**5.2. Residual Autocorrelation Function**

The sky plots shown above provide visual evidence that the
_{I}
= 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*,

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where _{m}
was defined in equation (23) and the sum is over
the *N*_{p}()
distinct pairs with *IRAS*-predicted
separation *d*_{ij} within
= 100 km s^{-1} of a given value
. This definition
makes ()
insensitive to the values of
_{TF}
and _{v}
(because
the _{m,i}
are themselves normalized using their maximum likelihood values for each
_{I}),
but sensitive to the residual correlations that signal a poor fit.

In Figure 17, we
plot ()
versus
for the *IRAS* plus quadrupole models, with
_{I}
= 0.5, 0.1, and 1.0, as well as
the _{I}
= 0.6, no-quadrupole model. The error bars are described below. The model
that fits best according to the likelihood statistic,
_{I}
= 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*(*m*,
*cz*) are independent, and thus validates the VELMOD likelihood
statistic _{forw}.

The other models shown in
Figure 17 all exhibit significant residual
correlations.
The _{I}
= 0.6, no-quadrupole model has noticeable correlations on small and large
scales, as does
the _{I}
= 0.1 plus quadrupole model. Indeed, several of the values of
() for
_{I} =
0.1 are so large that they are off-scale on the plot.
The _{I}
= 1.0 plus quadrupole model exhibits strong correlations for
2000
km s^{-1}, although it is well behaved on large scales.