Adjusted Euclidean Distance for Dodecad Eurasia7

[Palisto]

I thought it might be useful to have a tool that is better than just calculating the Euclidean distance between different genomes. The Euclidean distance calculation is one of the most accurate methods that we can use here. But the problem of the Euclidean distance calculation is that it does not take into account that the different components (West_Asian, South_Asian, Atlantic_Baltic, etc.) have different distances to each other. For instance, we know that the West_Asian component is closer to Atlantic_Baltic than to the Sub_Saharan component.

Example:
Individual#1: 80% West_Asian, 10% Atlantic_Baltic, 10% Sub_Saharan, rest all 0%
Individual#2: 10% West_Asian, 80% Atlantic_Baltic, 10% Sub_Saharan, rest all 0%
Individual#3: 10% West_Asian, 10% Atlantic_Baltic, 80% Sub_Saharan, rest all 0%

Intuitively, we know that Individual#1 and #2 are closer related than both to Individual#3, because Sub_Saharan component is so different compared to the other two components. However, the Euclidean distance would not see that because it is treating each component equally.

Euclidean distance #1 vs #2: ((10%-80%)^2+(80%-10%)^2+(10%-10%)^2)^0.5=98.99
Euclidean distance #1 vs #3: ((80%-10%)^2+(10%-10%)^2+(10%-80%)^2)^0.5=98.99
Euclidean distance #2 vs #3 ((10%-10%)^2+(80%-10%)^2+(10%-80%)^2)^0.5=98.99

Thus, I set-up a new method (distance calculator) that is taking into account how related the components are, when it is used to calculate the adjusted distance.

Normal Euclidean and adjusted Euclidean distance for Individual#1 (sorted by adjusted Euclidean distance):

# ID Normal Distance Adjusted distance
1 Individual#1 0.0 0.0
2 Individual#2 99.0 89.0
3 Individual#3 99.0 124.5

Normal Euclidean and adjusted Euclidean distance for Individual#2 (sorted by adjusted Euclidean distance):
# ID Normal Distance Adjusted distance
1 Individual#2 0.0 0.0
2 Individual#1 99.0 89.0
3 Individual#3 99.0 125.2

Normal Euclidean and adjusted Euclidean distance for Individual#3 (sorted by adjusted Euclidean distance):
# ID Normal Distance Adjusted distance
1 Individual#3 0.0 0.0
2 Individual#1 99.0 124.5
3 Individual#2 99.0 125.2

---------- Post added 2011-11-03 at 16:10 ----------

As a test run, I used the Eurasia7 results and the Fst distances between the components that were provided by Dienekes here.
http://dodecad.blogspot.com/2011/10/...alculator.html


Here are the TOP20 normal and the adjusted Euclidean distances for some reference populations (using Eurasia7 data).

TOP20 normal and adjusted Euclidean distance for Kurds_D (sorted by adjusted Euclidean distance):
# ID Normal Distance Adjusted distance
1 Kurd_D 0.0 0
2 DOD774 1.2 1.3
3 Kurds_Y 1.3 1.3
4 DOD767 2.4 2.4
5 DOD768 2.7 2.7
6 DOD834 3.4 3.1
7 DOD731 3.3 3.3
8 DOD313 3.2 3.3
9 DOD067 3.7 3.5
10 DOD403 4.1 4.0
11 DOD728 4.5 4.4
12 Iranian_D 4.7 4.5
13 DOD839 5.0 4.6
14 DOD766 5.4 4.6
15 DOD434 5.9 5.4
16 DOD294 5.9 5.4
17 DOD743 5.6 5.8
18 DOD833 6.2 6.1
19 DOD010 6.7 6.2
20 DOD600 6.8 6.2

Just reference populations:
# ID Normal Distance Adjusted distance
1 Kurd_D 0.0 0.0
2 Kurds_Y 1.3 1.3
3 Iranian_D 4.7 4.5
4 Iranians 6.4 6.6
5 Uzbekistan_Jews 8.0 7.9
6 Armenians_Y 9.4 8.6
7 Armenian_D 9.7 8.9
8 Armenians 10.8 9.6
9 Azerbaijan_Jews 10.5 10.2
10 Assyrian_D 11.7 11.5
11 Turks 13.1 11.9
12 Iranian_Jews 14.2 14.1
13 Georgians 16.3 14.6
14 Georgia_Jews 15.4 15.1
15 Turkish_D 16.8 15.1
16 Abhkasians_Y 17.4 15.8
17 Iraq_Jews 16.2 16.2
18 Kumyks_Y 17.6 17.0
19 North_Ossetians_Y 18.9 18.2
20 Adygei 19.6 18.6

TOP20 normal and adjusted Euclidean distance for Iranian_D (sorted by adjusted Euclidean distance):
# ID Normal Distance Adjusted distance
1 Iranian_D 0.0 0.0
2 DOD067 1.3 1.4
3 DOD313 2.7 2.4
4 DOD766 3.5 3.4
5 DOD401 3.6 3.6
6 DOD403 4.1 3.7
7 DOD010 4.1 3.8
8 DOD731 4.3 3.8
9 DOD774 4.6 4.3
10 Kurd_D 4.7 4.5
11 DOD728 5.4 4.8
12 DOD743 5.6 5.2
13 Kurds_Y 5.4 5.2
14 DOD833 6.7 6.0
15 DOD767 6.3 6.2
16 Iranians 6.0 6.7
17 DOD294 7.5 6.8
18 DOD768 7.0 6.9
19 DOD834 7.8 7.3
20 DOD839 8.0 7.5

Just reference populations:
# ID Normal Distance Adjusted distance
1 Iranian_D 0.0 0.0
2 Kurd_D 4.7 4.5
3 Kurds_Y 5.4 5.2
4 Iranians 6.0 6.7
5 Uzbekistan_Jews 11.6 11.6
6 Turks 13.4 12.0
7 Armenians 13.4 12.0
8 Armenians_Y 13.9 13.0
9 Armenian_D 14.2 13.3
10 Azerbaijan_Jews 14.8 14.5
11 Kumyks_Y 16.2 15.0
12 Turkish_D 17.2 15.4
13 Assyrian_D 16.1 15.7
14 Georgians 18.3 16.0
15 Turkmens_Y 16.0 16.2
16 Abhkasians_Y 19.0 16.8
17 North_Ossetians_Y 18.2 16.8
18 Adygei 18.9 17.3
19 Balkars_Y 19.5 17.9
20 Iranian_Jews 18.3 18.0

TOP20 normal and adjusted Euclidean distance for Druze (sorted by adjusted Euclidean distance):

# ID Normal Distance Adjusted distance
1 Druze 0.0 0.0
2 DOD452 3.8 3.8
3 Georgia_Jews 5.5 5.4
4 DOD386 6.1 5.9
5 Syrians 6.7 7.0
6 DOD599 7.6 7.5
7 DOD675 7.7 7.7
8 DOD135 8.0 7.9
9 Iraq_Jews 8.9 8.2
10 Lebanese 8.5 8.6
11 DOD207 9.3 9.3
12 Samaritians 9.4 9.5
13 DOD722 10.2 9.6
14 DOD634 10.5 10.2
15 DOD163 10.9 10.6
16 DOD026 10.9 10.6
17 Assyrian_D 11.1 10.6
18 DOD027 11.0 10.7
19 DOD601 11.4 10.9
20 DOD781 11.0 11.1


Just reference populations:
# ID Normal Distance Adjusted distance
1 Druze 0.0 0.0
2 Georgia_Jews 5.5 5.4
3 Syrians 6.7 7.0
4 Iraq_Jews 8.9 8.2
5 Lebanese 8.5 8.6
6 Samaritians 9.4 9.5
7 Assyrian_D 11.1 10.6
8 Iranian_Jews 12.1 11.2
9 Cypriots 12.4 11.3
10 Azerbaijan_Jews 12.2 11.7
11 Uzbekistan_Jews 12.4 12.2
12 Palestinian 12.9 13.6
13 Jordanians 12.6 14.4
14 Armenian_D 14.9 14.5
15 Armenians_Y 15.5 15.0
16 Armenians 18.5 18.3
17 Turkish_D 19.4 19.2
18 Iranians 20.2 19.6
19 Turks 19.5 19.8
20 Kurd_D 20.3 19.9

TOP20 normal and adjusted Euclidean distance for Assyrian_D (sorted by adjusted Euclidean distance):

# ID Normal Distance Adjusted distance
1 Assyrian_D 0.0 0.0
2 DOD027 0.8 0.8
3 Azerbaijan_Jews 1.4 1.4
4 DOD163 1.8 1.6
5 DOD040 1.8 1.6
6 DOD026 2.1 1.9
7 DOD634 2.1 1.9
8 DOD801 2.7 2.4
9 DOD601 2.8 2.4
10 DOD104 2.6 2.5
11 DOD037 2.7 2.6
12 DOD793 3.2 3.1
13 DOD836 3.3 3.1
14 DOD218 3.7 3.2
15 DOD135 3.5 3.3
16 DOD460 3.7 3.3
17 DOD028 3.5 3.3
18 DOD011 3.6 3.4
19 DOD599 3.9 3.6
20 DOD209 4.1 3.7

Just reference populations:
# ID Normal Distance Adjusted distance
1 Assyrian_D 0.0 0.0
2 Azerbaijan_Jews 1.4 1.4
3 Iranian_Jews 5.0 4.7
4 Armenian_D 5.2 5.1
5 Uzbekistan_Jews 5.7 5.2
6 Armenians_Y 5.6 5.5
7 Georgia_Jews 5.9 5.6
8 Iraq_Jews 6.0 5.9
9 Druze 11.1 10.6
10 Kurds_Y 11.6 11.4
11 Kurd_D 11.7 11.5
12 Armenians 12.6 12.1
13 Iranians 14.2 13.8
14 Syrians 14.3 14.1
15 Iranian_D 16.1 15.7
16 Turks 16.7 16.2
17 Lebanese 17.4 16.9
18 Cypriots 19.4 17.6
19 Turkish_D 18.9 17.7
20 Samaritians 19.4 18.9

TOP20 normal and adjusted Euclidean distance for Turkish_D (sorted by adjusted Euclidean distance):
# ID Normal Distance Adjusted distance
1 Turkish_D 0.0 0.0
2 DOD284 3.4 3.5
3 Turks 4.6 4.3
4 DOD838 4.8 4.4
5 DOD433 4.9 4.9
6 DOD843 6.6 6.2
7 DOD259 5.9 6.2
8 DOD759 6.6 6.7
9 DOD477 8.8 8.0
10 DOD320 8.8 8.1
11 DOD309 8.1 8.2
12 DOD756 10.7 9.6
13 Armenians 10.5 9.7
14 DOD784 10.1 10.1
15 DOD434 12.3 11.1
16 DOD590 12.2 11.3
17 DOD833 12.9 11.8
18 DOD623 13.2 12.0
19 DOD722 11.7 12.1
20 DOD010 13.9 12.4

Just the reference populations:
# ID Normal Distance Adjusted distance
1 Turkish_D 0.0 0.0
2 Turks 4.6 4.3
3 Armenians 10.5 9.7
4 Uzbekistan_Jews 15.0 13.9
5 Cypriots 14.2 14.5
6 Kumyks_Y 15.0 14.8
7 Kurds_Y 16.6 14.9
8 Kurd_D 16.8 15.1
9 Iranian_D 17.2 15.4
10 Armenian_D 17.1 15.7
11 Armenians_Y 17.3 15.9
12 Iranians 18.0 16.3
13 Georgia_Jews 17.2 16.7
14 Balkars_Y 17.0 16.7
15 Adygei 17.7 17.2
16 Azerbaijan_Jews 18.7 17.4
17 Assyrian_D 18.9 17.7
18 Syrians 17.5 17.8
19 Lebanese 17.7 18.2
20 North_Ossetians_Y 19.6 18.9

TOP20 normal and adjusted Euclidean distance for Armenian_D (sorted by adjusted Euclidean distance):
# ID Normal Distance Adjusted distance
1 Armenian_D 0.0 0.0
2 Armenians_Y 0.6 0.6
3 DOD221 1.8 1.7
4 DOD208 1.9 1.8
5 DOD842 2.0 1.8
6 DOD011 2.0 1.9
7 DOD439 2.3 2.2
8 DOD798 2.3 2.2
9 DOD799 2.8 2.6
10 DOD793 2.7 2.6
11 DOD104 2.8 2.7
12 DOD797 2.7 2.7
13 DOD830 2.9 2.7
14 DOD209 2.9 2.8
15 DOD792 3.4 3.0
16 DOD802 3.4 3.1
17 DOD796 3.6 3.2
18 DOD801 3.3 3.3
19 DOD600 3.8 3.6
20 DOD028 4.1 3.8

Just the reference populations:
# ID Normal Distance Adjusted distance
1 Armenian_D 0.0 0.0
2 Armenians_Y 0.6 0.6
3 Azerbaijan_Jews 4.7 4.4
4 Assyrian_D 5.2 5.1
5 Uzbekistan_Jews 6.7 5.8
6 Armenians 8.6 8.1
7 Kurds_Y 9.1 8.4
8 Kurd_D 9.7 8.9
9 Iranian_Jews 9.6 9.2
10 Georgia_Jews 9.6 9.3
11 Iraq_Jews 11.2 10.9
12 Iranians 14.1 13.0
13 Iranian_D 14.2 13.3
14 Turks 14.5 13.8
15 Druze 14.9 14.5
16 Georgians 15.8 15.4
17 Turkish_D 17.1 15.7
18 Abhkasians_Y 17.5 17.1
19 Syrians 17.6 17.1
20 Cypriots 20.7 19.0

TOP20 normal and adjusted Euclidean distance for Finnish_D (sorted by adjusted Euclidean distance):
# ID Normal Distance Adjusted distance
1 Finnish_D 0.0 0.0
2 DOD652 0.6 0.5
3 FIN 0.9 0.9
4 DOD811 1.6 1.6
5 DOD263 1.4 1.6
6 DOD262 1.9 1.7
7 DOD003 2.2 2.0
8 DOD776 2.5 2.4
9 DOD131 2.6 2.4
10 DOD038 2.6 2.4
11 DOD657 2.8 2.5
12 DOD334 2.8 2.6
13 DOD200 2.8 2.6
14 DOD005 3.5 3.6
15 DOD050 3.5 3.7
16 DOD222 4.9 4.6
17 DOD662 6.3 6.2
18 DOD820 7.5 7.0
19 DOD426 7.5 7.3
20 DOD688 7.7 7.4

Just the reference populations:
# ID Normal Distance Adjusted distance
1 Finnish_D 0.0 0.0
2 FIN 0.9 0.9
3 Russian 10.6 9.5
4 Lithuanian_D 8.9 9.7
5 Russian_D 10.7 9.9
6 Lithuanians 9.3 10.0
7 Swedish_D 9.4 10.2
8 Norwegian_D 9.8 10.7
9 Russian_B 11.7 10.8
10 Belorussian 10.7 11.0
11 Polish_D 12.4 12.6
12 Mordovians_Y 14.3 12.9
13 Orkney_1KG 13.1 13.8
14 Orcadian 13.5 14.3
15 Argyll_1KG 13.9 14.4
16 Irish_D 14.2 14.9
17 Ukranians_Y 15.8 15.1
18 British_Isles_D 14.8 15.5
19 British_D 15.2 15.9
20 Kent_1KG 15.7 16.3

TOP20 normal and adjusted Euclidean distance for Russian_D (sorted by adjusted Euclidean distance):

# ID Normal Distance Adjusted distance
1 Russian_D 0.0 0.0
2 DOD233 1.4 1.3
3 Russian_B 1.9 1.8
4 DOD214 1.8 2.0
5 DOD466 2.1 2.0
6 DOD343 2.3 2.2
7 DOD691 2.5 2.3
8 DOD699 3.1 3.1
9 DOD244 3.3 3.2
10 DOD365 3.0 3.2
11 DOD739 3.0 3.3
12 DOD537 3.2 3.3
13 DOD408 3.4 3.3
14 DOD281 3.4 3.6
15 DOD770 3.6 3.6
16 DOD690 3.8 3.7
17 DOD277 3.9 3.8
18 DOD820 4.2 3.9
19 DOD779 4.5 4.3
20 DOD662 5.2 4.7

Just the reference populations:
# ID Normal Distance Adjusted distance
1 Russian_D 0.0 0.0
2 Russian_B 1.9 1.8
3 Russian 4.6 5.0
4 Belorussian 5.2 5.4
5 Mordovians_Y 5.6 5.6
6 Ukranians_Y 5.7 5.8
7 Polish_D 5.4 5.9
8 Swedish_D 8.1 7.9
9 Argyll_1KG 7.3 8.0
10 Norwegian_D 8.8 8.6
11 Orkney_1KG 8.4 8.9
12 Orcadian 8.5 9.1
13 Irish_D 8.8 9.4
14 German_D 8.8 9.5
15 FIN 10.1 9.5
16 Finnish_D 10.7 9.9
17 Dutch_D 9.3 10.0
18 British_Isles_D 9.5 10.1
19 British_D 9.5 10.2
20 Kent_1KG 9.6 10.3


TOP20 normal and adjusted Euclidean distance for Georgians (sorted by adjusted Euclidean distance):
# ID Normal Distance Adjusted distance
1 Georgians 0.0 0.0
2 Abhkasians_Y 2.0 2.0
3 DOD832 3.3 3.2
4 DOD790 6.6 6.3
5 DOD049 9.8 9.6
6 DOD229 10.6 10.5
7 DOD800 12.4 11.7
8 DOD728 13.5 11.8
9 DOD287 12.5 12.3
10 DOD794 13.2 12.5
11 DOD623 13.8 13.0
12 DOD797 13.6 13.1
13 DOD731 14.8 13.2
14 DOD830 13.5 13.2
15 North_Ossetians_Y 13.7 13.3
16 DOD403 15.3 13.5
17 DOD798 14.2 13.6
18 DOD792 13.8 13.7
19 DOD294 15.5 13.8
20 DOD439 14.2 13.8


Just the reference populations:
# ID Normal Distance Adjusted distance
1 Georgians 0.0 0.0
2 Abhkasians_Y 2.0 2.0
3 North_Ossetians_Y 13.7 13.3
4 Kurds_Y 15.4 13.8
5 Armenians 15.5 14.4
6 Kurd_D 16.3 14.6
7 Adygei 16.0 14.8
8 Armenians_Y 15.3 14.9
9 Armenian_D 15.8 15.4
10 Iranian_D 18.3 16.0
11 Chechens_Y 17.7 16.5
12 Kumyks_Y 17.9 16.5
13 Balkars_Y 18.1 16.9
14 Lezgins 18.1 17.0
15 Azerbaijan_Jews 19.8 19.2
16 Uzbekistan_Jews 20.7 19.5
17 Turks 21.4 19.5
18 Iranians 22.1 20.0
19 Assyrian_D 20.7 20.2
20 Turkish_D 24.8 22.6


TOP20 normal and adjusted Euclidean distance for Greek_D (sorted by adjusted Euclidean distance):
# ID Normal Distance Adjusted distance
1 Greek_D 0.0 0.0
2 DOD039 1.4 1.4
3 DOD227 1.3 1.4
4 DOD167 3.2 2.9
5 DOD458 3.1 3.0
6 DOD014 3.2 3.1
7 DOD765 3.5 3.2
8 DOD372 3.6 3.6
9 DOD019 4.1 3.7
10 DOD318 4.3 4.1
11 DOD701 4.0 4.1
12 DOD264 4.7 4.3
13 DOD718 4.3 4.3
14 DOD138 4.3 4.3
15 DOD823 4.4 4.4
16 DOD604 4.9 4.5
17 DOD844 4.6 4.5
18 DOD693 4.6 4.6
19 DOD289 4.8 4.6
20 DOD344 4.6 4.7



Just the reference populations:
# ID Normal Distance Adjusted distance
1 Greek_D 0.0 0.0
2 S_Italian_Sicilian_D 5.9 5.9
3 C_Italian_D 6.8 6.2
4 Ashkenazy_Jews 6.5 6.3
5 Sicilian_D 6.4 6.5
6 S_Italian_D 6.9 6.7
7 Ashkenazi_D 8.0 7.8
8 O_Italian_D 10.6 9.6
9 Tuscan 13.5 12.1
10 TSI 14.9 13.5
11 OTHERS_D 14.8 14.1
12 Sephardic_Jews 15.1 14.6
13 Morocco_Jews 17.9 17.8
14 Bulgarian_D 19.7 18.4
15 Romanians 20.0 18.8
16 Bulgarians_Y 20.7 19.3
17 N_Italian_D 24.3 22.0
18 North_Italian 24.8 22.3
19 Cypriots 24.9 22.9
20 Turkish_D 25.4 23.1

[StarDS9]

I think the reason why Iranians_D are get closer to Kurds, it might be do that the Iranians on Behar are from Southern Iran and likely from the same regions most likely South western iran.

[Pot-Kettle]

interesting.

[Loxias]

What mathematical formula do you use to adjust Euclidian distances to the FST distance??
I've been trying to do it for a while, but I lack some quite basic maths knowledge to do it properly.

I ended up doing an approximation by calculating the coordinates on each of the 6f dimensions for groups (based on the approximate locations of components on the graphs Dienekes provided), but that's a bit clunky and still an approximation, so I'd be really interested in your methodes.

[Palisto]

What mathematical formula do you use to adjust Euclidian distances to the FST distance??
I've been trying to do it for a while, but I lack some quite basic maths knowledge to do it properly.

I ended up doing an approximation by calculating the coordinates on each of the 6f dimensions for groups (based on the approximate locations of components on the graphs Dienekes provided), but that's a bit clunky and still an approximation, so I'd be really interested in your methodes.

I will try to explain it with an example, not easy though.

I.
In order to calculate the distance of two points in 2 dimensions (x and y) you can use the Pythagorean theorem (a^2+b^2=c^2 or (a^2+b^2)^0.5=c):
Point1:
x1=3
y1=0

Point2:
x2=0
y2=4

((x1-x2)^2+(y1-y2)^2)^0.5

((3-0)^2+(0-4)^2)^0.5
=(9+16)^0.5
=(25)^0.5
=5

The distance between point1 and point2 is 5.

II.
In order to calculate the distance of two points in 3 dimensions (x, y and z) you can use the extended Pythagorean theorem or Euclidean distance (a^2+b^2+c^2=d^2 or (a^2+b^2+c^2)^0.5=d):
Point1:
x1=3
y1=0
z1=1

Point2:
x2=0
y2=4
z2=2

((x1-x2)^2+(y1-y2)^2+(z1-z2)^2)^0.5

((3-0)^2+(0-4)^2+(1-2)^2)^0.5
=(9+16+1)^0.5
=(26)^0.5
=5.099

The distance between point1 and point2 is 5.099.

III.
There is an alternative way to get the correct distance result (5.099) of II.

Imagine you have these two points (point1 and point2) in a 3-dimensional matrix, but you only can see the distance based on two dimension at a time. You have to look at it from 3 different position, each giving you a different distance.

Point1:
x1=3
y1=0
z1=1

Point2:
x2=0
y2=4
z2=2

So, you would 3 distances:
distance1=((x1-x2)^2+(y1-y2)^2)^0.5
=(25)^0.5
=5
distance2=((x1-x2)^2+(z1-z2)^2)^0.5
=(10)^0.5
=3.16

distance3=((y1-y2)^2+(y1-y2)^2)^0.5
=(17)^0.5
=4.12

Just by knowing these three distances (i.e. 5, 3.16, and 4.12) in 2-dimensional matrices for the two 3-dimensional points, you can calculate the total distance in the 3-dimensional matrix.

(((Distance1)^2+(Distance2)^2+(Distance3)^2)/(number of total dimensions-1))^0.5
=((5^2+3.16^2+4.12^2)/(3-1))^0.5
=((25+10+17)/2)^0.5
=(52/2)^0.5
=(26)^2
=5.099

IV.

In order to calculate the distance of two points in 4 dimensions you can do the same as in III.. Of course, now you have to look at it even more positions, when only 2-dimensional distances are given, precisely 6 different positions, each giving you a different distance.

Point1:
w1=5
x1=3
y1=0
z1=1

Point2:
w2=0
x2=0
y2=4
z2=2

So, you would 6 distances:
distance1=((x1-x2)^2+(y1-y2)^2)^0.5
=(25)^0.5
=5
distance2=((x1-x2)^2+(z1-z2)^2)^0.5
=(10)^0.5
=3.16

distance3=((y1-y2)^2+(y1-y2)^2)^0.5
=(17)^0.5
=4.12

distance4=((w1-w2)^2+(x1-x2)^2)^0.5
=(34)^0.5

distance5=((w1-w2)^2+(y1-y2)^2)^0.5
=(41)^0.5

distance6=((w1-w2)^2+(z1-z2)^2)^0.5
=(26)^0.5

Just by knowing these 6 distances in 2-dimensional matrices for the two 4-dimensional points, you can calculate the total distance in the 4-dimensional matrix.

(((Distance1)^2+(Distance2)^2+(Distance3)^2)+(Dist ance4)^2+(Distance5)^2+(Distance6)^2/(number of total dimensions-1))^0.5
=((25+10+17+34+41+26)/(4-1))^0.5
=((153)/3)^0.5
=51^0.5
=7.14

V.
You can add up more dimensions the same way I did in III. and IV.

VI.
Example:
For points in a 7-dimensional matrix you will need a total of 21 two-dimensional distances.
http://3.bp.blogspot.com/-aIGsQu-slx.../s1600/fst.jpg

VII.

See the components of Dodecad as dimensions, see the individual Dodecad Eurasia7 results as points in a 7-dimensional matrix (e.g. w=Atlantic_Baltic, x=South_Asian, y=East_Asian, etc.).
First, we assume that all components are equally related to each other (all Fst value are equal)

IIX.
Calculate all the 21 two-dimensional distances of two individual Dodecad Eurasia7 results and then use these 21 distances to calculate the overall distance in the 7-dimensional matrix, just like in IV (but with more distances (21) and dimensions (7-1=6) in the formula.

IX.

Now, we assume that all components are not equally related to each (all Fst different are different)

Do the same as in IIX., but multiply each 2-dimensional distance with the corresponding Fst distance from Dienekes table.


You have now adjusted the total distance by the Fst values provided by Dienekes.

X.

Because you multiplied a factor (Fst) to each 2D-distance, you number is off by the mean Fst value. For Eurasia7 it is 0.108 or 1/9.261981398.
Thus, you have to multiply the final result from IX. with 9.261981398 to normalize the data.

[Polako]

This is amazing. It's a much better way of looking at affinity via ADMIXTURE clusters.

Can you come up with a simple tool that takes in data from different ADMIXTURE runs? I could make a blog post about it, because this is the correct way of doing things, but no one's doing it.

Also, can you make a list for the Poles?

[Palisto]

This is amazing. It's a much better way of looking at affinity via ADMIXTURE clusters.

Can you come up with a simple tool that takes in data from different ADMIXTURE runs? I could make a blog post about it, because this is the correct way of doing things, but no one's doing it.

Also, can you make a list for the Poles?

Thanks.

Here is the TOP20 list for the Poles.

TOP20 normal and adjusted Euclidean distances for Polish_D using Dodecad Eurasia7 (sorted by adjusted Euclidean distance):

# ID Normal Distance Adjusted distance
1 Polish_D 0.0 0.0
2 DOD700 1.0 1.0
3 DOD636 1.2 1.2
4 DOD606 2.1 1.9
5 DOD066 2.1 2.2
6 DOD064 2.4 2.2
7 DOD446 2.3 2.2
8 DOD436 2.4 2.3
9 Belorussian 2.4 2.4
10 DOD597 2.8 2.6
11 DOD648 3.0 2.8
12 Argyll_1KG 2.8 2.9
13 DOD377 3.0 2.9
14 DOD771 2.9 2.9
15 DOD041 3.2 3.0
16 DOD638 3.2 3.1
17 DOD340 3.3 3.1
18 DOD083 3.5 3.2
19 DOD298 3.4 3.3
20 DOD120 3.5 3.4

Just reference populations:
# ID Normal Distance Adjusted distance
1 Polish_D 0.0 0.0
2 Belorussian 2.4 2.4
3 Argyll_1KG 2.8 2.9
4 Orcadian 3.9 4.0
5 Orkney_1KG 4.1 4.0
6 Swedish_D 5.0 4.5
7 Irish_D 4.5 4.5
8 Ukranians_Y 5.2 4.8
9 Norwegian_D 5.5 4.9
10 German_D 5.5 5.5
11 British_Isles_D 5.4 5.5
12 British_D 5.6 5.7
13 Russian_B 5.5 5.9
14 Kent_1KG 5.8 5.9
15 Russian_D 5.4 5.9
16 Dutch_D 6.2 6.1
17 Cornwall_1KG 6.3 6.3
18 CEU 6.3 6.4
19 Mixed_Germanic_D 8.0 7.9
20 Lithuanians 9.9 9.2

I have done it now for Dodecad V3 K=12, too.
http://3.bp.blogspot.com/-Pw7x-HD7ON.../s1600/fst.png

TOP20 normal and adjusted Euclidean distances for Polish_D using Dodecad V3 K=12 (sorted by adjusted Euclidean distance):

# Normal Distance Adjusted Distance ID Ethnicity
# ID Ethnicity Normal Distance Adjusted Distance
1 Dodecad Polish_D 0.0 0.0
2 DOD597 2.2 2.0
3 DOD140 3.3 3.0
4 DOD064 3.4 3.1
5 DOD293 3.4 3.2
6 DOD636 3.5 3.3
7 Dodecad Mixed_Slav_D 3.4 3.3
8 DOD083 3.8 3.6
9 DOD598 3.9 3.9
10 DOD446 4.4 4.2
11 DOD606 5.1 4.8
12 DOD281 5.0 4.9
13 DOD206 5.3 5.1
14 DOD738 5.5 5.1
15 DOD615 5.5 5.2
16 DOD408 5.4 5.5
17 DOD160 5.9 5.6
18 DOD214 6.8 6.7
19 DOD625 7.3 6.8
20 DOD739 7.9 7.7


Just reference populations:

# ID Ethnicity Normal Distance Adjusted Distance
1 Dodecad Polish_D 0.0 0.0
2 Dodecad Mixed_Slav_D 3.4 3.3
3 Dodecad Russian_D 9.0 9.1
4 Behar Belorussian 13.1 12.4
5 HGDP Russian 14.2 14.5
6 Ukranians_Y Yunusbayev 14.5 13.6
7 Mordovians_Y Yunusbayev 15.1 15.1
8 Behar Lithuanians 18.2 17.3
9 Behar Hungarians 20.6 19.5
10 Dodecad Lithuanian_D 21.0 20.0
11 Dodecad Finnish_D 22.8 22.1
12 Xing Slovenian 24.5 23.1
13 Balkans_D Balkans_D 25.7 24.1
14 1000Genomes FIN 27.9 26.6
15 Dodecad German_D 33.5 31.5
16 Bulgarians_Y Yunusbayev 34.7 32.5
17 HapMap CEU 39.6 37.3
18 HGDP Orcadian 40.5 38.2
19 1000Genomes Orkney_1KG 40.8 38.4
20 Dodecad Swedish_D 44.7 41.9

Can you come up with a simple tool that takes in data from different ADMIXTURE runs?

I could make an Excel file with Makro.

---------- Post added 2011-11-04 at 01:34 ----------

I think the reason why Iranians_D are get closer to Kurds, it might be do that the Iranians on Behar are from Southern Iran and likely from the same regions most likely South western iran.

Yes, I agree. When Zack from Harappa Ancestry Project analyzed the data of Behar, he saw elevated African admixture (>10%) in 3 Iranians, most likely from the South.

The big difference between Harappa Project Iranians and Behar et al Iranians is the African admixture. Only one Harappa Iranian (HRP0046) has 1% African admixture, while three Behar Iranians have more than 10%.

[Loxias]

Thanks, Palisto, this absolutely awesome, I just have a little question

III.
There is an alternative way to get the correct distance result (5.099) of II.

Imagine you h,ave these two points (point1 and point2) in a 3-dimensional matrix, but you only can see the distance based on two dimension at a time. You have to look at it from 3 different position, each giving you a different distance.

Point1:
x1=3
y1=0
z1=1

Point2:
x2=0
y2=4
z2=2

So, you would 3 distances:
distance1=((x1-x2)^2+(y1-y2)^2)^0.5
=(25)^0.5
=5
distance2=((x1-x2)^2+(z1-z2)^2)^0.5
=(10)^0.5
=3.16

distance3=((y1-y2)^2+(y1-y2)^2)^0.5
=(17)^0.5
=4.12

Just by knowing these three distances (i.e. 5, 3.16, and 4.12) in 2-dimensional matrices for the two 3-dimensional points, you can calculate the total distance in the 3-dimensional matrix.

(((Distance1)^2+(Distance2)^2+(Distance3)^2)/(number of total dimensions-1))^0.5
=((5^2+3.16^2+4.12^2)/(3-1))^0.5
=((25+10+17)/2)^0.5
=(52/2)^0.5
=(26)^2
=5.099

IV.

In order to calculate the distance of two points in 4 dimensions you can do the same as in III.. Of course, now you have to look at it even more positions, when only 2-dimensional distances are given, precisely 6 different positions, each giving you a different distance.

Point1:
w1=5
x1=3
y1=0
z1=1

Point2:
w2=0
x2=0
y2=4
z2=2

So, you would 6 distances:
distance1=((x1-x2)^2+(y1-y2)^2)^0.5
=(25)^0.5
=5
distance2=((x1-x2)^2+(z1-z2)^2)^0.5
=(10)^0.5
=3.16

distance3=((y1-y2)^2+(y1-y2)^2)^0.5
=(17)^0.5
=4.12

distance4=((w1-w2)^2+(x1-x2)^2)^0.5
=(34)^0.5

distance5=((w1-w2)^2+(y1-y2)^2)^0.5
=(41)^0.5

distance6=((w1-w2)^2+(z1-z2)^2)^0.5
=(26)^0.5

Just by knowing these 6 distances in 2-dimensional matrices for the two 4-dimensional points, you can calculate the total distance in the 4-dimensional matrix.

(((Distance1)^2+(Distance2)^2+(Distance3)^2)+(Dist ance4)^2+(Distance5)^2+(Distance6)^2/(number of total dimensions-1))^0.5
=((25+10+17+34+41+26)/(4-1))^0.5
=((153)/3)^0.5
=51^0.5
=7.14

Did you actually mean, for distance3 the following:
distance3=((y1-y2)^2+(z1-z2)^2)^0.5
??

Otherwise there is something I don't understand.

[Loxias]

Also, if you don't mind giving the results for dod332 as well as the French.

[Palisto]

Thanks, Palisto, this absolutely awesome, I just have a little question


Did you actually mean, for distance3 the following:
distance3=((y1-y2)^2+(z1-z2)^2)^0.5
??

Otherwise there is something I don't understand.

Thanks for catching this mistake, yes I mean:
distance3=((y1-y2)^2+(z1-z2)^2)^0.5

---------- Post added 2011-11-04 at 11:06 ----------

Also, if you don't mind giving the results for dod332 as well as the French.

TOP20 of DOD332:
# ID Ethnicity Normal Distance Adjusted Distance
1 DOD332 (Loxias) 0.0 0.0
2 DOD828 8.4 7.8
3 DOD393 9.1 8.7
4 DOD825 10.0 9.3
5 DOD195 10.2 9.4
6 DOD205 10.1 9.5
7 DOD161 10.0 9.5
8 DOD826 10.4 9.8
9 DOD133 10.5 10.0
10 DOD146 10.9 10.0
11 DOD246 Portugal 10.8 10.1
12 DOD082 11.1 10.1
13 DOD074 11.0 10.3
14 DOD614 Spaniards 11.1 10.4
15 Dodecad Portuguese_D 11.3 10.6
16 DOD628 11.4 10.6
17 DOD081 11.3 10.7
18 DOD052 11.9 11.1
19 Dodecad Spanish_D 11.7 11.1
20 1000Genomes IBS 11.8 11.2

TOP20 of DOD332, just refernce populations:
# ID Ethnicity Normal Distance Adjusted Distance
1 Dodecad Portuguese_D 11.3 10.6
2 Dodecad Spanish_D 11.7 11.1
3 1000Genomes IBS 11.8 11.2
4 Behar Spaniards 12.1 11.5
5 N_Italian_D N_Italian_D 13.3 12.1
6 HGDP North_Italian 15.4 14.2
7 Dodecad French_D 15.6 14.6
8 HGDP French 17.1 16.0
9 HGDP French_Basque 17.3 16.2
10 TSI (HapMap) TSI 21.0 18.9
11 HGDP Sardinian 20.7 19.5
12 Tuscan (HGDP) Tuscan 22.2 20.3
13 HapMap CEU 22.1 20.8
14 Xing Slovenian 22.5 21.3
15 Bulgarians_Y Yunusbayev 23.9 21.9
16 Behar Hungarians 23.2 22.0
17 HGDP Orcadian 24.2 22.7
18 1000Genomes Orkney_1KG 24.7 23.2
19 Dodecad German_D 24.7 23.3
20 Balkans_D Balkans_D 26.7 24.9

TOP20 French_D:
# ID Ethnicity Normal Distance Adjusted Distance
1 Dodecad French 0.0 0.0
2 HGDP French 2.3 2.1
3 DOD440 2.6 2.4
4 DOD706 3.8 3.5
5 DOD594 4.2 3.9
6 DOD184 4.2 3.9
7 DOD687 4.9 4.5
8 VXP001 5.0 4.7
9 VXP001 5.0 4.7
10 DOD618 5.2 4.9
11 DOD337 Portugal 7.1 6.7
12 DOD551 7.4 6.9
13 DOD052 7.5 6.9
14 DOD275 Spanish 7.9 7.3
15 DOD202 Portuguese_D 7.9 7.4
16 DOD619 8.4 7.6
17 DOD595 8.2 7.7
18 DOD818 8.5 8.0
19 DOD504 Spanish_D 8.5 8.0
20 DOD409 IBS 8.8 8.2

TOP20 French_D, just reference populations:
# ID Ethnicity Normal Distance Adjusted Distance
1 Dodecad French_D 0.0 0.0
2 HGDP French 2.3 2.1
3 HapMap CEU 11.4 10.8
4 HGDP French_Basque 12.7 12.0
5 HGDP Orcadian 12.7 12.1
6 1000Genomes Orkney_1KG 13.1 12.4
7 Dodecad Dutch_D 15.1 14.2
8 Dodecad Mixed_Germanic_D 15.8 14.9
9 Behar Spaniards 16.6 15.6
10 1000Genomes Kent_1KG 16.8 15.8
11 Dodecad Portuguese_D 17.2 15.9
12 Dodecad German_D 17.4 16.6
13 Dodecad British_D 18.1 17.0
14 Dodecad Spanish_D 18.2 17.1
15 1000Genomes IBS 18.6 17.4
16 1000Genomes Cornwall_1KG 18.9 17.7
17 Dodecad British_Isles_D 18.8 17.7
18 N_Italian_D N_Italian_D 22.0 20.5
19 Dodecad Irish_D 22.2 20.8
20 Xing Slovenian 24.1 22.8