meeting 2024 02 07 n315 - JacobPilawa/TriaxSchwarzschild_wiki_5 GitHub Wiki
02/08 Update
I've made a few plots to prepare for the meeting which I think visualize the convex hull a bit better/illustrate the issues I was trying to describe.
First, here are a set of convex hulls (notably only fit to the 2 dimensions in the plot, thought I don't think it matters) for various cutoffs to the data.
Cutoff
50
100
150
200
250
300
350
400
450
500
Plot
And here are all of the hulls plotted on top of one another. On the left is the full range, and the right zooms into the low Tmaj region.
In this plot, the colors represent the different hulls from above (red = 500, blue = 50).
Additionally, I've marked the location of the current best fitting model as a small red star. It's a bit tough to see depending on the panel we're looking at.
No Zoom
Zoomed
Context
Here are some diagnostics/more information on the convex hull fitting routine applied to the two grids of models that we have now.
Previously, we were noticing that the convex hull applied to the first grid of models had very, very little volume in the region we care about (specifically with Tmin>0.6 or so).
I've re-run the fitting routine and some of our diagnostics on the new full set of all models + all scalings (86611 + 7297 = 14,629 models in total).
As a first set of tests, I've applied the convex hull routine to **all points** within various cutoffs of the minimum chi2. Importantly, this makes no cuts on Tmin/any parameter in particular and simply feeds everything below a given threshold in to be fit and sampled from. In these plots, the black points show the set of all scalings that are used to build the hull. The blue histograms are the proposed model points coming from sampling from inside the hull. I think these in large part are similar to what we saw the other day -- without restricting the range of Tmin being input, we end up skewing the distribution toward low Tmin/away from the minima in the shapes. If anything, this suggests that the initial cutoff of 500 we were playing around with was likely far too large.
Cutoff
50
100
250
500
Plot
Here's a second test, similar to the one from Monday. Here, I'm taking all models within the cutoff, and further dividing the models into bins of Tmin with stepsize of 0.2.
Tmin=[0.0,0.2]
[0.2,0.4]
[0.4,0.6]
[0.6,0.8]
[0.8,1.0]
Cut=100
Cut=150
Cut=250
Cut=500
To my eye and given the results above, I think it makes sense to first cut out all models with Tmin<0.4, effectively using that as the new "lower bound" of the grid. I think that the cutoff of 500 was far too high and was causing the relative volume of the well-fitting region to be much smaller, so something like 100-200 seems to work as a new cutoff. With this in mind, here's the result of making the cut on Tmin<0.4 and also applying the different K cuttoffs:
Cutoff
K=50
K=100
K=150
K=200
K=250
K=300
K=500
Plot
I think this set of plots is a bit more informative than the ones in the previous bullet. Here, I'm plotting the distribution of proposed points for the Tmin=[0.4,1.0] cases above, but with different cutoffs in the points we are using to build the convex hull. I've added a vertical line showing the location of the current best fit model (in terms of chi2). It looks to me like the K = 150/200 case seems to nicely sample both sides of the minimum with an ample number of models. **HOWEVER,** I noticed looking at this plot that the distribution of Tmaj is quite weird, and we are left with almost **zero** points at Tmaj<0.013 or so, which is the location of our current minimum. This is pretty concerning and made me suspicious of the set of input points we are using to construct the convex hull (see next bullet).
Plot
Given the above distributions, I wanted to investigate the low Tmaj set of points we have to see what might be going on, and I think there's a few competing factors here. First, all of our grids so far have had a hard boundary of Tmaj = 0.01 for the lower bound, so there are 0 models below this Tmaj threshold. Our current best chi2 model has Tmaj = 0.0127, so we're already really, really close to the lower boudary of Tmaj itself. As you might expect, there are **very few models** even searching the region between [0.01, 0.0127], and even fewer models when we apply the cutoff in chi2 to the points going in to build the convex hull. The plot in this bullet helps visualize this. In this plot, I am plotting the number of models less than a given Tmaj (effectively a CDF of our Tmaj points), plotted when considering different cutoffs in the chi2. For the low cutoff cases, you can see the single best fit model appear around Tmaj~0.0127. For larger cutoffs, you can see we only have a handful of models with Tmaj<0.0127 (at most, if we take all models we have, we're left with 79/15,000 scaled models in the region [0.01, 0.0127]. Only maybe 10 of these points have a reasonable chi2.
