3.4. Properties of VELMOD Likelihood
The mock catalogs may be used to illustrate some important features of the VELMOD analysis also. An example of these is shown in Figure 3. The left-hand panel shows forw versus I for one of the 20 catalogs. The right-hand panels show how three other quantities vary with I in the same VELMOD run: the amplitude of the LG random velocity wLG (top panel), the velocity noise v (middle panel), and the TF scatter TF for each of the two mock TF samples (A82 and MAT) considered (bottom panel). Note first that the amplitude of the LG velocity vector is minimized near the true value of I. This generally was seen in the mock catalogs; it reflects the fact that the fits at the wrong values of I try to compensate for wrong peculiar velocity predictions with Local Group motion. If wLG was held fixed at its maximum likelihood value, or set equal to zero, the forw versus I curves would have sharper minima, and the -uncertainty would be reduced (cf. Section 4.5). Unfortunately, we cannot do this for the real universe because we do not know wLG a priori. Nevertheless, there we will find similar behavior; wLG has a minimum near the best-fit value of I for the real universe.
Figure 3. Some of the parameters obtained from running VELMOD on a single mock catalog. The left-hand panel shows the likelihood statistic along with the cubic fit used to determine its minimum. The right-hand panels show the amplitude of the Local Group random velocity vector, the velocity noise v, and the TF scatters for the mock A82 and mock MAT samples. Note that the Local Group velocity vector has its minimum amplitude for I 1. Note also that the TF scatters do not track the likelihood curve, primarily because the velocity noise v also measures the inaccuracy of the fit. This demonstrates that minimizing TF scatter is not equivalent to maximizing likelihood, as it is for standard Method II.