1. Diffusion kernels on SNP data embedded in a
non-Euclidean metric
Animal Breeding & Genomics Seminar
Gota Morota
April 10, 2012
1 / 37

2. Kernel functions
Definition
A kernel is a weighting function which provides a similarity metric
1. define a function that measures distance (metric) for
genotypes
2. compute a similarity based on this metric space
⇓
function of a distance under certain metric space f(||x − x ||)
• Euclidean distance
• Manhattan distance
• Mahalanobis distance
• Minkowski distance
2 / 37

3. Metric (Distance function)
Definition
A function which defines a distance between two points
If one picks Euclidean metric, the Mat´ern covariance function
offers flexible kernels
K(x, x ) = σ2
K
21−ν
Γ(ν)
√
2ν(||x − x ||/h)ν
K(||x − x ||/h)
• Gaussian Kernel: ν = ∞, exp(−θ(||x − x ||2
))
• Exponentail Kernel: ν = 1
2 , exp(−θ(||x − x ||))
⇓
A choice of a metric determines characteristics of a kernel
3 / 37

4. Euclidean Metric
Definition
The distance function given by the Pythagorean theorem
(a2
+ b2
= c2
)
Euclidean distance on R2
xi = (xi1, xi2), xj = (xj1, xj2)
||xi − xj|| = (xi1 − xj1)2 + (xi2 − xj2)2
Figure 1: Euclidean distance
between two points A and B
Euclidean distance on Rp
||xi − xj|| = (xi1 − xj1)2 + · · · + (xik − xjk )2 + · · · + (xip − xjp)2
4 / 37

5. Euclidean space
Euclidean distance is a metric on a metric space callled Euclidean
space
Figure 2: 3-dimensional Euclidean
space. −∞ ≤ (X, Y, Z) ≤ ∞
Suppose, we observed two
individuals with 3 SNP
genotypes.
• ID1 = x1 = (0,2,2)
• ID2 = x2 = (2,1,0)
Euclidean distance on R3
||x1 − x2|| = (0 − 2)2 + (2 − 1)2 + (2 − 0)2 = 3
5 / 37

6. Metric on graphs
A graph is consisted of vertices and edges
0 1 2
012
0
1
2
1st Genotype
2ndGenotype
3rdGenotype
6 / 37

7. Metric on graphs (continue)
0 1 2
012
0
1
2
1st Genotype
2ndGenotype
3rdGenotype
(2,1,2)
(2,0,1)
(0,1,2)
(0,2,0)
(0,1,0)
(0,1,1)
(1,0,0) (2,0,0)
(1,1,0) (2,1,0)
(1,2,0) (2,2,0)
(1,0,1)
(1,1,1) (2,1,1)
(0,2,1) (1,2,1) (2,2,1)
(0,2,2) (1,2,2) (2,2,2)
(1,0,2)
(1,1,2)
7 / 37

8. Metric on graphs (continue)
Two individuals with 3 SNP genotypes previously shown.
• ID1 = x1 = (0,2,2), ID2 = x2 = (2,1,0)
0 1 2
012
0
1
2
1st Genotype
2ndGenotype
3rdGenotype
(2,1,0)
(0,2,2)
8 / 37

9. The purpose of this study
1. Is the Euclidean distance adequate for genotypes?
2. The metric on graphs seems to be given by the Manhattan
distance, but how to express the degree of similarity?
• Embed SNP data in a non-Euclidean metric space
• Define a metric for discrete genotypes on graphs and
construct a kernel on this metric
⇓
Develope a kernel that is suited for all kinds of kernel-based
genomic analyses
9 / 37

10. Diffusion on one-dimensional graphs (Z1
3)
We have three possible genotypes, 0 (aa), 1 (Aa) and 2 (’AA’).
0 − 1 − 2 (1)
0 − 1
/
2
(2)
10 / 37

11. Diffusion on one-dimensional graphs (Z1
3)
We have three possible genotypes, 0 (aa), 1 (Aa) and 2 (’AA’).
0 − 1 − 2 (1)
0 − 1
/
2
(2)
1. Graph (1) path graph
• genotype 1’s (’Aa’) influence diffuses to genotype 0 (’aa’) and
2 (’AA’)
• genotype 0’s (’aa’) influence diffuses to only genotype 1 (’Aa’)
• genotype 2’s (’AA’) influence diffuses to only genotype 1 (’Aa’)
2. Graph (2) complete graph
• the distance from genotype 0 (’aa’) to genotype 2 (’AA’) is the
same as that from 0 (’aa’) to 1 (’Aa’).
10 / 37

12. Diffusion on one-dimensional graphs (Z1
3)
We have three possible genotypes, 0 (aa), 1 (Aa) and 2 (’AA’).
0 − 1 − 2 (1)
0 − 1
/
2
(2)
1. Graph (1) path graph
• genotype 1’s (’Aa’) influence diffuses to genotype 0 (’aa’) and
2 (’AA’)
• genotype 0’s (’aa’) influence diffuses to only genotype 1 (’Aa’)
• genotype 2’s (’AA’) influence diffuses to only genotype 1 (’Aa’)
2. Graph (2) complete graph
• the distance from genotype 0 (’aa’) to genotype 2 (’AA’) is the
same as that from 0 (’aa’) to 1 (’Aa’).
• more reasonable to assume that genotype ’Aa’ is closer than
’aa’ to ’AA’ which has two copies of the ’A’ allele.
• genotype 0 (’aa’) requires two mutations to become genotype
2 (’AA’), while genotype 1 (’Aa’) requires only one mutation 10 / 37

13. Diffusion on two-dimensional graphs (Z2
3)
Two-dimensional graphs are given by the Cartesian graph product
( ) of the 2 one-dimensional graphs 0 - 1 - 2.
0 − 1 − 2 0 − 1 − 2 (3)
Let Γ1 and Γ2 be two graphs. Consider a graph with vertex set
V(Γ1) × V(Γ2), with vertices (x, x ) ∈ V(Γ1) and (y, y ) ∈ V(Γ2).
Cartesian graph product
The Cartesian graph product connects two vertices (x, y) and
(x , y ) if only if x = x , y ∼ y or y = y , x ∼ x , where “∼” means
connected.
11 / 37

14. Example of the Cartesian graph product ( )
Cartesian graph product of the 2 one-dimensional graphs
0 − 1 − 2 0 − 1 − 2
Fisrt, list all possible configuration of vertices
02 12 22
01 11 21
00 10 20
12 / 37

15. Example of the Cartesian graph product ( ) (continue)
Cartesian graph product of the 2 one-dimensional graph
0 − 1 − 2 0 − 1 − 2
The Cartesian graph product connects two vertices (x, y) and
(x , y ) if only if x = x , y ∼ y or y = y , x ∼ x , where “∼” means
connected.
• 0 = 0, 0 ∼ 1 → connected
• 0 = 0, 1 ∼ 2 → connected
02 12 22
01 11 21
00 10 20
⇒
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16. Example of the Cartesian graph product ( ) (continue)
Cartesian graph product of the 2 one-dimensional graph
0 − 1 − 2 0 − 1 − 2
The Cartesian graph product connects two vertices (x, y) and
(x , y ) if only if x = x , y ∼ y or y = y , x ∼ x , where “∼” means
connected.
• 0 = 0, 0 ∼ 1 → connected
• 0 = 0, 1 ∼ 2 → connected
02 12 22
01 11 21
00 10 20
⇒
13 / 37

17. Example of the Cartesian graph product ( ) (continue)
Cartesian graph product of the 2 one-dimensional graph
0 − 1 − 2 0 − 1 − 2
The Cartesian graph product connects two vertices (x, y) and
(x , y ) if only if x = x , y ∼ y or y = y , x ∼ x , where “∼” means
connected.
• 0 = 0, 0 ∼ 1 → connected
• 0 = 0, 1 ∼ 2 → connected
02 12 22
01 11 21
00 10 20
⇒
02 12 22
|
01 11 21
|
00 10 20
13 / 37

18. Example of the Cartesian graph product ( ) (continue)
Cartesian graph product of the 2 one-dimensional graphs
0 − 1 − 2 0 − 1 − 2
The Cartesian graph product connects two vertices (x, y) and
(x , y ) if only if x = x , y ∼ y or y = y , x ∼ x , where “∼” means
connected.
• 0 = 0, 0 ∼ 1 → connected
• 0 1, 0 1 → not connected
• 0 1, 0 2 → not connected
02 12 22
|
01 11 21
|
00 10 20
⇒
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19. Example of the Cartesian graph product ( ) (continue)
Cartesian graph product of the 2 one-dimensional graphs
0 − 1 − 2 0 − 1 − 2
The Cartesian graph product connects two vertices (x, y) and
(x , y ) if only if x = x , y ∼ y or y = y , x ∼ x , where “∼” means
connected.
• 0 = 0, 0 ∼ 1 → connected
• 0 1, 0 1 → not connected
• 0 1, 0 2 → not connected
02 12 22
|
01 11 21
|
00 10 20
⇒
14 / 37

20. Example of the Cartesian graph product ( ) (continue)
Cartesian graph product of the 2 one-dimensional graphs
0 − 1 − 2 0 − 1 − 2
The Cartesian graph product connects two vertices (x, y) and
(x , y ) if only if x = x , y ∼ y or y = y , x ∼ x , where “∼” means
connected.
• 0 = 0, 0 ∼ 1 → connected
• 0 1, 0 1 → not connected
• 0 1, 0 2 → not connected
02 12 22
|
01 11 21
|
00 10 20
⇒
02 12 22
|
01 11 21
|
00 − 10 20
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21. Diffusion on two-dimensional graphs (Z2
3) (continue)
A graph from the Cartesian graph product between path graphs of
any size takes the form of a grid.
02 − 12 − 22
| | |
01 − 11 − 21
| | |
00 − 10 − 20
A SNP grid of p loci is a p dimensional grid with vertices in Zp
3
, with
two vertices x and x adjacent if and only if
p
i=1
|xi − xi | = 1.
i.e., two vertices are adjacent if and only if just one SNP locus
differs by 1.
15 / 37

22. Diffusion on three-dimensional graphs (Z3
3
)
Cartesian graph product of the 3 one-dimensional graphs.
0 − 1 − 2 0 − 1 − 2 0 − 1 − 2
In general, the p-dimensional SNP grid graph is
p
i=1
Γ, where
Γ = 0 − 1 − 2.
0 1 2
012
0
1
2
1st Genotype
2ndGenotype
3rdGenotype
(2,1,2)
(2,0,1)
(0,1,2)
(0,2,0)
(0,1,0)
(0,1,1)
(1,0,0) (2,0,0)
(1,1,0) (2,1,0)
(1,2,0) (2,2,0)
(1,0,1)
(1,1,1) (2,1,1)
(0,2,1) (1,2,1) (2,2,1)
(0,2,2) (1,2,2) (2,2,2)
(1,0,2)
(1,1,2)
16 / 37

23. Graph Laplacians
The Laplacian of a graph 0 − 1 − 2 is
L(Γ) = −A(Γ) + Λ
= −


0 1 0
1 0 1
0 1 0


+


1 0 0
0 2 0
0 0 1


=


1 −1 0
−1 2 −1
0 −1 1


where A is an adjacency matrix and Λ is a diagonal matrix with
Λii = n
j=1 Aij.
17 / 37

24. Graph Laplacians (continue)
The Laplacian of a graph 0 − 1 − 2 0 − 1 − 2 is a square matrix of
dimension 32
× 32
.
L(Γ) =


200 −1 0 −1 0 0 0 0 0
−1 301 −1 0 −1 0 0 0 0
0 −1 202 0 0 −1 0 0 0
−1 0 0 310 −1 0 −1 0 0
0 −1 0 −1 411 −1 0 −1 0
0 0 −1 0 −1 312 0 0 −1
0 0 0 −1 0 0 220 −1 0
0 0 0 0 −1 0 −1 321 −1
0 0 0 0 0 −1 0 −1 222


18 / 37

25. Diffusion on graphs at time t
• kx is a function which measures the spread of ’influence’ of
the genotype x over other genotypes.
• k˜x(0, x) = 1x=˜x(x), at time 0.
• define the time t diffusion of the ’influence’ of genotype ˜x on
genotype x to be
k˜x(t, x) = k˜x(t − 1, x) +
|x−x |=1
α(k˜x(t − 1, x ) − k˜x(t − 1, x))
19 / 37

26. Diffusion on graphs at time t (continue)
k˜x(t, x) = k˜x(t − 1, x) +
|x−x |=1
α(k˜x(t − 1, x ) − k˜x(t − 1, x))
• x = (0, 1, 2) is the genotype code, α = (0.1, 0.2) is the
diffusion rate.
• k˜x(t, x) is the time t diffusion of the influence of genotype ˜x on
genotype x.
α= 0.1 α = 0.2
x = 0 1 2 x= 0 1 2
k1(0, x) 0 1 0 k1(0, x) 0 1 0
k1(1, x) 0.1 0.8 0.1 k1(1, x) 0.2 0.6 0.2
k1(2, x) 0.17 0.66 0.17 k1(2, x) 0.28 0.44 0.28
k1(3, x) 0.219 0.562 0.219 k1(3, x) 0.312 0.376 0.312
k1(15, x) 0.331 0.336 0.331 k1(15, x) 0.333 0.333 0.333
20 / 37

27. Diffusion on graphs at time t (continue)
Writing in vector form, with k˜x(t, x) = [k˜x(t)]x, we get
k˜x(t) = k˜x(t − 1) + αHk˜x(t − 1)
= (I + αH)k˜x(t − 1)
= (I + αH)t
k˜x(0)
• H is the negative of the graph Laplacian
• in order to make ’time’ continuous, let α = θh (θ > 0) and
t = 1/h.
• by using a small h, we can achieve a discretization of the
’diffusion time’
lim
h→0
(I + θhH(Γ))1/h
= exp(θH)
=
∞
k=0
θk
k!
Hk
= I + θH +
θ2
2
H2
+
θ3
3!
H3
+ · · · +
θn
n!
Hn
+ · · ·
21 / 37

28. Diffusion kernels
Definition
Suppose a graph Γ with a graph Laplacian L(Γ). Then exp(θH(Γ))
or exp(−θL(Γ)) is called the diffusion kernel or heat kernel for
graph Γ, where θ is a rate of diffusion.
Here putting K = exp(θH) and taking the derivative with respect to
θ gives,
d
dθ
K = HK (4)
which is a diffusion equation (heat equation) on a graph with
H = −L(Γ).
22 / 37

29. Gaussian kernels
Definition
A Gaussian kernel is a space continuous diffusion kernel
• in order to make ’space’ continuous, we create an infinite
number of ’fake’ genotypes between and outside of 0 and 2
• i.e., consider genotypes such as 1.23 or −10.5.
• each genotype x is connected to only two genotypes, x + dx
and x − dx for some infinitesimal dx.
• H becomes an infinite matrix, and H(x, x ) is −2 for x = x
and 1 for x + dx, x − dx.
H(Γ) =


−1 1 0
1 −2 1
0 1 −1


⇒
Infinite matrix with diagonal
elements equal to -2 and 1 for its
neighbors and 0 otherwise
23 / 37

30. Gaussian kernels (continue)
• a vector of genotypes: x = (−∞, · · · , x − dx, x, x + dx, · · · , ∞)
• an influence function:
f = (f(−∞), · · · , f(x − dx), f(x), f(x + dx), · · · , f(∞))
• Approximating dx by h, and dividing H by h2
, HfT
/h2
indexed
by the genotype x will be
1
h2
[H(x, ·)fT
] =
f(x + h) − 2f(x) + f(x − h)
h2
=
f(x+h)−f(x)
h −
f(x)−f(x−h)
h
h
f (x)
• Thus, with space continuity, H acts like d
dx2 . Using this analogy
back in (4), we get the heat equation.
d
dθ
Kθ(x) =
d
dx2
Kθ(x)
24 / 37

31. Gaussian kernels (continue)
• The solution to this partial differential equation (PDE) with
Dirac delta initial condition of concentration on x = 0,
k0(x) = 1x=0, is given by
Gθ(x) =
1
√
4πθ
exp −
x2
4θ
• This is a Gaussian density in one dimensional space with
θ = σ2
e/2.
• With the initial condition K0(x) = f(x), the solution to this PDE
is
Kθ(x) =
R
f(x )Gθ(x − x )dx
This kernel gθ(x, x ) = G(x − x ) is the Gaussian kernel with
bandwidth θ.
25 / 37

32. Gaussian kernels (continue)
• For example, allowing additional genotypes (0.25, 0.50, 0.75,
1.25, 1.50, 1.75).
• now, x ∈ R9
instead of x ∈ Z3
0 1 2
012
0
1
2
1st Genotype
2ndGenotype
3rdGenotype
(0,1.75,2)
26 / 37

33. Computation of diffusion kernels
Kernel notation:
• K as the kernel matrix indexed by the observed covariates
• K for the infinite dimensional kernel for the Gaussian, and the
3p
× 3p
dimensional kernel for the diffusion kernel
Gaussian kernels
• K: infinite dimensional
kernel
• K = exp(−θ(||x − x ||2
))
• we have a closed form for K,
so no need to deal with K
Diffusion kernels
• K: 3p
× 3p
dimensional
kernel
• K: is there any way to
directly compute K so that
we don’t need to deal with
K?
• closed form for K?
27 / 37

34. Computation of diffusion kernels (continue)
Let K1(θ) and K2(θ) be the kernels for the two graphs Γ1 and Γ2.
The diffusion kernel for Γ = Γ1 Γ2 is
K1(θ) ⊗ K2(θ).
were ⊗ is the tensor product.
Suppose, Γ1 = 0 − 1 − 2, K(Γ1) is a diffusion kernel on Γ1.
SNP grid graph on p dimensions
• p
i=1
Γ1
SNP grid kernel on p dimensions
•
p
i=1 K(Γ1)
⇓
We just need to compute K(Γ1) = exp(−θL(Γ1)) and take the
tensor product p times!
28 / 37

35. Matrix exponentiation
Γ1 = 0 − 1 − 2
H =


1 −1 0
−1 2 −1
0 −1 1


We make use of matrix diagonalization H = TDT−1
to obtain
Kθ = exp(θH)
= T exp(θD)T−1
=
1
6


e−3θ + 3e−θ + 2 −2e−3θ + 2 e−3θ − 3e−θ + 2
−2e−3θ + 2 4e−3θ + 2 −2e−3θ + 2
e−3θ − 3e−θ + 2 −2e−3θ + 2 e−3θ + 3e−θ + 2


Here, exp(θD) becomes simple componentwise exponentiation
because D is a diagonal matrix of eigenvalues.
29 / 37

36. Diffusion kernels indexed by the observed covariates
Symmetric property
Kθ(x, x ) =



−2e−3θ + 2 if |xi − xi
| = 1
e−3θ − 3e−θ + 2 if |xi − xi
| = 2
e−3θ + 3e−θ + 2 if xi = xi
, x 1
4e−3θ + 2 if xi = xi
= 1
Thus,
K
⊗p
θ (x, x ) ∝
p
i=1



(e−3θ
− 3e−θ
+ 2)δ|xi−xi
|=2 + (−2e−3θ
+ 2)δ|xi−xi
|=1
+ (e−3θ
+ 3e−θ
+ 2)δxi=xi
1 + (4e−3θ
+ 2)δxi=xi
=1



30 / 37

37. Diffusion kernels indexed by the observed covariates
(continue)
• let x and x be an SNP data for p loci; ns be the number of loci
for which |xi − xi
| = s
• let m11 be the number of loci for which xi = xi
= 1, i.e., m11 is
the number of loci that two individuals share heterozygous
states.
Using the fact that
n1 + n0 + n2 = p,
K
⊗p
θ (x, x ) =(−2e−3θ
+ 2)n1
(e−3θ
− 3e−θ
+ 2)n2
(e−3θ
+ 3e−θ
+ 2)n0−m11
(4e−3θ
+ 2)m11
∝
(−2e−3θ + 2)n1 (e−3θ − 3e−θ + 2)n2 (4e−3θ + 2)m11
(e−3θ + 3e−θ + 2)n1+n2+m11
We obtain a SNP grid kernel.
31 / 37

38. Example of computing a diffusion kernel
Two individuals with 3 SNP genotypes previously shown.
• ID1 = x1 = (0,2,2)
• ID2 = x2 = (2,1,0)
0 1 2
012
0
1
2
1st Genotype
2ndGenotype
3rdGenotype
(2,1,0)
(0,2,2)
Since
Kθ(x, x ) =



−2e−3θ + 2 if |xi − xi
| = 1
e−3θ − 3e−θ + 2 if |xi − xi
| = 2
e−3θ + 3e−θ + 2 if xi = xi
, x 1
4e−3θ + 2 if xi = xi
= 1
Similarity between ID1 and ID2 is
K⊗3
θ (x, x ) =
(−2e−3θ + 2)1
(e−3θ − 3e−θ + 2)2
(e−3θ + 3e−θ + 2)1+2
32 / 37

39. Diffusion kernels for binary genotypes
Here, x ∈ Zp
2
Γ = 0 − 2
L(Γ) = −H(Γ)
=
1 −1
−1 1
K
⊗p
θ (x, x ) ∝
1 − exp(−2θ)
1 + 2 exp(−2θ)
d(x,x )
where d(x, x ) is the Hamming distance, that is, number of
coordinates at which x and x differ.
33 / 37

40. Applications
A SNP kernel can be used in DNA-based genomic analyses
including
• regressions
• classifications
• kernel association studies
• kernel principal component analyses
Application of using the diffusion kernel on real data
• 7902 Holstein bulls (USDA-ARS AIPL)
• 43382 SNPs
34 / 37

41. Diffusion kernels based on for different θ
K(i,i')
Frequency
0.10 0.15 0.20 0.25 0.30 0.35
0.0e+001.0e+07
θ = 10
K(i,i')
Frequency
0.45 0.50 0.55 0.60 0.65 0.70
0.0e+001.0e+07
θ = 11
K(i,i')
Frequency
0.74 0.78 0.82 0.86
0.0e+006.0e+061.2e+07
θ = 12
K(i,i')
Frequency
0.90 0.91 0.92 0.93 0.94 0.95
0.0e+006.0e+061.2e+07
θ = 13
Figure 8: Elements of four diffusion kernels based on four different
bandwidth parameters (θ).
35 / 37

42. Conclusion
Diffusion kernels
• various graph structures can be used to represent sets of
discrete random variables, such as genotypes
• defines the distance between two vertices, and projects this
information into a more interpretable Rn
• matrix exponentiation of the graph Laplacian
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43. Conclusion
Diffusion kernels
• various graph structures can be used to represent sets of
discrete random variables, such as genotypes
• defines the distance between two vertices, and projects this
information into a more interpretable Rn
• matrix exponentiation of the graph Laplacian
• which senario, the Gaussian can approximate the diffusion
kernel well?
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44. Conclusion
Diffusion kernels
• various graph structures can be used to represent sets of
discrete random variables, such as genotypes
• defines the distance between two vertices, and projects this
information into a more interpretable Rn
• matrix exponentiation of the graph Laplacian
• which senario, the Gaussian can approximate the diffusion
kernel well?
R package ’dkDNA’ will be available on CRAN soon
• SNP grid kernel
• binary grid kernel
• other DNA structures/polymorphisms in future
• written in Fortran
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45. Acknowledgments
• Daniel Gianola
• Grace Wahba
• Masanori Koyama
• Chen Yao
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