Random Walks

Presented By Cindy Xiaotong Lin

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).

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

nn XXXS 10

0XdR 1, nX n

dR 1: nSS n

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.

0X nXdI

1, nSn

),...,( 1 dxxX ),...,( 11 ddxxY 1

1k 1

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.

Example: sequence alignment modeled as RW

| | | ||| || |||ggagactgtagacagctaatgctatagaacgccctagccacgagcccttatc

Simple scoring schemes:at a position: +1, same nucleotides -1, different nucleotides

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Example (cont.): simple RW

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.

Ladder point

Example (cont.): General RW

Consider arbitrary scoring scheme (e.g. substitution matrix)

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 ( )?

Primary Study of RW: 1-d simple RW

h

qX n )1Pr(

hm

pX n )1Pr(

hu

Methods

The Difference Equation Approach Classical

The Moment-Generating Function Approach Ready to generate to more complicate

walk

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)

Difference Equation Approach (M1)

h

11 hhh qp

1,0 ba

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,

qp

ab

ah

hee

ee**

**

ab

hb

hee

eeu **

**

p

qlog*

M1 (cont.): mean number of steps

Difference Equation:

Initial Conditions:

Solution:

hm

111 hhh qmpmm

0 ba mm

ab

ah

hee

ee

pq

ab

pq

ahm **

**

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)

)()()( yPeeEm Y

y

y

Y

nX

peqem )(

*

1)( * m

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:

)0(,1)(**

Npeqe N

hb ha h

hu

1)1(** )()( ha

hhb

h eae

M2 (cont.)

Therefore, we have

Thus,

Which is identical to (7.9), the solution from difference equation approach.

1)1(** )()( ha

hhb

h eae

**

**

ab

ah

hee

ee

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

N

j jN ST1

1))(( NTN emE

hN mSETE )()(

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

hN mSETE )()(

)()()( hauhbTE hhN

qpSE )(

M2(cont.)

The mean of number of steps until the walk stops,

Which is agree with the result from difference equation approach

qp

hbhaum hh

h

)()(

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,

bah ;1;0

b

pqm

10

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,

ddcc ,1,...,0,...,1, dcc ppp ,...,1,

0)(

d

cjjjpSE

d

cj

jjepm )(

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.

*

1*

d

cj

jjep

0y

.1,...,,...,1, dyycc

1)(*

NTeE NT

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

111

**

dy

yk

kk

ck

kk ePeP

kP

A 0m

d

cj j

c

j jN

jp

jR

SE

TEA 1

)(

)(

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.

)(yFunrY

ykQ

k

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,

d

Y

y

kkY

QQQQ

kyFQQyFunrunr

...1

);()(

21

_

0

_

),,(

,))(1(lim

*_

*

k

yY

y

QQfV

VeyFunr

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 .

y

)(* yF Y

)()()( *

1

** jyFRyFyF unrunr Y

c

jjYY

jRj

General Walk: restricted

Then,

d

k

kk

c

j

jj

yY

ekQe

eRQ

C

CeyYyF

1

1

_

*

))(1(

)1(

,~)Pr()(

**

*

*

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.

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.

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.

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Discussion

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