Summarizing the MLE Method
By Elijah Bernstein-Cooper, July 24, 2015, 0 comments.

1. Mask all but faintest 10% of pixels.

2. Fit a model $A_V$ using a linear least squares fit, deriving a DGR and an intercept.

3. Unmask the next faintest 5% of pixels.

4. Mask the residuals by fitting residuals $% $ mag with a Gaussian. Mask all pixels more than 3 times the standard deviation of the fitted Gaussian.

5. Repeat steps 2 through 5 until the fitted DGR converges to within 1%. Supply the mask from step 4 to step 2.

Perseus

Taurus

California

Figure 1. - Masked maps of each cloud for each iteration. Left: The fractional-masked map from step 3. Right: The residual-masked map from step 4. The residual mask begins excluding most added pixels from the fractional mask after just a few iterations.

We examine the residual masking more closely by plotting the distribution of the residuals and the fitted Gaussian performed in step 4. Plotted below are kernel density PDFs with the fitted Gaussian, and the residual cut-off for each iteration.

Kernel density plots are PDFs created by smoothing each data point by some kernel, then adding the contributions of each data point together. Here is a simple outline of why to use a KD plot. Here is a more in-depth description. The choice of the kernel shape can be tricky, but there is plenty of literature and well-established methods. The advantage of a Kernel density plot is that there is always a unique answer, quite unlike histograms.

Perseus

Taurus

California

Figure 2. - Residual kernel density plots for each cloud at each iteration. The residual PDF is in black, and the fitted Gaussian in purple. The residual cut-off is the dashed black line. These make it obvious that the majority of data points are masked by residual masking.

## Likelihoods

In the previous post the MLE calculation found $HI$ from a chance intermediate-velocity cloud along the line of sight in Perseus reproduced the observed $A_V$ in the Lee+12 dataset. The image below shows the $HI$ channel at -45 km/s. Overplotted is the $A_V = 1$ mag contour, roughly tracing the outline of the mask.

Figure 3. - Perseus $HI$ emission at -45 km/s overlaid with mask contour.

IVCs are present in the north-west and south-west corners of the region. This additional $HI$ emission, at -45 km/s, would only be included in the Perseus $N(HI)$ map if the velocity width were about 100 km/s. This is exactly what the MLE code finds in the Lee+12 data. There are two peaks in the likelihood space, one at 40 km/s and one at 100 km/s. This means that the dust column density along the line of sight includes some dust contributed by these IVCs.

This likely means the code is doing its job: finding the $HI$ which best traces the dust column density. It is up to us to exclude widths which are unrealistic for GMCs.

Figure 3. - Taurus $HI$ emission at -33 km/s overlaid with mask contour.

Imara et al. (2011a) searched for cloud $HI$ emission across an $HI$ width of $\pm 20$ km/s around the CO line center for clouds in the Milky Way. In their study of M33 clouds (Imara et al. 2011b), they consider a larger range, $\pm 25$ km/s, over which to search for a GMC’s $HI$ extent.