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t tt s Prr P rst Prs ts

slide-1
SLIDE 1

❉◆❆ ❡✈♦❧✉t✐♦♥✱ ❆✉t♦♠❛t❛ ❛♥❞ ❈❧✉♠♣s

P✐❡rr❡ ◆✐❝♦❞è♠❡ ▲■P◆ ❚❡❛♠ ❈❆▲■◆✱ ❯♥✐✈❡rs✐t② P❛r✐s ✶✸✱ ❱✐❧❧❡t❛♥❡✉s❡

slide-2
SLIDE 2

Pr♦❜❧❡♠ s❡tt✐♥❣

◮ ❆❧♣❤❛❜❡t A = {A, C, G, T}

✭❉◆❆✮ t✐♠❡ = 0 Sn(0) = ❨❨❨❨❨❨❨✳✳✳✳✳✳❨❨❨❨❨❨❨❨ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ t✐♠❡ = T Sn(T) = ❨❨❨❨✳✳✳❋❋✳✳❋❋✳✳❨❨❨❨❨❨

◮ n = ❧❡♥❣t❤ ♦❢ r❛♥❞♦♠ s❡q✉❡♥❝❡s Sn(0), . . . , Sn(T) (n ≈ 2000) ◮ b = ❋❋✳✳❋❋ ∈ Ak ❚r❛♥s❝r✐♣t✐♦♥ ❋❛❝t♦r (5 ≤ k = |b| ≤ 10)

◮ b ❞♦❡s ♥♦t ♦❝❝✉r ✐♥ Sn(0) ◮ b ♦❝❝✉rs ❢♦r t❤❡ ✜rst t✐♠❡ ❜② ❡✈♦❧✉t✐♦♥ ❛t t✐♠❡ T ✐♥ ❛

s❡q✉❡♥❝❡ ❡✈♦❧✈✐♥❣ ❢r♦♠ Sn(0)

❆✐♠✿ ❈♦♠♣✉t❡ T

▼❛② ✸✵✱ ✷✵✶✸

slide-3
SLIDE 3

■♥✐t✐❛❧ ν(α) ❛♥❞ ❙✉❜st✐t✉t✐♦♥ Pr♦❜❛❜✐❧✐t✐❡s πα→β

α ν(α) ❆ ✵✳✷✸✽✽✾ ❈ ✵✳✷✻✷✹✷

  • ✵✳✷✺✽✻✺

❚ ✵✳✷✹✵✵✹

  • s✉❜st✐t✉t✐♦♥ ♣r♦❜✳

P(1) = πα→β ❢♦r ♦♥❡ ❣❡♥❡r❛t✐♦♥ ✭20 ②❡❛rs✮ ❆

  • ❆

0.9999999763 ❆

  • ❈

4.54999994943 × 10−9 ❆

  • 1.57499995613 × 10−8

❆

  • ❚

3.40000001733 × 10−9 ❈

  • ❆

6.14999993408 × 10−9 ❈

  • ❈

0.99999996495 ❈

  • 7.14999984731 × 10−9

❈

  • ❚

2.17499993935 × 10−8

  • ❆

2.17499993935 × 10−8

  • ❈

7.14999984731 × 10−9

  • 0.99999996495
  • ❚

6.14999993408 × 10−9 ❚

  • ❆

3.40000001733 × 10−9 ❚

  • ❈

1.57499995613 × 10−8 ❚

  • 4.54999994943 × 10−9

❚

  • ❚

0.9999999763

▼❛② ✸✵✱ ✷✵✶✸

slide-4
SLIDE 4

P♦✇❡rs ♦❢ P(1) r❡♠❛✐♥s ❝❧♦s❡ t♦ t❤❡ ■❞❡♥t✐t② ▼❛tr✐①

P(1) ≈     1 − 3m m m m m 1 − 3m m m m m m 1 − 3m m m m 1 − 3m     ✇✐t❤ m ≈ 10−8 PN(1) ≈     1 − 3mN mN mN mN mN 1 − 3mN mN mN mN mN mN 1 − 3mN mN mN mN 1 − 3mN    +O(m2N) ❚❤❡r❡❢♦r❡ P N(1) × ν ≈ ν ❢♦r N ≈ 106 ❛♥❞ N < 106 P ∞(1) × ν = (0.25, 0.25, 0.25, 0.25)t

▼❛② ✸✵✱ ✷✵✶✸

slide-5
SLIDE 5
  • ❡♦♠❡tr✐❝ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ❲❛✐t✐♥❣ ❚✐♠❡

❇② st❛t✐♦♥♥❛r✐t② ♦❢ ν✱ ❛ss✉♠✐♥❣ T ∈ N P

  • ♥♦ b ✐♥ Sn(j + 1) | ♥♦ b ✐♥ Sn(j)
  • = P
  • ♥♦ b ✐♥ Sn(1) | ♥♦ b ✐♥ Sn(0)
  • = 1 − P
  • b ♦❝❝✉rs ✐♥ Sn(1) | ♥♦ b ✐♥ Sn(0)
  • ♦❝❝✉rs ✐♥

♥♦ ✐♥ ♦❝❝✉rs ✐♥ ♥♦ ✐♥ ❙❡tt✐♥❣ ♦❝❝✉rs ✐♥ ♥♦ ✐♥ ✱

▼❛② ✸✵✱ ✷✵✶✸

slide-6
SLIDE 6
  • ❡♦♠❡tr✐❝ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ❲❛✐t✐♥❣ ❚✐♠❡

❇② st❛t✐♦♥♥❛r✐t② ♦❢ ν✱ ❛ss✉♠✐♥❣ T ∈ N P

  • ♥♦ b ✐♥ Sn(j + 1) | ♥♦ b ✐♥ Sn(j)
  • = P
  • ♥♦ b ✐♥ Sn(1) | ♥♦ b ✐♥ Sn(0)
  • = 1 − P
  • b ♦❝❝✉rs ✐♥ Sn(1) | ♥♦ b ✐♥ Sn(0)
  • P
  • b ♦❝❝✉rs ✐♥ Sn(T) | ♥♦ b ✐♥ Sn(T − 1)
  • = P
  • b ♦❝❝✉rs ✐♥ Sn(1) | ♥♦ b ✐♥ Sn(0)
  • ❙❡tt✐♥❣

♦❝❝✉rs ✐♥ ♥♦ ✐♥ ✱

▼❛② ✸✵✱ ✷✵✶✸

slide-7
SLIDE 7
  • ❡♦♠❡tr✐❝ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ❲❛✐t✐♥❣ ❚✐♠❡

❇② st❛t✐♦♥♥❛r✐t② ♦❢ ν✱ ❛ss✉♠✐♥❣ T ∈ N P

  • ♥♦ b ✐♥ Sn(j + 1) | ♥♦ b ✐♥ Sn(j)
  • = P
  • ♥♦ b ✐♥ Sn(1) | ♥♦ b ✐♥ Sn(0)
  • = 1 − P
  • b ♦❝❝✉rs ✐♥ Sn(1) | ♥♦ b ✐♥ Sn(0)
  • P
  • b ♦❝❝✉rs ✐♥ Sn(T) | ♥♦ b ✐♥ Sn(T − 1)
  • = P
  • b ♦❝❝✉rs ✐♥ Sn(1) | ♥♦ b ✐♥ Sn(0)
  • ❙❡tt✐♥❣ pn = P
  • b ♦❝❝✉rs ✐♥ Sn(1) | ♥♦ b ✐♥ Sn(0)
  • ✱

E(T) ≈ i

  • i≥0

(1 − pn)i × pn = 1 pn

▼❛② ✸✵✱ ✷✵✶✸

slide-8
SLIDE 8

❘❡♥❡✇✐♥❣ t❤❡ ❆✐♠ ❲❡ ♥❡❡❞ ♥♦✇ ❝♦♠♣✉t✐♥❣

pn = P

  • b ♦❝❝✉rs ✐♥ Sn(1) | ♥♦ b ✐♥ Sn(0)
  • = P
  • b ♦❝❝✉rs ✐♥ Sn(1) ❆◆❉ ♥♦ b ✐♥ Sn(0)
  • P(no b ✐♥ Sn(0))

▼❛② ✸✵✱ ✷✵✶✸

slide-9
SLIDE 9

❉✐✛❡r❡♥t ❝♦♠♣✉t❛t✐♦♥s ♦❢ pn

✶✳ ❇❡❤r❡♥s✲❱✐♥❣r♦♥ ✭✷✵✶✵✮

◮ ❆♣♣r♦❛❝❤ ♥❡❣❧❡❝t✐♥❣ ✇♦r❞s ❝♦rr❡❧❛t✐♦♥✳ ◮ ❊✣❝✐❡♥t ❝♦♠♣✉t❛t✐♦♥ ♦❢ pn ✇✐t❤ r❡s♣❡❝t t♦ t❤✐s ❛ss✉♠♣t✐♦♥✳

✷✳ ❇❡❤r❡♥s✲◆✐❝❛✉❞✲◆ ✭✷✵✶✷✮

◮ ❘✐❣♦r♦✉s ❛♥❞ ❡✣❝✐❡♥t ❛♣♣r♦❛❝❤ ❜② ❛✉t♦♠❛t❛✳

✸✳ ◆ ✭◆❈▼❆✷✵✶✷✮

◮ ❍❡✉r✐st✐❝ ❛♣♣r♦❛❝❤ ❜② ❝❧✉♠♣ ❛♥❛❧②s✐s✱ ❡✐t❤❡r ❜②

❝♦♠❜✐♥❛t♦r✐❝s ♦❢ ✇♦r❞s ♦r ❜② ❛✉t♦♠❛t❛ ❛♥❞ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥s✳

✹✳ ◆ ✭✷✵✶✸✮

◮ ❍❡✉r✐st✐❝ ❛♣♣r♦❛❝❤✱ ❛❞❛♣t❛t✐♦♥ ♦❢ t❤❡ ❘é❣♥✐❡r✲❙③♣❛♥❦♦✇s❦✐

❡q✉❛t✐♦♥s ❛♥❞ ❡①♣❧✐❝✐t ❢♦r♠✉❧❛ ❛♣♣r♦①✐♠❛t✐♥❣ pn

▼❛② ✸✵✱ ✷✵✶✸

slide-10
SLIDE 10

❉✐✛❡r❡♥t ❝♦♠♣✉t❛t✐♦♥s ♦❢ pn

t✐♠❡ = 0 Sn(0) = ❨❨❨❨❨❨❨✳✳✳✳✳✳❨❨❨❨❨❨❨❨ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ t✐♠❡ = 1 Sn(1) = ❨❨❨❨✳✳✳❋❋✳✳❋❋✳✳❨❨❨❨❨❨

◮ ❇❡❤r❡♥s✲❱✐♥❣r♦♥ ❝♦♠♣✉t❡ t❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t b ♦❝❝✉rs ✐♥

Sn(1) ✭✇✐t❤♦✉t ❛❧❧♦✇✐♥❣ ♦✈❡r❧❛♣s ♦❢ ♦❝❝✉rr❡♥❝❡s✮✱ ❛♥❞ t❤❡♥ t❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t Sn(0) ❡✈♦❧✈❡s t♦ Sn(1) ❇❡❤r❡♥s✲◆✐❝❛✉❞✲◆ ✉s❡ ❛♥ ❛✉t♦♠❛t♦♥ ♦♥ t❤❡ ❛❧♣❤❛❜❡t t❤❛t s❝❛♥s s✐♠✉❧t❛♥❡♦✉s❧② ❛♥❞ ✳ ❚❤✐s ❛✉t♦♠❛t♦♥ ✐s ❛ ❦✐♥❞ ♦❢ ♣r♦❞✉❝t ♦❢ t✇♦ ❑♥✉t❤✲▼♦rr✐s✲Pr❛tt ❛✉t♦♠❛t❛✳ ◆ ✭✷✵✶✷✮ ❛ss✉♠❡s t❤❛t ❛ s✐♥❣❧❡ ♠✉t❛t✐♦♥ ♦❝❝✉rr❡❞ ❛♥❞ ❝♦♥s✐❞❡rs t❤❡ ❝❧✉♠♣s ♦❢ ♥❡✐❣❤❜♦rs ♦❢ ❛t ❞✐st❛♥❝❡ ✐♥ ✳

▼❛② ✸✵✱ ✷✵✶✸

slide-11
SLIDE 11

❉✐✛❡r❡♥t ❝♦♠♣✉t❛t✐♦♥s ♦❢ pn

t✐♠❡ = 0 Sn(0) = ❨❨❨❨❨❨❨✳✳✳✳✳✳❨❨❨❨❨❨❨❨ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ t✐♠❡ = 1 Sn(1) = ❨❨❨❨✳✳✳❋❋✳✳❋❋✳✳❨❨❨❨❨❨

◮ ❇❡❤r❡♥s✲❱✐♥❣r♦♥ ❝♦♠♣✉t❡ t❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t b ♦❝❝✉rs ✐♥

Sn(1) ✭✇✐t❤♦✉t ❛❧❧♦✇✐♥❣ ♦✈❡r❧❛♣s ♦❢ ♦❝❝✉rr❡♥❝❡s✮✱ ❛♥❞ t❤❡♥ t❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t Sn(0) ❡✈♦❧✈❡s t♦ Sn(1)

◮ ❇❡❤r❡♥s✲◆✐❝❛✉❞✲◆ ✉s❡ ❛♥ ❛✉t♦♠❛t♦♥ ♦♥ t❤❡ ❛❧♣❤❛❜❡t A × A

t❤❛t s❝❛♥s s✐♠✉❧t❛♥❡♦✉s❧② Sn(0) ❛♥❞ Sn(1)✳ ❚❤✐s ❛✉t♦♠❛t♦♥ ✐s ❛ ❦✐♥❞ ♦❢ ♣r♦❞✉❝t ♦❢ t✇♦ ❑♥✉t❤✲▼♦rr✐s✲Pr❛tt ❛✉t♦♠❛t❛✳ ◆ ✭✷✵✶✷✮ ❛ss✉♠❡s t❤❛t ❛ s✐♥❣❧❡ ♠✉t❛t✐♦♥ ♦❝❝✉rr❡❞ ❛♥❞ ❝♦♥s✐❞❡rs t❤❡ ❝❧✉♠♣s ♦❢ ♥❡✐❣❤❜♦rs ♦❢ ❛t ❞✐st❛♥❝❡ ✐♥ ✳

▼❛② ✸✵✱ ✷✵✶✸

slide-12
SLIDE 12

❉✐✛❡r❡♥t ❝♦♠♣✉t❛t✐♦♥s ♦❢ pn

t✐♠❡ = 0 Sn(0) = ❨❨❨❨❨❨❨✳✳✳✳✳✳❨❨❨❨❨❨❨❨ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ t✐♠❡ = 1 Sn(1) = ❨❨❨❨✳✳✳❋❋✳✳❋❋✳✳❨❨❨❨❨❨

◮ ❇❡❤r❡♥s✲❱✐♥❣r♦♥ ❝♦♠♣✉t❡ t❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t b ♦❝❝✉rs ✐♥

Sn(1) ✭✇✐t❤♦✉t ❛❧❧♦✇✐♥❣ ♦✈❡r❧❛♣s ♦❢ ♦❝❝✉rr❡♥❝❡s✮✱ ❛♥❞ t❤❡♥ t❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t Sn(0) ❡✈♦❧✈❡s t♦ Sn(1)

◮ ❇❡❤r❡♥s✲◆✐❝❛✉❞✲◆ ✉s❡ ❛♥ ❛✉t♦♠❛t♦♥ ♦♥ t❤❡ ❛❧♣❤❛❜❡t A × A

t❤❛t s❝❛♥s s✐♠✉❧t❛♥❡♦✉s❧② Sn(0) ❛♥❞ Sn(1)✳ ❚❤✐s ❛✉t♦♠❛t♦♥ ✐s ❛ ❦✐♥❞ ♦❢ ♣r♦❞✉❝t ♦❢ t✇♦ ❑♥✉t❤✲▼♦rr✐s✲Pr❛tt ❛✉t♦♠❛t❛✳

◮ ◆ ✭✷✵✶✷✮ ❛ss✉♠❡s t❤❛t ❛ s✐♥❣❧❡ ♠✉t❛t✐♦♥ ♦❝❝✉rr❡❞ ❛♥❞

❝♦♥s✐❞❡rs t❤❡ ❝❧✉♠♣s ♦❢ ♥❡✐❣❤❜♦rs ♦❢ b ❛t ❞✐st❛♥❝❡ 1 ✐♥ Sn(0)✳

▼❛② ✸✵✱ ✷✵✶✸

slide-13
SLIDE 13

❆♥ ✉♥❡①♣❡❝t❡❞ ❜❡❤❛✈✐♦✉r

n n pn π × P(♥♦ b ✭♦r b′) ✐♥ Sn(0)) P(♥♦ b ✭♦r b′) ✐♥ Sn(0))

b = ❆❈❆❈ b′ = ❆❆❈❈, ν(❆) = ν(❈) = 1 2, π = πA→C = πC→A r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥ ♥♦ ✐♥ ♥♦ ✐♥

▼❛② ✸✵✱ ✷✵✶✸

slide-14
SLIDE 14

❆♥ ✉♥❡①♣❡❝t❡❞ ❜❡❤❛✈✐♦✉r

n n pn π × P(♥♦ b ✭♦r b′) ✐♥ Sn(0)) P(♥♦ b ✭♦r b′) ✐♥ Sn(0))

b = ❆❈❆❈ b′ = ❆❆❈❈, ν(❆) = ν(❈) = 1 2, π = πA→C = πC→A

pn π n

r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥ ♥♦ ✐♥ ♥♦ ✐♥

▼❛② ✸✵✱ ✷✵✶✸

slide-15
SLIDE 15

❆♥ ✉♥❡①♣❡❝t❡❞ ❜❡❤❛✈✐♦✉r

n n pn π × P(♥♦ b ✭♦r b′) ✐♥ Sn(0)) P(♥♦ b ✭♦r b′) ✐♥ Sn(0))

b = ❆❈❆❈ b′ = ❆❆❈❈, ν(❆) = ν(❈) = 1 2, π = πA→C = πC→A

pn π n

F(z, t) r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥ P

  • b ∈ Sn(1) | b ∈ Sn(0)
  • = pn × P(♥♦ b ✐♥ Sn(0)) = [zn] ∂F(z, t)

∂t

  • t=1

P(♥♦ b ✐♥ Sn(0)) = [zn]F(z, 1)

▼❛② ✸✵✱ ✷✵✶✸

slide-16
SLIDE 16

❋♦r♠❛❧ ▲❛♥❣✉❛❣❡s ❆♣♣r♦❛❝❤ ✲ ✭◆ ✷✵✶✸✮

✭❆ss✉♠✐♥❣ ❛ s✐♥❣❧❡ ♠✉t❛t✐♦♥✮

b = AAAAA

S(0) = XXXX...XXXAAAAXAAAAXXXX..........XXX S(1) = XXXX...XXXAAAAAAAAAXXXX..........XXX = XX... − s❤♦rt ❝❧✉♠♣ ♦❢ AAAAA − ...XXX

◮ ❧❡♥❣t❤ ♦❢ s❤♦rt ❝❧✉♠♣ ♦❢ b ✐♥ S(1) ♠✉st ❜❡ ❧❡ss t❤❛♥

2 × |b| − 1✱

◮ ❡❧s❡ t❤❡r❡ ✐s ❛t ❧❡❛st ♦♥❡ ♦❝❝✉rr❡♥❝❡ ♦❢ b ✐♥ S(0) ◮ ♥♦ ♦❝❝✉rr❡♥❝❡s ♦❢ b ✐♥ t❤❡ XXX...XXX

✐❢ b ✇✐t❤♦✉t s❡❧❢✲♦✈❡r❧❛♣✱ s❤♦rt ❝❧✉♠♣❂b

▼❛② ✸✵✱ ✷✵✶✸

slide-17
SLIDE 17
  • ✉✐❜❛s✲❖❞❧②③❦♦ ❞❡❝♦♠♣♦s✐t✐♦♥ ✲ ♦❝❝✉rr❡♥❝❡s ♦❢ ❛ ✇♦r❞ b

b = ❛❝❛❝❛ ❛❝❛❝❛ ❛❝❛❝❛ ❛❝❛❝❛ ❛❝❛❝❛ ❛❝❛❝❛ ∈ R ∈ bM ∈ bM ∈ b U

❘✐❣❤t ✿ ❡t ▼✐♥✐♠❛❧ ✿ ❡t ❯❧t✐♠❛t❡ ✿ ❩❡r♦

▼❛② ✸✵✱ ✷✵✶✸

slide-18
SLIDE 18
  • ✉✐❜❛s✲❖❞❧②③❦♦ ❞❡❝♦♠♣♦s✐t✐♦♥ ✲ ♦❝❝✉rr❡♥❝❡s ♦❢ ❛ ✇♦r❞ b

b = ❛❝❛❝❛ ❛❝❛❝❛ ❛❝❛❝❛ ❛❝❛❝❛ ❛❝❛❝❛ ❛❝❛❝❛ ∈ R ∈ bM ∈ bM ∈ b U

◮ ❘✐❣❤t R✿ = { w = u.b ❡t ∃r, s, w = r.b.s } aaaaaacaca ⊂ R, cccccacacaca ⊂ R ▼✐♥✐♠❛❧ ✿ ❡t ❯❧t✐♠❛t❡ ✿ ❩❡r♦

▼❛② ✸✵✱ ✷✵✶✸

slide-19
SLIDE 19
  • ✉✐❜❛s✲❖❞❧②③❦♦ ❞❡❝♦♠♣♦s✐t✐♦♥ ✲ ♦❝❝✉rr❡♥❝❡s ♦❢ ❛ ✇♦r❞ b

b = ❛❝❛❝❛ ❛❝❛❝❛ ❛❝❛❝❛ ❛❝❛❝❛ ❛❝❛❝❛ ❛❝❛❝❛ ∈ R ∈ bM ∈ bM ∈ b U

◮ ❘✐❣❤t R✿ = { w = u.b ❡t ∃r, s, w = r.b.s } aaaaaacaca ⊂ R, cccccacacaca ⊂ R ◮ ▼✐♥✐♠❛❧ M✿ = { w, b.w = u.b ❡t ∃r, s, b.w = r.b.s } acaca ccaca ccaca aaaaacaca ⊂ M caccccccccacaca ⊂ M ca ⊂ M ❯❧t✐♠❛t❡ ✿ ❩❡r♦

▼❛② ✸✵✱ ✷✵✶✸

slide-20
SLIDE 20
  • ✉✐❜❛s✲❖❞❧②③❦♦ ❞❡❝♦♠♣♦s✐t✐♦♥ ✲ ♦❝❝✉rr❡♥❝❡s ♦❢ ❛ ✇♦r❞ b

b = ❛❝❛❝❛ ❛❝❛❝❛ ❛❝❛❝❛ ❛❝❛❝❛ ❛❝❛❝❛ ❛❝❛❝❛ ∈ R ∈ bM ∈ bM ∈ b U

◮ ❘✐❣❤t R✿ = { w = u.b ❡t ∃r, s, w = r.b.s } aaaaaacaca ⊂ R, cccccacacaca ⊂ R ◮ ▼✐♥✐♠❛❧ M✿ = { w, b.w = u.b ❡t ∃r, s, b.w = r.b.s } acaca ccaca ccaca aaaaacaca ⊂ M caccccccccacaca ⊂ M ca ⊂ M ◮ ❯❧t✐♠❛t❡ U✿ = {w, ∃r, s, b.w = r.b.s} acaca ccaca aacccaccccccc ⊂ U caccccccc ⊂ U ❩❡r♦

▼❛② ✸✵✱ ✷✵✶✸

slide-21
SLIDE 21
  • ✉✐❜❛s✲❖❞❧②③❦♦ ❞❡❝♦♠♣♦s✐t✐♦♥ ✲ ♦❝❝✉rr❡♥❝❡s ♦❢ ❛ ✇♦r❞ b

b = ❛❝❛❝❛ ❛❝❛❝❛ ❛❝❛❝❛ ❛❝❛❝❛ ❛❝❛❝❛ ❛❝❛❝❛ ∈ R ∈ bM ∈ bM ∈ b U

◮ ❘✐❣❤t R✿ = { w = u.b ❡t ∃r, s, w = r.b.s } aaaaaacaca ⊂ R, cccccacacaca ⊂ R ◮ ▼✐♥✐♠❛❧ M✿ = { w, b.w = u.b ❡t ∃r, s, b.w = r.b.s } acaca ccaca ccaca aaaaacaca ⊂ M caccccccccacaca ⊂ M ca ⊂ M ◮ ❯❧t✐♠❛t❡ U✿ = {w, ∃r, s, b.w = r.b.s} acaca ccaca aacccaccccccc ⊂ U caccccccc ⊂ U ◮ ❩❡r♦ Z := A⋆ − A⋆.b.A⋆ = {w, ∃r, s, w = r.b.s}

▼❛② ✸✵✱ ✷✵✶✸

slide-22
SLIDE 22

❘é❣♥✐❡r✲❙③♣❛♥❦♦✇s❦✐ ❊q✉❛t✐♦♥s ✭s❡❡ ▲♦t❤❛✐r❡✮

◮ A⋆ = U + MA⋆ ◮ A⋆b = R.C + R.A⋆.b ◮ M+ = A⋆.b + C − ǫ ◮ Z.σ = R + Z − ǫ

  • ❡♥❡r❛t✐♥❣ ❋✉♥❝t✐♦♥s ♦❢ t❤❡ ▲❛♥❣✉❛❣❡s

R(z) = P(b)z|b| D(z) , M(z) = 1 − 1 − z D(z) , U(z) = 1 D(z), Z(z) = C(z) D(z),

  • ✇✐t❤ D(z) = (1−z)C(z)+P(b)z|b|,

C ❛✉t♦❝♦rr❡❧❛t✐♦♥ s❡t ♦❢ t❤❡ ✇♦r❞ b C = {w; b.w = u.b, 0 ≤ |w| < |b|} C(z) =

  • w∈C

P(w)z|w|

▼❛② ✸✵✱ ✷✵✶✸

slide-23
SLIDE 23

❲❤❛t ❞♦ ✇❡ ♥❡❡❞ ✐♥ Sn(1)❄

b = ACACA ❆❈❆❈❆❈❆❈❆ ∈ R ∈ C ∈ U ❇✉t ♥♦t ❛♥② ♣♦s✐t✐♦♥ ♦❢ t❤❡ ❝❧✉♠♣ ❝❛♥ ♠✉t❛t❡ ACACACACA NNNNYNNNN ACACACA NNYYYNN ACACA YYYYY

◮ t♦ ❛✈♦✐❞ ❛♥ ♦❝❝✉rr❡♥❝❡ ♦❢ b ✐♥ S(0) ◮ ✐❢ t❤❡ s❤♦rt ❝❧✉♠♣ ✐s b.c ✇✐t❤ c ∈ C ◮ ♦♥❧② t = |b| − |c| ♣♦s✐t✐♦♥s ❝❛♥ ♠✉t❛t❡ ◮ t❤❡s❡ ♣♦s✐t✐♦♥s ❛r❡ t❤❡ t ❧❛st ♣♦s✐t✐♦♥s ♦❢ b

▼❛② ✸✵✱ ✷✵✶✸

slide-24
SLIDE 24

❚❤❡ r✐❣❤t ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥

b = ACACA ❆❈❆❈❆❈❆❈❆ ∈ R ∈ C ∈ U

  • ❡♥✳❋✉♥✳

♦❢ s❡q✉❡♥❝❡s ✇✐t❤ ♦♥❡ s❤♦rt ❝❧✉♠♣

  • ❡♥✳❋✉♥

♦❢ s❡q✉❡♥❝❡s ✇✐t❤ ♥♦ ♦❝❝✉rr❡♥❝❡s ♦❢ ❛♥❞ ❤❛✈❡ t❤❡ s❛♠❡ ❞♦♠✐♥❛♥t s✐♥❣✉❧❛r✐t②

▼❛② ✸✵✱ ✷✵✶✸

slide-25
SLIDE 25

❚❤❡ r✐❣❤t ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥

b = ACACA ❆❈❆❈❆❈❆❈❆ ∈ R ∈ C ∈ U

◮ ●❡♥✳❋✉♥✳ F(z) ♦❢ s❡q✉❡♥❝❡s ✇✐t❤ ♦♥❡ s❤♦rt ❝❧✉♠♣

F(z) = R(z) ×

  • c∈C

P(c)z|c| × U(z) =

  • c∈C

P(b.c)z|b.c| D2(z)

  • ❡♥✳❋✉♥

♦❢ s❡q✉❡♥❝❡s ✇✐t❤ ♥♦ ♦❝❝✉rr❡♥❝❡s ♦❢ ❛♥❞ ❤❛✈❡ t❤❡ s❛♠❡ ❞♦♠✐♥❛♥t s✐♥❣✉❧❛r✐t②

▼❛② ✸✵✱ ✷✵✶✸

slide-26
SLIDE 26

❚❤❡ r✐❣❤t ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥

b = ACACA ❆❈❆❈❆❈❆❈❆ ∈ R ∈ C ∈ U

◮ ●❡♥✳❋✉♥✳ F(z) ♦❢ s❡q✉❡♥❝❡s ✇✐t❤ ♦♥❡ s❤♦rt ❝❧✉♠♣

F(z) = R(z) ×

  • c∈C

P(c)z|c| × U(z) =

  • c∈C

P(b.c)z|b.c| D2(z)

◮ ●❡♥✳❋✉♥ Z(z) ♦❢ s❡q✉❡♥❝❡s ✇✐t❤ ♥♦ ♦❝❝✉rr❡♥❝❡s ♦❢ b

Z(z) = C(z) D(z) ❛♥❞ ❤❛✈❡ t❤❡ s❛♠❡ ❞♦♠✐♥❛♥t s✐♥❣✉❧❛r✐t②

▼❛② ✸✵✱ ✷✵✶✸

slide-27
SLIDE 27

❚❤❡ r✐❣❤t ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥

b = ACACA ❆❈❆❈❆❈❆❈❆ ∈ R ∈ C ∈ U

◮ ●❡♥✳❋✉♥✳ F(z) ♦❢ s❡q✉❡♥❝❡s ✇✐t❤ ♦♥❡ s❤♦rt ❝❧✉♠♣

F(z) = R(z) ×

  • c∈C

P(c)z|c| × U(z) =

  • c∈C

P(b.c)z|b.c| D2(z)

◮ ●❡♥✳❋✉♥ Z(z) ♦❢ s❡q✉❡♥❝❡s ✇✐t❤ ♥♦ ♦❝❝✉rr❡♥❝❡s ♦❢ b

Z(z) = C(z) D(z)

◮ D(z) = (1 − z)C(z) + P(b)z|b| ◮ F(z) ❛♥❞ Z(z) ❤❛✈❡ t❤❡ s❛♠❡ ❞♦♠✐♥❛♥t s✐♥❣✉❧❛r✐t② ω

▼❛② ✸✵✱ ✷✵✶✸

slide-28
SLIDE 28

❆s②♠♣t♦t✐❝s ♦❢ qn ✭❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ pn✮ qn = [zn]F(z) [z]Z(z) (P(ǫ) = 1) ω ❞♦♠✐♥❛♥t s✐♥❣✉❧❛r✐t② ♦❢ D(z)

qn = P(b) C(ω)D′(ω) ×

  • c∈C

(|b| − |c|)P(c)ω|b.c| ×

  • β∈{b[|c|+1|],...,b[|b|]}

α=β

P(α) P(β) × πα→β ×

  • (n − |b.c| + 1)ω−1 + D′′(ω)

D′(ω)

  • + o(P(b)).

▼❛② ✸✵✱ ✷✵✶✸

slide-29
SLIDE 29

❆♥ ❡✈❡♥ ♠♦r❡ ❛♣♣r♦①✐♠❛t❡❞ r❡s✉❧t D(z) = (1 − z)C(z) + P(b)z|b| ❜② ❜♦♦tstr❛♣♣✐♥❣ ω ≈ 1 + P(b) C(1) + |b|P(b) ≈ 1 ❯s✐♥❣ ω ≈ 1 ❣✐✈❡s q(approx)

n

= P(b) C2(1) ×

  • c∈C

(|b| − |c|)P(c)(n − |b.c| + 1) ×

  • β∈{b[|c|+1|],...,b[|b|]}

α=β

P(α) P(β) × πα→β

▼❛② ✸✵✱ ✷✵✶✸

slide-30
SLIDE 30

❚❤❡♦r❡♠❬◆ ✷✵✶✸❪✳ ❚❤❡ ❝♦♥❞✐t✐♦♥❡❞ ♣r♦❜❛❜✐❧✐t② pn t❤❛t ❛ r❛♥❞♦♠ s❡q✉❡♥❝❡ ♦❢ ❧❡♥❣t❤ n t❤❛t ❞♦❡s ♥♦t ❝♦♥t❛✐♥ ❛ k✲♠❡r b ❛t t✐♠❡ 0 ❡✈♦❧✈❡s ❛t t✐♠❡ 1 t♦ ❛ r❛♥❞♦♠ s❡q✉❡♥❝❡ t❤❛t ❝♦♥t❛✐♥s t❤✐s k✲♠❡r ✈❡r✐✜❡s pn = qn × (1 + O(nψ)) + O(n2ψ2) ✇❤❡r❡ qn = P(b) C(ω)D′(ω) ×

  • c∈C

(|b| − |c|)P(c)ω|b.c| ×

  • β∈{b[|c|+1|],...,b[|b|]}

α=β

P(α) P(β) × πα→β ×

  • (n − |b.c| + 1)ω−1 + D′′(ω)

D′(ω)

  • + o(P(b)).

ψ = maxα,β∈A;α=β pα→β minα∈A pα→α

▼❛② ✸✵✱ ✷✵✶✸

slide-31
SLIDE 31

◆✉♠❡r✐❝❛❧ ✈❛❧✐❞❛t✐♦♥

A = {A, C, G, T} ✲ ✉♥✐❢♦r♠ ❇❡r♥♦✉❧❧✐ ♠♦❞❡❧ ❢♦r S(0)✳

b = AAAAA ❛♥❞ ❢♦r α = β✱ pα→β = 10−8 ▲❡♥❣t❤ n pn × 106 hn × 106 qn × 106 q(approx)

n

× 106 10000 1.03335528 1.03335588 1.03335587 1.02703244 100000 10.3368481 10.3369021 10.3369021 10.2742439 10000000 1033.19278 1033.72699 1033.72698 1027.46750

◮ pn ✲ ❊①❛❝t r❡s✉❧t ❜② ❛✉t♦♠❛t❛ ✭❇❡❤r❡♥s✲◆✐❝❛✉❞✲◆ ✷✵✶✷✮ ◮ ❍❡✉r✐st✐❝ ♦❢ ❛ s✐♥❣❧❡ ♠✉t❛t✐♦♥

◮ hn ❝❧✉♠♣s ♦❢ ♥❡✐❣❤❜♦rs ❛t ❞✐st❛♥❝❡ 1 ♦❢ b ✐♥ Sn(0) ✭◆ ✷✵✶✷✮ ◮ qn✱ q(approx)

n

s❤♦rt ❝❧✉♠♣ ❛♣♣r♦❛❝❤ ♦♥ Sn(1) ✭◆ ✷✶✵✸✮

▼❛② ✸✵✱ ✷✵✶✸

slide-32
SLIDE 32

❇r✉♥♦ r❛✐s❡❞ ✈✐❣♦r♦✉s ❛♥❞ ❥✉st✐✜❡❞ ❝r✐t✐❝s t♦ ❛ ♣r❡✈✐♦✉s ✈❡rs✐♦♥ ♦❢ t❤✐s t❛❧❦✳ ■ ♠♦❞✐✜❡❞ t❤❡ t❛❧❦ ❝♦♥s❡q✉❡♥t❧② ❙♦✱ ✐❢ ②♦✉ ❞✐❞ ♥♦t ❧✐❦❡ ✐t✱ ♣❧❡❛s❡ ❝♦♠♣❧❛✐♥ t♦ ❇r✉♥♦

▼❛② ✸✵✱ ✷✵✶✸

slide-33
SLIDE 33

❇r✉♥♦ r❛✐s❡❞ ✈✐❣♦r♦✉s ❛♥❞ ❥✉st✐✜❡❞ ❝r✐t✐❝s t♦ ❛ ♣r❡✈✐♦✉s ✈❡rs✐♦♥ ♦❢ t❤✐s t❛❧❦✳ ■ ♠♦❞✐✜❡❞ t❤❡ t❛❧❦ ❝♦♥s❡q✉❡♥t❧② ❙♦✱ ✐❢ ②♦✉ ❞✐❞ ♥♦t ❧✐❦❡ ✐t✱ ♣❧❡❛s❡ ❝♦♠♣❧❛✐♥ t♦ ❇r✉♥♦

▼❛② ✸✵✱ ✷✵✶✸

slide-34
SLIDE 34

❇r✉♥♦ r❛✐s❡❞ ✈✐❣♦r♦✉s ❛♥❞ ❥✉st✐✜❡❞ ❝r✐t✐❝s t♦ ❛ ♣r❡✈✐♦✉s ✈❡rs✐♦♥ ♦❢ t❤✐s t❛❧❦✳ ■ ♠♦❞✐✜❡❞ t❤❡ t❛❧❦ ❝♦♥s❡q✉❡♥t❧② ❙♦✱ ✐❢ ②♦✉ ❞✐❞ ♥♦t ❧✐❦❡ ✐t✱ ♣❧❡❛s❡ ❝♦♠♣❧❛✐♥ t♦ ❇r✉♥♦

▼❛② ✸✵✱ ✷✵✶✸