1. Presented By Cindy Xiaotong Lin
Random Walks
Presented By Cindy Xiaotong Lin

2. Why Random Walks?
A random walk (RW) is a useful model in understanding stochastic processes across a variety of scientific disciplines.
Random walk theory supplies the basic probability theory behind BLAST ( the most widely used sequence alignment theory).

3. What is a Random Walk?
An Intuitive understanding: A series of movement which direction and size are randomly decided (e.g., the path a drunk person left behind).
Formal Definition: Let a fixed vector in the d-dimensional Euclidean space and a sequence of independent, identically distributed (i.i.d.) real-valued random variables in . The discrete-time stochastic process defined by
is called a d-dimensional random walk

4. Definitions (cont.)
If and RVs take values in , then is called d-dimensional lattice random walk.
In the lattice walk case, if we only allow the jump from to where or , then the process is called d-dimensional sample random walk.

5. Definitions (cont.)
A random walk is defined as restricted walk if the walk is limited to the interval [a, b].
The endpoints a and b are called absorbing barriers if the random walk eventually stays there forever;
or reflecting barriers if the walk reaches the endpoint and bounces back.

6. Example: sequence alignment modeled as RW
| | | ||| || |||
ggagactgtagacagctaatgctata
gaacgccctagccacgagcccttatc
Simple scoring schemes:
at a position: +1, same nucleotides
-1, different nucleotides *

7. Example (cont.): simple RW
Ladder point
Ladder Point (LP):the point in the walk lower than any previously reached points.
Excursion: the part of the walk from a LP until the highest point attained before the next LP.
Excursions in Fig: 1, 1, 4, 0, 0, 0, 3;
BLAST theory focused on the maximum heights achieved by these excursions.

8. Example (cont.): General RW
Consider arbitrary scoring scheme
(e.g. substitution matrix)

9. Primary Study of RW: 1-d simple RW
RW: Consider a 1-d simple RW starting at h,
restricted to the interval [a, b], where a
and b are absorbing barriers,
and
Problems: I. (Absorption Probabilities) what is the
probability that eventually the walk
finishes at b (or a) rather than a (or
b), i.e., (or )?
II. What is the mean number of steps
taken until the walk stops ( )?

10. Methods The Difference Equation Approach Classical
The Moment-Generating Function Approach
Ready to generate to more complicate walk

11. Difference Equation Approach (M1)
Assume: the probability that the simple random walk eventually finishes (absorbed) at b.
Difference Equation obtained by comparing the situation just before and after the first step of the walk:
(7.4)
Initial Conditions:
(7.5)

12. M1 (cont.): solutions
Solve Equ 7.4, using the theory of homogeneous difference equations
when :
The same procedure can be used to obtain the probability that the walk ends at a,

13. M1 (cont.): mean number of steps
Difference Equation:
Initial Conditions:
Solution:

14. Moment-Generating function Approach (M2)
Recall the definition of mgf of a random variable Y:
In our case, mgf of random variable is:
According to Theorem 1.1, there exists a unique nonzero value of such that
(7.12)

15. M2 (cont.)
The mgf of the total displacement after N steps is from (2.17)
When the walk has just finished, the total displacement is either
or with the probabilities of
or respectively:

16. M2 (cont.) Therefore, we have Thus,
Which is identical to (7.9), the solution from difference equation approach.

17. M2(cont.): Mean number of steps until the walk stops
Assume the total displacement after N steps is
Theorem 7.1(Wald’s Identity) states:
Derivative with respect to on both sides, and obtain

18. M2(cont.) In , (7.24) The mean of displacement in N steps
The mean of step size
Which states: the mean value of the final total displacement of the walk, is the mean size of each step multipled by the mean number of steps taken until the walk stops

19. M2(cont.) The mean of number of steps until the walk stops,
Which is agree with the result from difference equation approach

20. An Asymptotic case: a walk BLAST concerns
The walks BLAST concerns are,
a walk without upper boundary and ending at -1.
Applying the previous results and
We get the following Asymptotic results:
The probability distribution of the maximum value that the walk ever achieves before reaching -1 is in the form of the geometric-like probability.
The mean number of steps until the walk stops,

21. General Walk Suppose generally the possible step sizes are,
and their respective
probabilities are,
The mean of step size is negative, i.e.,
The mgf of S(step size) is,

22. General Walk (cont.)
According to Theorem 1.1, there exists unique positive , such that,
To consider the walk that start at 0, with stopping boundary at -1 and without upper boundary, impose an artificial barrier at
The possible stopping points can be,
And Wald’s Identity states,
where, is the total displacement when the walk stops.

23. General Walk
Thus,
Where, is the probability that the walk finishes at the point k.
The mean of number of steps until the walk stops or would be

24. General Walk: unrestricted
Objective: Find the probability distribution of the maximum value that the walk ever achieves before reaching -1 or lower.
Define:
the probability that in the unrestricted walk, the maximum upward excursion is or less;
is the probability that the walk visits the positive value before reaching any other positive value.

25. General Walk: unrestricted
Therefore,
The event that in the unrestricted walk the maximum upward excursion is y or less is the union of the event that the maximum excursion never reaches positive values and the events the first positive value achieved by the excursion is k, k=1,2,…y, then the walk never achieves a further height exceeding y-k
Applying the Renewal Theorem, we have,

26. General Walk: restricted
Consider general walk starting at 0, lower barrier at -1.
The size of an excursion of the unrestricted walk can exceed the value either before or after reaching negative value, i.e.,
Where, the probability that the size of an excursion in the restricted walks exceeds the value up y is the probability that the first negative value reached by the walk is

27. General Walk: restricted
Then,

28. Application: BLAST
BLAST is the most frequently used method for assessing which DNA or protein sequences in a large database have significant similarity to a given query sequence;
a procedure that searches for high-scoring local alignments between sequences and then tests for significance of the scores found via P-value.
The null hypothesis to be test is that for each aligned pair of animo acids, the two amino acids were generated by independent mechanism.

29. BLAST (cont.) : modeling
The positions in the alignment are numbered from left to right as 1, 2,…, N. A score S(j, k) is allocated to each position where the aligned amino acid pair (j,k) is observed, where S(j,k) is the (j,k) element in the substitution matrix chosen.
An accumulated score at position i is calculated as the sum of the scores for the various amino acid comparison at position 1, 2,…,i. As i increases, the accumulated score undergoes a random walk.

30. BLAST (cont.) : calculating parameters
Let Y1, Y2,… be the respective maximum heights of the excursions of this walk after leaving one ladder point and before arriving the next, and let Ymax be the maximum of these maxima. It is in effect the test statistic used in BLAST. So it is necessary to find its null hypothesis distribution.
The asymptotic probability distribution of any Yi is shown to be the geometric-like distribution. The values of C and in this distribution depend on the substitution matrix used and the amino acid frequencies {pj} and {pj’}. The probability distribution of Ymax also depends on n, the mean number of ladder points in the walk.

31. Discussion
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