Next Contents Previous

3.1. Accuracy of beta-Determination

The IRAS velocity field reconstructions may be produced using a variety of smoothing scales, and we have used 300 and 500 km s-1 Gaussian smoothing. We found, however, that at 500 km s-1 smoothing, VELMOD returned a mean betaI biased high by ~ 20%; the predicted peculiar velocities were too small, and a too-large betaI was needed to compensate. Our discussion from this point on will refer to 300 km s-1 smoothing, which, as we now describe, we found to yield correct peculiar velocities and an unbiased estimate of betaI.

VELMOD was run on the 20 mock catalogs, and likelihood (curlyLforw) versus betaI curves were generated for each. As with the real data (Section 4), we used the A82 and MAT TF samples only; we limited the analysis to cz leq 3500 km s-1. (10) The curves were fitted with a cubic equation of the form

Equation 17 (17)

to determine betamin, the value of betaI for which curlyLforw is minimized. This is the maximum likelihood value of betaI. Four representative curlyLforw versus betaI plots are shown in Figure 2, along with the cubic fits. We estimate the 1 sigma errors deltabeta± in our maximum likelihood estimate by noting the values beta ± deltabeta± at which curlyL = curlyL0 + 1. Given the presence of the cubic term in equation (17), this is not necessarily rigorous, but we can test our errors by defining the chi2-like statistic

Equation 18 (18)

where deltabeta+ was used if betamin leq 1, and deltabeta- was used if betamin > 1. For the 20 mock catalogs, it was found that chi2 = 21.2. Thus, our tests were consistent with the statement that the error estimates obtained from the change in the likelihood statistic near its minimum are true 1 sigma error estimates. Although we formally derive two-sided error bars, the upper and lower errors differ little, and when we discuss the real data (Section 4), we will give only the average of the two. The weighted mean value of betamin over the mock catalogs was 0.984, with an error in the mean of ~ 0.08 / (20)1/2 = 0.017. Thus, the mean betamin is within ~ 1 sigma from the true answer. We conclude that there is no statistically significant bias in the VELMOD estimate of betaI. The results of this and other tests that we carried out using the mock catalogs are summarized in Table 1.


Quantity Input Value Mock Results a Typical Error b

betaI 1.0 0.984 ± 0.017 0.08
sigmav 147 149 ± 5 20 km s-1
wLG,x c 89 ± 8 77 ± 12 54 km s-1
wLG,y c -51 ± 10 -50 ± 14 63 km s-1
wLG,z c -57 ± 9 -55 ± 10 45 km s-1
bA82 10.0 10.12 ± 0.08 0.36
AA82 -13.40 d -13.44 ± 0.02 0.09
sigmaTF,A82 0.45 0.460 ± 0.006 0.026
bMAT 6.71 6.68 ± 0.05 0.22
AMAT -5.86 d -5.92 ± 0.02 0.09
sigmaTF,MAT 0.42 0.419 ± 0.003 0.013

a The errors given are in the mean.
b Errors in a single realization.
c Cartesian coordinates defined by Galactic coordinates.
d These true zero points differ from those reported by Kolatt et al. 1996, their Table 1, because they measured distances in units of megaparsecs, whereas we measure in units of km s-1.

Figure 2

Figure 2. Plots of the likelihood statistic, curlyLforw, vs. betaI for VELMOD runs using four of the mock catalogs. (The true value of betaI for the mock catalogs is unity, as discussed in the text.) Also indicated on the plots are betamin, the maximum likelihood values for betaI, and the average of its two one-sided errors deltabeta±. The solid lines drawn through the points are the cubic fits used to determine betamin and deltabeta±.

10 The real data analysis extended only to 3000 km s-1, but because there are fewer nearby TF galaxies in the mock catalogs, we extended the mock analysis to a slightly larger distance. Back.

Next Contents Previous