About BOSS Data from Percival

Got word from Will Percival over the weekend:
He said the mask for the data/randoms is here:

https://trac.sdss3.org/browser/repo/boss/bosslss/trunk/geometry/current_boss_geometry.ply

He said:
I used the current_boss_geometry, choosing sectors with > 70% completeness. I don't have this separately in a file I'm afraid, but you could easily create it by placing the random catalog within the current_boss_geometry file - taking the quoted numbers as the completeness.

I said:
Thanks for the response. In terms of the 'current_boss_geometry' file. Is this as old as the randoms/galaxy files? Because I could imagine a scenario where the geometry file covers MORE area than the data files (because more of the sky has been observed) and then when I apply this "mask" to my other data it would let in regions of the sky which are not in the BOSS data files.

He said:
The current_boss_geometry file will always cover a larger area than that of the galaxies observed with completeness > 70%. But you can only use those polygons that contain one of the randoms. This will cut it back.

I said:
Also, how are you dealing with the spectroscopic completeness when generating the randoms? Are you basically filtering for each sky polygon based on the completeness in the current_boss_geometry file? For instance if a polygon has 80% completeness, then only 80% of randoms generated in that polygon are kept?

He said:
Yes, this is what I'm doing, based on the number of observed galaxies over the number of targets.

Learning About HEALPix

I've been playing with Shirley's LRG catalog and mask. The mask uses something called HEALPix which I've never used before. It basically breaks up the spherical sky into equal area pixels:



The map is in the form of weights for each pixel. Shirley says that if the weight is greater than 0.2 she considers that pixel in the mask.

So to create a random catalog, I just go through the weights, and if it is greater than 0.2 I place objects in that pixel.

The key is to now convert from a Healpix mask to ra/dec.
There are two routines that do this that I have found:

DPF_PIX2ANG_NEST
pix2ang_nest

The problem is that these routines return the ra/dec of the center of the pixel. And to avoid putting extra power at the separations of the pixels, we want to select random points within the pixel. However, as you can see below, the ra range of these pixels changes dramatically as a function of cos (theta) (i.e. large at the poles and small at the equator) and so it isn't a simple procedure to fill in the area of a pixel evenly with random points:


I've searched to see if there is an existing package that does this, and I can't seem to find anything. In the HEALPix documentation there is a comment on random number generation, it doesn't seem to be what I am interested in here. I also have a email out to Shirley about this.

There are many functions that deal with HEALPix on their web page and well as in IDLUTILS. I'm currently combing through them to see if there is a function that gives me the ra/dec ranges for a given pixel, or even better, automatically picks a random point given a pixel.

Some questions:

1. How do I assign a random place in a HEALPix pixel and convert this to a ra/dec position?
2. The mask contains a "weight" for each HEALPix pixel, how to do factor in the weight into the random catalog. My intuition is that pixel's with higher weight should have more objects in them, but will this add clustering signal to the random catalog that we don't want?
   
3. How do we distinguish what is signal because of observation effects (we only looked at that patch of the sky once, or we didn't target as many objects there) and what is signal due to clustering of the data (there is a huge galaxy cluster in that part of the sky, and that is why there is more data there)?
   

Davis to the Rescue

I was at the cosmology seminar today and I was complaining to Nic Ross about my correlation function woos and Marc Davis happened to be in earshot and told me that I was thinking about this all wrong. Basically what I should do is take my galaxies with coordinates of Ra and Dec and calculate their x,y,z coordinates (projecting them onto a unit sphere). Then I can calculate the 3D correlation function on these objects (which I already have code to do), where I can translate between their physical separation in x, y, z coordinates to their angular separation by the following:

Ra₁, Dec₁ → x₁, y₁, z₁
Ra₂, Dec₂ → x₂, y₂, z₂

Separation of objects (s) in Cartesian coordinates:

s = [(x₁ - x₂)² + (y₁ - y₂)² + (z₁ - z₂)²]^½

Geometry relates the separation angle (γ) to Cartesian separation (s):

s = 2sin(γ/2)

or using the dot product of the two vectors:

x₁x₂ + y₁y₂ + z₁z₂ = cos( γ)

Therefore, the only change I need to make is after I grid objects, and select which objects are 'close by' using above conversion from s to γ, is to convert the 3D separation of the objects to the angular separation. This shouldn't be very difficult to do using the 3D code I already have.

Thank you Marc Davis for making my life a lot better today!

Useful links for writing equations in HTML (used above):
http://www.w3schools.com/tags/ref_entities.asp
http://www.tizag.com/htmlT/htmlsuperscript.php
http://htmlhelp.com/reference/html40/entities/symbols.html

Problems with Geometry

While watching the Superbowl with a bunch of mathematicians, I realized that I am not doing the angular correlation functions correctly. I've been calculating the angular separation (γ) incorrectly. The actual formula is as follows:

cos γ = cos(90 - Dec1)cos(90 - Dec2) + sin(90 - Dec1)sin(90 - Dec2)cos(RA1 - RA2)

I need to fix this and the fact that I am not doing a periodic boundary condition around 0 -> 360.

Rage!

Who says football isn't intellectual?

Populating the Sphere

I am trying to populate the randoms in the same way that the data is populated over the sphere. This means that in right ascension I populate the randoms with a uniform distribution over the range of the data values. However, for declination I populate using a cosine function (because we want to density per volume element to be constant). I am doing the following:

oo = 1;
while(oo){
   thisdec = drand48()*mask;  // Choose a declination inside the mask
   theta = drand48();  //Choose a random value for sin(thisdec)
   /*If theta is "below the sin(thisdec) curve",
   keep thisdec, otherwise repeat above process. */
   if(theta < sin(thisdec){
     Rb[nn].pos[i] = thisdec;
     oo = 0;
  }
}

This uses a rejection sampling method to populate the declinations. It seem to work, here are plots of the random distributions compared with the data, you can see that the distributions match:

Matching distributions of data (red) and randoms (blue)

Just to make sure that the data still falls in the same regions as the randoms here are more plots:

The data (red) falls in the same physical region as the randoms (blue)

Now my correlation function matches Alexia's again (as opposed to my October 30th post):

Matching 3D correlation functions!