Chaos Game Representationppt

7/25/2019 Chaos Game Representationppt

1/35

PlanChaos Game Representation (CGR)

Digital Search Tree (DST) and CGRMain ResultsPerspectives

Digital Search Trees andChaos Game Representation

Peggy Cenac1

in collaboration with Brigitte Chauvin2, Nicolas Pouyanne2 and

Stephane Ginouillac2

Journees ALEA 2006 - CIRM Luminy

1INRIA Rocquencourt

2University of Versailles Saint Quentin

Peggy Cenac Digital Search Trees and Chaos Game Representation


2/35



Plan

1 Chaos Game Representation (CGR)DefinitionStochastic properties of the CGR

2 Digital Search Tree (DST) and CGRThe CGR-treeConstruction of the CGR-treeExampleRelation between the CGR-treeand DSTState of the art

3 Main ResultsAssumptions and notationsAsymptotic resultsNumerical experimentsGuidelines for the proofs

4 PerspectivesPeggy Cenac Digital Search Trees and Chaos Game Representation


3/35



DefinitionStochastic properties of the CGR

Chaos Game Representation (CGR)



4/35




Definition

Graphical representation of DNA in a bounded set.

Storage toolPattern visualization

Sequences comparison (local/global)Iterative mapping technique

DNA sequence U= (ui)i=1,...,n, where ui {A, C, G, T}.The Chaos Game Representation ofU, on the unit square S isa sequence {X0, . . . , Xn} defined by

X0 = (12 ,

12 )

Xi+1 = 12

Xi+ui+1

,

A= (0, 0), C = (0, 1), G = (1, 1), T = (1, 0).


Pl


5/35




Examples (1)

CGR of the word ATGCGAGTGT.Peggy Cenac Digital Search Trees and Chaos Game Representation

Pl


6/35




Examples (2)

CGR of 200000 nucleotides of Chromosome 2 of Homo Sapiens (on the

left) and of Bacteroides Thetaiotaomicron (on the right).


Plan


7/35




A(0,0)

C(0,1)

T(1,0)

G(1,1)

Sa

Sc

St

Sg

A(0,0)

C(0,1)

T(1,0)

G(1,1)

Saa

Sca

Sac

Scc

Sta

Sga

Stc

Sgc

Sat

Sct

Sag

Scg

Stt

Sgt

Stg

Sgg

Sw def=i

k=11

2ik+1vk+

12i

S, where w is the word v1. . . vi.

Counting pointsin Sw counting occurrencesofw.

Each point contains thewholesequencehistory.Peggy Cenac Digital Search Trees and Chaos Game Representation

Plan


8/35




Stochastic properties of the CGR

U is supposed to be a stationary ergodic sequence.

(Xn)n0 is a Markov chain of order 1, and converges almostsurely to a random vector Xwith distribution .

When U is i.i.d. and uniformly distributed, is the Lebesguemeasure on S. Whenever U is not uniformly distributed, iscontinuous, singular with respect to the Lebesgue measure.

The law of large number holds, and the empirical measuresconverge.


Plan The CGR-tree


9/35



The CGR-treeConstruction of the CGR-treeExampleRelation between the CGR-treeand DSTState of the art

Digital Search Tree (DST) and CGR


Plan The CGR-tree


10/35



The CGR treeConstruction of the CGR-treeExampleRelation between the CGR-treeand DSTState of the art

The CGR-tree

In the CGR of a sequence U=U1. . . Ui. . . , one successivelyrepresents

U1

U1U2...

U1U2. . . Undef=U(n)

U(i) Swis equivalent to Ui|w|+1. . . Ui1Ui =w.

We define a representation of a DNA sequence U as aquaternary tree, the CGR-tree, in which one can visualizerepetitions of subwords.


Plan The CGR-tree


11/35



The CG t eeConstruction of the CGR-treeExampleRelation between the CGR-treeand DSTState of the art

Construction

We adopt the classical order (A, C, G, T) on letters.

Let Tbe the complete infinite 4-ary tree. Each node ofT has4 branches corresponding to the letters (A, C, G, T) orderedin the same way.

The CGR-tree ofU is anincreasing sequenceT1 T2. . . Tn . . .of finite subtrees ofT, each Tn

having n nodes.Successively insertthe reverted words Ui. . . U1.


Plan The CGR-tree


12/35

Chaos Game Representation (CGR)Digital Search Tree (DST) and CGR

Main ResultsPerspectives

Construction of the CGR-treeExampleRelation between the CGR-treeand DSTState of the art

Example

Construction of the tree for U=GAGCACAGTGGAAGGG :GAGCACAGTGGAAGGG


Plan The CGR-tree


13/35




GAGCACAGTGGAAGGG


Plan The CGR-tree


14/35




GAGCACAGTGGAAGGG


PlanCh G R i (CGR)

The CGR-treeC i f h CGR


15/35




GAGCACAGTGGAAGGG


PlanCh G R t ti (CGR)

The CGR-treeC t ti f th CGR t


16/35




GAGCACAGTGGAAGGG



The CGR-treeConstruction of the CGR tree


17/35




Representation of 16 nucleotides ofMus Musculus

GAGCACAGTGGAAGGG in the CGR-tree (on the left) and in the

normalized CGR (on the right).



The CGR-treeConstruction of the CGR tree


18/35




Remarks

A CGR-tree without its labels is equivalent toa list of wordsin the sequence without their order.

Shapeof CGR-tree Representation in the unit square.Each nodeof the tree w=w1. . . wd is associated with thepoint

Xwdef=

d

k=1

wk

2dk+1

+X0

2d

,

thecenter of the corresponding square Sw.



The CGR-treeConstruction of the CGR-tree


19/35




Chaos Game Representation (on the left) and normalized CGR (on the

right) of the first 400000 nucleotides of Chromosome 2 ofHomo Sapiens.





20/35



Construction of the CGR treeExampleRelation between the CGR-treeand DSTState of the art

Relation between the CGR-treeand DST

Proposition

The CGR-tree of a random sequence U=U1U2. . .is a DigitalSearch Tree (DST), obtained by inserting in a quartenary tree thesuccessive reverted prefixes.

W(1) = U1,

W(2) = U2U1,...

W(n) = UnUn1. . . U1,...





21/35

p ( )Digital Search Tree (DST) and CGR


ExampleRelation between the CGR-treeand DSTState of the art

State of the art

In the Bernoulli model : the trees are binary, built withindependent successive sequences having the samedistribution ; the two letters have the same probability 12 . Several results are known (see chap. 6 in Mahmoud

(1992)), concerning the height, the insertion depth andthe profile.

Aldous and Shields (1998) prove by embedding incontinuous time that the height satisfies

Hn log2n Pn

0.

The height is concentrated (Drmota (2002)).For DSTs built from independent sequences on an alphabetwith m letters, withnonsymmetric i.i.d or Markovian sources,Pittel (1985) gets asymptotic results on the insertion depth

and on the height. Peggy Cenac Digital Search Trees and Chaos Game Representation




22/35

p ( )Digital Search Tree (DST) and CGR



Theorem (Pittel, 1985)

Let us denoten (resp. Ln) the length of the shortest (resp.longest) branches, then we have :

n

ln n

a.s.n

1

h+

, and Ln

ln n

a.s.n

1

h

.

Moreover, in probability :

Dnln n

Pn

1

h,

h+, h and h are some constants depending on the distribution ofthe source.

In the CGR-tree, the successive inserted words are stronglydependentfrom each other.





23/35



The overlapping structure

The main difficulty is thestrong dependencybetween thewords inserted in the CGR-tree, due to theiroverlapping

structure.

We need classical results on the distribution of wordoccurences in a random sequences. Generating functions

Markov chains embedding methods Martingale approach (Penney game)



( )

Assumptions and notationsAsymptotic results


24/35


Asymptotic resultsNumerical experimentsGuidelines for the proofs

Main Results



Di i l S h T (DST) d CGR



25/35


Asymptotic resultsNumerical experimentsGuidelines for the proofs

Assumptions and notations (1)

U=U1. . . Un is supposed to be aMarkov chainof order 1,with transition matrix Qt and invariant measure as initial

distribution.

Let us denote s(j) def=s1. . . sj, where sidenotes the i

th letter ofthe infinite sequence s.

p(s(j)) can be defined as p(s(j)) def= P(U1 =sj, . . . , Uj=s1).



Di it l S h T (DST) d CGR



26/35


y pNumerical experimentsGuidelines for the proofs


We define the constants

h+def= lim

n+

1

nmax

ln

1

p

s(n)

: p

s(n)

>0

,

hdef

= limn+

1

nmin

ln 1

p

s(n)

:p

s(n)

>0

,

h def

= limn+

1

nE

ln 1

p

s(n).

Due to an argument ofsub-additivity, these limits are well

defined. Moreover, Pittel shows that there exists two infinitesequences denoted here by s+ and s such that

h+= limn

1

nln

1

ps(n)+

, and h= lim

n

1

nln

1

ps(n)

.






27/35


y pNumerical experimentsGuidelines for the proofs


Tjdef=Tj(w) : the finite tree with jnodes (without counting

the root), built from the jfirst sequences W(1), . . . , W(j),which are thesuccessive suffixes of the reversed sequence U

n.

n (resp. Ln) denotes thelengthof theshortest path(resp.thelongest path) from the root to a feasible external node ofthe tree Tn1(w).

Dn denotes theinsertion depthofW(n) in Tn1 to build Tn.

Mn is the length of a path ofTn, randomly and uniformlychosen in the n possible paths.






28/35


Numerical experimentsGuidelines for the proofs

Asymptotic results

Theorem

For a CGR-tree built on a markovian sequence U of order1,

n

ln n

a.s.n

1

h+

and Ln

ln n

a.s.n

1

hDnln n

Pn

1

h and lim

n

Mnln n

Pn

1

h.

Remark

For an i.i.d. sequence U, in the case when the random variables Ui are

not equiprobable, Dnln n does not converge a.s. since

lim supn

Dnln n

1

h >

1

h+= lim inf

n

Dnln n

.




Assumptions and notationsAsymptotic resultsN


29/35



Numerical experiments




Assumptions and notationsAsymptotic resultsN i l i


30/35






Assumptions and notationsAsymptotic resultsN i l i t


31/35

g ( )Main ResultsPerspectives





Assumptions and notationsAsymptotic resultsNumerical experiments


32/35

g ( )Main ResultsPerspectives


Guidelines for the proofs

We define

for adeterministicinfinite sequence s, the random variable

Xj(s) def=

0 ifs1 is not in Tjmax{k: the word s(k) is already inserted in Tj}

Tk(s) def= min{j :Xj(s) =k}.

Xj(s) and Tk(s) are induality: {Xj(s)k}={Tk(s) j}






33/35



Lemma

Let s be such that

limn+

1

nln

1

p

s(n)

= h(s)>0.

Then we have Xn(s)

ln na.s.

n

1

h(s).

Corollary

Xn(v)

ln na.s.

n

1

ln 1p

,

where p def= P(Ui=v).






34/35



We decompose

Tk(s) =

kr=1

(Tr(s) Tr1(s)) def

=

kr=1

Zr(s)

The random variables (Zr(s))r areindependent.The proofs are based on the generating functions ofZr(s).


PlanChaos Game Representation (CGR)Digital Search Tree (DST) and CGR

M i R lt


35/35


Perspectives

Second order in the asymptotic behaviour

Convergence in L1

Central Limit Theorem


Chaos Game Representationppt

Documents

Transcript of Chaos Game Representationppt