Double Match Theorem 1 - Sun, 4 Dec 2016

When I think of something in the middle of the night, I just have to write it down as soon as I get up. My mind’s background processor has been absorbed in figuring out how to get the output from Double Match Triangulator to display the two halves of each Chromosome pair.

It’s tricky. Currently, DMT is determining Double Match Groups by looking for overlapping matching segments. These Double Match Groups themselves may overlap and you can have groups within groups. Any relative used as Person b in Double Matching may match to Person a on both halves, meaning they are connected in more than one way, both of them through both their parents.

What I realized last night, and what I’m going to present as a Theorem (going back to my mathematical days) is:

Double Match Theorem 1: Any segment where Person a and Person b Triangulate with any other person must be on the other half of the Chromosome pair of any overlapping Missing a-b segments.

Well that’s a bit of gobbledygook. What it means is that I might be able to start separating out Person a’s Chromosome pairs. Take a look for example at the following map produced by DMT:

To refresh your memory, the green X’s are Double Matches, where both Person a matches Person c and Person b matches Person c. The pink a’s are where only Person a matches Person c, and the blue b’s are where only Person b matches Person c.

The green X’s in the yellow line are where Person a matches Person b. Any Double Matches that overlap will have the third leg needed to Triangulate. Those that do are designated as “Full Triangulation”.

Notice there are 7 lines that are marked as Missing a-b Match. Their Double Match area (two green X’s) end in the 142,000,000 to 142,999,999 address range just before the Base a-b begins. Their Double Match area does not intersect with the a-b match and therefore they do not Triangulate.

However, notice that 5 of the matches have those pink a’s extending to the right into the Triangulation area. This means that Person a matches Person c over that area. But Person b does not match Person c over that area or those would not have been pink a’s but green X’s. Yet, we know that Person a matches Person b on one of the Chromosome pair over that region. Therefore, these matches must be on the other Chromosome of the pair.  Q.E.D.

(Wow. I don’t think I’ve used Q.E.D. at the end of a theorem in like 40 years)

As a result of this, I am going to separate the Full Triangulations from the Missing a-b Matches prior to the determination of Double Match Groups. This will result in pure Triangulation Groups and pure Double Match Groups. I’ll have to see what this does and how it works and after a bit of experience with it, maybe it will lead to the possible enhancement and ultimately to full identification of both halves of each Chromosome pair.

And while I’m at it, let’s bring up another term from my early mathematical days:

Double Match Corollary 1: Any segment where Person a and Person b Triangulate on both halves of the Chromosome pair will not have any overlapping Missing a-b segments.

For example, for siblings. Each matches half of their father and half of their mother. When comparing one with the other, they will on average not match each other over 25% of their DNA, they will match on one parent over 50% of their DNA, and they will on average match on both their parents over 25% of their DNA.

This also can apply to anyone who matches someone else through their father AND through their mother on the same segment. Cousins can fully match just as siblings do, but the double match area would cover a much smaller percentage of their DNA.

GEDmatch for example, color codes these:

Matches to one parent are colored yellow. Matches to both parents are colored green.

What this Corollary really means for Double Match Triangulator is that some Triangulation Groups might be triangulating to two different ancestors, one on Person a’s father’s side, and one on Person a’s mother’s side.

If there are any overlapping Missing a-b segments, then the Triangulation is guaranteed to be on one half only.

If there are not any overlapping Missing a-b segments, then the Trangulation may be over both halves.

This Corollary will provide additional help in identifying the matches that belong to each of half of the Chromosome pair.