Omitting Variance Estimation of Binned Images
Instead of recalculating the errors from the binned data as outlined in section 3.2.5 in the paper, we can try using the initial errors estimating in the binned image. The errors in each bin were added in quadrature. Below are likelihood spaces from Perseus in the initial MLE, before using the best-fit model to estimate the variance of the binned image.
Figure 1. - Perseus likelihoods from Planck data. The likelihood spaces look quite realistic. The MLE width = 13 km/s, DGR cm mag, intercept = mag. The likelihood spaces do not well represent the MLE of each parameter because each likelihood space is marginalized over another parameter.
The variances of the MLE parameters were unphysically tight in earlier analysis when we were masking at full resolution, and then binning the unmasked pixels together. Now we are binning all pixels, and then masking. This allows for much better estimates of the variance of each binned pixel, because there are more pixels with errors in each bin.
Evaluating the Fits
We can plot vs. to examine the distribution of points and the fitted relationship. I have included the MLE fit as well as a polynomial fit to the data. We can see for cloud the masking process has successfully masked outliers which do not follow a linear correlation between and . That is, for the masked data, is linearly dependent on . However, the MLE fit is incorrect, it should agree with the polynomial fit.
Figure 2. - Perseus vs. . The top shows the data points which the MLE of the parameters were fitted to, i.e., the masked data and corresponding fit. The bottom shows all the data. Each plot shows the model line determined by the MLE method, as well as a polynomial fit to the data. The MLE calculation has the parameter resolution to correctly estimate the intercept, however is incorrect. This is not consistent.
Figure 3. - Taurus vs. .
Figure 3. - California vs. . The MLE fit is so far off that it doesn’t show.