Sequence Comparison.pdf

96 6 Anatomy of Spaced Seeds
Pr[s] to denote the probability that s occurs at a position k ≥|s|. For any i, j, and k
such that 1 ≤ i, j ≤ m, 1≤ k ≤|π|, we define
⎧
⎨ Pr[W j [k,|π|−1]] if W i [|π|−k,|π − 1]=W j [0,k − 1];
p (ij)
k
= 1 k = |π| & i = j;
⎩
0 otherwise.
It is easy to see that p (ij)
k
is the conditional probability that W j hits at the position
n + k given that W i hits at position n for k < |π| and n.
Theorem 6.1. Let p j = Pr[W j ] for W j ∈ W π (1 ≤ j ≤ m). Then, for any n ≥|π|,
p j
¯Π n =
Proof. For each 1 ≤ j ≤ m,
|π|−1
∑
k=1
m
∑
i=1
m
|π|
∑ ∑
i=1 k=1
π (i)
n+k p(ij) k
, j = 1,2,...,m. (6.6)
p j
¯Π n
]
= Pr
[Ā0 Ā 1 ···Ā n−1 B ( j)
n+|π|−1
|π|−1
]
]
= ∑ Pr
[Ā0 Ā 1 ···Ā n+k−2 A n+k−1 B ( j)
n+|π|−1
+ Pr
[Ā0 Ā 1 ···Ā n+|π|−2 B ( j)
n+|π|−1
k=1
]
= Pr
[Ā0 Ā 1 ···Ā n+k−2 B (i) j)
n+k−1B( n+|π|−1
+ π ( j)
n+|π|
=
=
=
|π|−1
∑
k=1
|π|−1
∑
k=1
m
m
∑
i=1
m
∑
i=1
|π|
∑ ∑
i=1 k=1
] [
]
Pr
[Ā0 Ā 1 ···Ā n+k−2 B (i)
n+k−1
Pr B ( j)
n+|π|−1 |B(i) n+k−1
+ π ( j)
n+|π|
π (i)
n+k p(ij) k
+ π ( j)
n+|π|
π (i)
n+k p(ij) k
.
This proves formula (6.6) .
Example 6.3. Let π = 1 a ∗1 b , a ≥ b ≥ 1. Then, |π| = a+b+1 and W π = {W 1 ,W 2 } =
{1 a 01 b ,1 a+b+1 }. Then we have
⊓⊔
p (11)
k
p (11)
k
p (11)
|π|
= 1,
= p |π|−k−1 q, k = 1,2,...,b,
= 0, k = b + 1,b + 2,...,|π|−1,