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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 psi(tau) from unity. Let us define xi(tau) ident Np(tau) psi(tau) (cf. eq. [25]). In Appendix C, we show that xi(tau) approximates a Gaussian random variable of mean zero and variance Np(tau), if indeed the VELMOD residuals are uncorrelated on scale tau. [This property was used to compute the error bars on psi(tau) above.] To the degree this approximation is a good one, the quantity

Equation 26 (26)

will be distributed approximately as a chi2 variable with M degrees of freedom, where M is the number of separate bins in which xi(tau) is calculated. In contrast, if the residuals are strongly correlated on any scale tau, chi2xi 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 tau, 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 chi2xi follows chi2 statistics. We computed it for each of the 20 mock catalog runs (Section 3) with betaI = 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 <chi2xi> = 27.83±1.82, which may be compared with an expected value of 32 for a true chi2 statistic. The rms scatter in chi2xi was 8.15, which is the same as that expected for a true chi2. The difference between the mean and expected values is 2.3 sigma, indicating that chi2xi is not exactly a chi2 statistic, for reasons discussed in Appendix C. However, because the departure from true chi2 statistics is small, chi2xi is a useful statistic for measuring goodness of fit when calibrated against the mock catalogs.

Before presenting chi2xi for the real data, we consider its variation with betaI for the mock catalogs. In Figure 18, we plot the average value of chi2xi over the 20 mock catalogs at each value of betaI for which VELMOD was run. Although the minimum is at betaI = 1, it is not nearly as sharp as is that of the likelihood as function of betaI (e.g., Fig. 2); this statistic does not have the power that the likelihood does for measuring betaI. Indeed, for a single realization (the open symbols), the statistic has several local minima. However, it is apparent that a chi2xi value much greater than its expected true value of ~ 28 will indicate a poor fit of the model to the data.

Figure 18

Figure 18. Residual correlation statistic chi2xi, defined by eq. (26), plotted as a function of betaI for the mock catalogs. The filled symbols show an average over 20 mock catalogs; the open symbols show the values obtained for a single mock catalog.

In Figure 19, we plot the statistic chi2xi as a function of betaI for the real data, with and without the quadrupole included. The horizontal lines indicate the expected value of chi2xi, and the 1 sigma and 3 sigma deviations from it. Note first that the no-quadrupole model does not provide an acceptable fit for any value of betaI. 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 betaI that are unambiguously ruled out are betaI = 0.1, 0.2, and 1.0. The best-fit model according to VELMOD, betaI = 0.5 plus quadrupole, also has the smallest value of chi2xi. Given the multiple minima seen for one mock realization in Figure 18, this is not necessarily deeply significant. The statistic chi2xi 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.

Figure 19

Figure 19. Residual autocorrelation statistic chi2xi, defined by eq. (26), plotted as a function of betaI for the real data, with and without the quadrupole modeled. The heavy solid line shows the expected value of the statistic, which was determined by averaging the derived value for 20 mock catalogs. The dot-dashed line and the dashed line show 1 sigma and 3 sigma deviations from this value, respectively. Note that when the quadrupole is not modeled, highly significant residual correlations are detected for all values of betaI. (The no-quadrupole points for betaI = 0.1 and 0.2 are not shown because their chi2xi values are too large.)

In summary, the VELMOD likelihood maximization procedure is the proper one for determining which value of betaI is better than others, but it cannot identify poor fits to our model. The residual correlation statistic chi2xi 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 betaI = 0.5, plus the external quadrupole, is both the best fit of those considered and an acceptable fit. Values of betaI > 0.9 and betaI < 0.3 are strongly ruled out.

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