will demonstrate coherence and incoherence of the residuals for "poor" and "good" models, respectively, by plotting residual autocorrelation functions. Motivated by these considerations, we will define and compute a statistic that measures goodness of fit.