**5.2.1. Using Residual Correlations to Identify Poor Fits
Quantitatively**

In order to compare the observed residual
correlations with the results from the mock catalogs, we would like to
define a single statistic that summarizes the deviation of
()
from unity. Let us define
()
*N*_{p}()
()
(cf. eq. [25]). In Appendix C, we show that
()
approximates a Gaussian random variable of mean zero and
variance *N*_{p}(),
if indeed the VELMOD residuals are uncorrelated on scale
. [This
property was used to compute the error bars on
()
above.] To the degree this approximation is a good one, the quantity

(26) |

will be distributed approximately as a
^{2}
variable with *M* degrees of freedom, where *M* is the number
of separate bins in which
()
is calculated. In contrast, if the residuals are strongly correlated on any
scale ,
^{2}_{}
will exceed its expected value significantly.

However, because a single galaxy
will appear in many different pairs in the correlation statistic, both
within and between bins in
, the
assumptions made above do not hold rigorously. In
Appendix C, we explore this issue
further. For now, we appeal to the mock
catalogs to assess how closely the quantity
^{2}_{}
follows ^{2}
statistics. We computed it for each of the 20 mock catalog runs
(Section 3)
with _{I}
= 1. We carried out the calculation to a maximum separation of 6400 km
s^{-1}, in bins of width 200 km s^{-1}, so that *M* =
32, and found a
mean value <^{2}_{}> = 27.83±1.82, which
may be compared with an expected value of 32 for a
true ^{2}
statistic. The rms scatter
in ^{2}_{}
was 8.15, which is the same as that expected for a true
^{2}.
The difference between the mean and expected values is 2.3
, indicating
that ^{2}_{}
is not exactly a ^{2}
statistic, for reasons discussed in Appendix C.
However, because the departure from true
^{2} statistics is
small, ^{2}_{}
is a useful statistic for measuring goodness of fit when calibrated against
the mock catalogs.

Before presenting
^{2}_{}
for the real data, we consider its variation with
_{I}
for the mock catalogs. In Figure 18,
we plot the average value
of ^{2}_{}
over the 20 mock catalogs at each value of
_{I}
for which VELMOD was run. Although the minimum is
at _{I}
= 1, it is not nearly as sharp as is that of the likelihood as function of
_{I}
(e.g., Fig. 2); this statistic
does not have the power that the likelihood does for measuring
_{I}.
Indeed, for a single realization (*the open symbols*),
the statistic has several local minima. However, it is apparent that
a ^{2}_{}
value much greater than its expected true value of ~ 28 will
indicate a poor fit of the model to the data.

In Figure 19, we plot the
statistic ^{2}_{} as a function
of _{I}
for the real data, with and without the quadrupole included. The horizontal
lines indicate the expected value
of ^{2}_{},
and the 1 and 3
deviations from it. Note first that *the no-quadrupole
model does not provide an acceptable fit for any value of*
_{I}.
This is not a conclusion we could have reached on the basis of
the likelihood analysis alone. When the quadrupole is included, the only
values of
_{I}
that are unambiguously ruled out
are _{I}
= 0.1, 0.2, and 1.0. The best-fit model according to VELMOD,
_{I}
= 0.5 plus quadrupole, also has the smallest value
of ^{2}_{}.
Given the multiple minima seen for one mock realization
in Figure 18, this is not necessarily deeply
significant. The
statistic ^{2}_{}
is suitable for identifying models that do *not* fit the data, but it
does not have the power of the likelihood statistic for discriminating
among those models that do fit.

In summary, the VELMOD likelihood
maximization procedure is the proper one for determining which value
of _{I}
is better than others, but it cannot identify poor fits to our model. The
residual correlation
statistic ^{2}_{}
can identify unacceptable fits but does not have the power to determine
which of the acceptable fits is best. We have found that the *IRAS*
velocity field with
_{I}
= 0.5, plus the external quadrupole, is both the best fit of those
considered and an acceptable fit. Values of
_{I}
> 0.9
and _{I}
< 0.3 are strongly ruled out.