A variant of the Wright-Fisher diffusion model coming from the theory of matrix-valued spherical functions
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The W igh -Fishe model A a ian o he W igh -Fishe model
A a ian o he W igh -Fishe
di usion model coming om he heo y
o ma ix- alued sphe ical unc ions
Manuel Dom´ınguez de la Iglesia
Depa amen o de An´alisis Ma em´a ico, Uni e sidad de Se illa
1s Join Con e ence o he Belgian, Royal Spanish and
Luxembou g Ma hema ical Socie ies
Li`ege, Belgium, June 7 h, 2012
The W igh -Fishe model A a ian o he W igh -Fishe model
Ou line
1The W igh -Fishe model
The o iginal p oblem
The di usion app oxima ion
Spec al me hods
2A a ian o he W igh -Fishe model
The coe icien s o he hyb id p ocess
P obabilis ic in e p e a ion
Spec al me hods
The W igh -Fishe model A a ian o he W igh -Fishe model
Ou line
1The W igh -Fishe model
The o iginal p oblem
The di usion app oxima ion
Spec al me hods
2A a ian o he W igh -Fishe model
The coe icien s o he hyb id p ocess
P obabilis ic in e p e a ion
Spec al me hods
The W igh -Fishe model A a ian o he W igh -Fishe model
The W igh -Fishe model
The W igh -Fishe model conside s a gene popula ion o cons an size M
composed o wo ypes Aand B. Call #A=i.
The nex gene a ion is de e mined by Mindependen binomial ials:
each ial esul s in Ao Bwi h p obabili ies
pi=i
M,qi= 1 −pi= 1 −i
M
The e o e we gene a e a disc e e- ime Ma ko chain {X(n)}whe e
X(n) = {#Ain he n- h gene a ion}
wi h s a e space S={0,1,...,M}and ansi ion p obabili y ma ix
P {X(n+ 1) = j|X(n) = i}=M
jpj
iqM−j
i
A mo e ealis ic model akes accoun o mu a ion p essu es
Aa
−→ B,Bb
−→ A,a,b>0
We ha e he same ansi ion p obabili y ma ix bu now
pi=i
M(1 −a) + 1−i
Mb,qi= 1 −pi
The W igh -Fishe model A a ian o he W igh -Fishe model
The W igh -Fishe model
The W igh -Fishe model conside s a gene popula ion o cons an size M
composed o wo ypes Aand B. Call #A=i.
The nex gene a ion is de e mined by Mindependen binomial ials:
each ial esul s in Ao Bwi h p obabili ies
pi=i
M,qi= 1 −pi= 1 −i
M
The e o e we gene a e a disc e e- ime Ma ko chain {X(n)}whe e
X(n) = {#Ain he n- h gene a ion}
wi h s a e space S={0,1,...,M}and ansi ion p obabili y ma ix
P {X(n+ 1) = j|X(n) = i}=M
jpj
iqM−j
i
A mo e ealis ic model akes accoun o mu a ion p essu es
Aa
−→ B,Bb
−→ A,a,b>0
We ha e he same ansi ion p obabili y ma ix bu now
pi=i
M(1 −a) + 1−i
Mb,qi= 1 −pi
The W igh -Fishe model A a ian o he W igh -Fishe model
The W igh -Fishe model
The W igh -Fishe model conside s a gene popula ion o cons an size M
composed o wo ypes Aand B. Call #A=i.
The nex gene a ion is de e mined by Mindependen binomial ials:
each ial esul s in Ao Bwi h p obabili ies
pi=i
M,qi= 1 −pi= 1 −i
M
The e o e we gene a e a disc e e- ime Ma ko chain {X(n)}whe e
X(n) = {#Ain he n- h gene a ion}
wi h s a e space S={0,1,...,M}and ansi ion p obabili y ma ix
P {X(n+ 1) = j|X(n) = i}=M
jpj
iqM−j
i
A mo e ealis ic model akes accoun o mu a ion p essu es
Aa
−→ B,Bb
−→ A,a,b>0
We ha e he same ansi ion p obabili y ma ix bu now
pi=i
M(1 −a) + 1−i
Mb,qi= 1 −pi
The W igh -Fishe model A a ian o he W igh -Fishe model
The W igh -Fishe model
The W igh -Fishe model conside s a gene popula ion o cons an size M
composed o wo ypes Aand B. Call #A=i.
The nex gene a ion is de e mined by Mindependen binomial ials:
each ial esul s in Ao Bwi h p obabili ies
pi=i
M,qi= 1 −pi= 1 −i
M
The e o e we gene a e a disc e e- ime Ma ko chain {X(n)}whe e
X(n) = {#Ain he n- h gene a ion}
wi h s a e space S={0,1,...,M}and ansi ion p obabili y ma ix
P {X(n+ 1) = j|X(n) = i}=M
jpj
iqM−j
i
A mo e ealis ic model akes accoun o mu a ion p essu es
Aa
−→ B,Bb
−→ A,a,b>0
We ha e he same ansi ion p obabili y ma ix bu now
pi=i
M(1 −a) + 1−i
Mb,qi= 1 −pi
The W igh -Fishe model A a ian o he W igh -Fishe model
The W igh -Fishe model
The W igh -Fishe model conside s a gene popula ion o cons an size M
composed o wo ypes Aand B. Call #A=i.
The nex gene a ion is de e mined by Mindependen binomial ials:
each ial esul s in Ao Bwi h p obabili ies
pi=i
M,qi= 1 −pi= 1 −i
M
The e o e we gene a e a disc e e- ime Ma ko chain {X(n)}whe e
X(n) = {#Ain he n- h gene a ion}
wi h s a e space S={0,1,...,M}and ansi ion p obabili y ma ix
P {X(n+ 1) = j|X(n) = i}=M
jpj
iqM−j
i
A mo e ealis ic model akes accoun o mu a ion p essu es
Aa
−→ B,Bb
−→ A,a,b>0
We ha e he same ansi ion p obabili y ma ix bu now
pi=i
M(1 −a) + 1−i
Mb,qi= 1 −pi
The W igh -Fishe model A a ian o he W igh -Fishe model
Example M= 15
A= b own eyes, B= blue eyes
bb
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
X(0) = 8
The W igh -Fishe model A a ian o he W igh -Fishe model
Example M= 15
A= b own eyes, B= blue eyes
bb
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
X(0) = 8
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
X(1) = 9
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
X(2) = 7
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
X(3) = 10
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
X(4) = 11
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
X(5) = 10
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
X(6) = 13
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
X(7) = 11
The W igh -Fishe model A a ian o he W igh -Fishe model
Example M= 15
A= b own eyes, B= blue eyes
bb
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
X(0) = 8
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
X(1) = 9
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
X(2) = 7
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
X(3) = 10
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
X(4) = 11
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
X(5) = 10
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
X(6) = 13
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
X(7) = 11
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
X(8) = 8
The W igh -Fishe model A a ian o he W igh -Fishe model
The di usion app oxima ion
Conside he p ocess
Y( ) = lim
M→∞
YM( ) = lim
M→∞
X([M ])
M
I we call h= 1/Mand x=i/M, we ha e ha
τ(x) = lim
h→0+
1
hE[YM( +h)−YM( )|YM( ) = x] = −γ1x+(1−x)γ2
σ2(x) = lim
h→0+
1
hEh(YM( +h)−YM( ))2|YM( ) = xi=x(1 −x)
whe e γ1=aM and γ2=bM a e he in ensi ies o mu a ion.
The e o e Y( ) is a con inuous- ime di usion p ocess wi h s a e
space S= [0,1], d i τ(x) and di usion coe icien σ2(x).
Y( ) = Y e ol es acco ding o he s ochas ic di e en ial equa ion
dY =τ(Y ) + σ(Y )dB
The W igh -Fishe model A a ian o he W igh -Fishe model
The di usion app oxima ion
Conside he p ocess
Y( ) = lim
M→∞
YM( ) = lim
M→∞
X([M ])
M
I we call h= 1/Mand x=i/M, we ha e ha
τ(x) = lim
h→0+
1
hE[YM( +h)−YM( )|YM( ) = x] = −γ1x+(1−x)γ2
σ2(x) = lim
h→0+
1
hEh(YM( +h)−YM( ))2|YM( ) = xi=x(1 −x)
whe e γ1=aM and γ2=bM a e he in ensi ies o mu a ion.
The e o e Y( ) is a con inuous- ime di usion p ocess wi h s a e
space S= [0,1], d i τ(x) and di usion coe icien σ2(x).
Y( ) = Y e ol es acco ding o he s ochas ic di e en ial equa ion
dY =τ(Y ) + σ(Y )dB
The W igh -Fishe model A a ian o he W igh -Fishe model
The di usion app oxima ion
Conside he p ocess
Y( ) = lim
M→∞
YM( ) = lim
M→∞
X([M ])
M
I we call h= 1/Mand x=i/M, we ha e ha
τ(x) = lim
h→0+
1
hE[YM( +h)−YM( )|YM( ) = x] = −γ1x+(1−x)γ2
σ2(x) = lim
h→0+
1
hEh(YM( +h)−YM( ))2|YM( ) = xi=x(1 −x)
whe e γ1=aM and γ2=bM a e he in ensi ies o mu a ion.
The e o e Y( ) is a con inuous- ime di usion p ocess wi h s a e
space S= [0,1], d i τ(x) and di usion coe icien σ2(x).
Y( ) = Y e ol es acco ding o he s ochas ic di e en ial equa ion
dY =τ(Y ) + σ(Y )dB
The W igh -Fishe model A a ian o he W igh -Fishe model
The di usion app oxima ion
Conside he p ocess
Y( ) = lim
M→∞
YM( ) = lim
M→∞
X([M ])
M
I we call h= 1/Mand x=i/M, we ha e ha
τ(x) = lim
h→0+
1
hE[YM( +h)−YM( )|YM( ) = x] = −γ1x+(1−x)γ2
σ2(x) = lim
h→0+
1
hEh(YM( +h)−YM( ))2|YM( ) = xi=x(1 −x)
whe e γ1=aM and γ2=bM a e he in ensi ies o mu a ion.
The e o e Y( ) is a con inuous- ime di usion p ocess wi h s a e
space S= [0,1], d i τ(x) and di usion coe icien σ2(x).
Y( ) = Y e ol es acco ding o he s ochas ic di e en ial equa ion
dY =τ(Y ) + σ(Y )dB
The W igh -Fishe model A a ian o he W igh -Fishe model
Bounda y beha io
The bounda ies 1,0 a e abso bing i 0 < γ1, γ2<1/2
and e lec ing i γ1, γ2≥1/2.
0 1 2 3 4 5
0
0.5
1
γ1=1/4,γ2=1/4
0 1 2 3 4 5
0
0.5
1
γ1=1,γ2=1/4
0 1 2 3 4 5
0
0.5
1
γ1=1/4,γ2=1
0 1 2 3 4 5
0
0.5
1
γ1=2,γ2=3/2
The W igh -Fishe model A a ian o he W igh -Fishe model
Spec al me hods
W i e γ1=1+β
2and γ2=1+α
2.
The in ini esimal ope a o Ao he p ocess Y is
A=x(1 −x)d2
dx2+ (1 + α−x(α+β+ 2)) d
dx , α, β > −1
The o hono mal Jacobi polynomials Pα,β
n(x) (o hogonal w. .
ω(x) = xα(1 −x)β) a e eigen unc ions o A, i.e.
APα,β
n(x) = λnPα,β
n(x), λn=−n(n+α+β+ 1)
We ha e wo impo an p ope ies:
Spec al ep esen a ion o he p obabili y densi y
p( ;x,y) =
∞
X
n=0
eλn Pα,β
n(x)Pα,β
n(y)yα(1 −y)β
In a ian dis ibu ion (α, β ≥0)
ψ(y) = lim
→∞
p( ;x,y) = Γ(α+ 1)Γ(β+ 1)
Γ(α+β+ 2) yα(1 −y)β
The W igh -Fishe model A a ian o he W igh -Fishe model
Spec al me hods
W i e γ1=1+β
2and γ2=1+α
2.
The in ini esimal ope a o Ao he p ocess Y is
A=x(1 −x)d2
dx2+ (1 + α−x(α+β+ 2)) d
dx , α, β > −1
The o hono mal Jacobi polynomials Pα,β
n(x) (o hogonal w. .
ω(x) = xα(1 −x)β) a e eigen unc ions o A, i.e.
APα,β
n(x) = λnPα,β
n(x), λn=−n(n+α+β+ 1)
We ha e wo impo an p ope ies:
Spec al ep esen a ion o he p obabili y densi y
p( ;x,y) =
∞
X
n=0
eλn Pα,β
n(x)Pα,β
n(y)yα(1 −y)β
In a ian dis ibu ion (α, β ≥0)
ψ(y) = lim
→∞
p( ;x,y) = Γ(α+ 1)Γ(β+ 1)
Γ(α+β+ 2) yα(1 −y)β
The W igh -Fishe model A a ian o he W igh -Fishe model
Spec al me hods
W i e γ1=1+β
2and γ2=1+α
2.
The in ini esimal ope a o Ao he p ocess Y is
A=x(1 −x)d2
dx2+ (1 + α−x(α+β+ 2)) d
dx , α, β > −1
The o hono mal Jacobi polynomials Pα,β
n(x) (o hogonal w. .
ω(x) = xα(1 −x)β) a e eigen unc ions o A, i.e.
APα,β
n(x) = λnPα,β
n(x), λn=−n(n+α+β+ 1)
We ha e wo impo an p ope ies:
Spec al ep esen a ion o he p obabili y densi y
p( ;x,y) =
∞
X
n=0
eλn Pα,β
n(x)Pα,β
n(y)yα(1 −y)β
In a ian dis ibu ion (α, β ≥0)
ψ(y) = lim
→∞
p( ;x,y) = Γ(α+ 1)Γ(β+ 1)
Γ(α+β+ 2) yα(1 −y)β
The W igh -Fishe model A a ian o he W igh -Fishe model
The coe icien s o he hyb id p ocess
We conside now a hyb id p ocess o he o m
{(Y ,R ) : ∈[0,+∞)}
whe e Y ∈[0,1] is a W igh -Fishe ype di usion p ocess and
R ∈ {1,2,...,N}is a con inuous- ime Ma ko chain ep esen ing N
di e en phases o which he coe icien s o he p ocess Y may change.
These p ocesses a e also known as di usions wi h Ma ko ian swi ching.
Ou p ocess e ol es acco ding o he s ochas ic di e en ial equa ion
dY =τR (Y ) + σR (Y )dB
τi(x) = α+ 1 + N−i−x(α+β+ 2 + N−i), σ2
i(x) = 2x(1 −x)
Obse e ha he in ensi ies o mu a ions depend on he phase so
A
β+1
2
−−→ Band B
α+N−i+1
2
−−−−−→ A,i= 1,2,...,N
A phase Nwe eco e he o iginal W igh -Fishe model.
B→Ag ows as we ge close o he i s phases.
The W igh -Fishe model A a ian o he W igh -Fishe model
The ansi ion o phases
The con inuous- ime p ocess R (depending also on he posi ion Y )
e ol es acco ding o a bi h-and-dea h p ocess whose in ini esimal
ope a o is gi en by an N×N idiagonal ma ix Q(x) whe e
Qi,i−1(x) = 1
1−x(N−i)(i+β−k),Qi,i+1(x) = x
1−x(i−1)(N−i+k)
Qi,i(x) = −(Qi,i−1(x) + Qi,i+1(x)),0<k< β + 1
Q(x) only depends on βand a new pa ame e k.
Fo example: N= 3 phases, β= 1,k= 3/2:
Q(x) =
−1
1−x
1
1−x0
5x
2(1 −x)
−3−5x
2(1 −x)
3
2(1 −x)
03x
1−x−3x
1−x
,x∈(0,1)
The W igh -Fishe model A a ian o he W igh -Fishe model
The ansi ion o phases
The con inuous- ime p ocess R (depending also on he posi ion Y )
e ol es acco ding o a bi h-and-dea h p ocess whose in ini esimal
ope a o is gi en by an N×N idiagonal ma ix Q(x) whe e
Qi,i−1(x) = 1
1−x(N−i)(i+β−k),Qi,i+1(x) = x
1−x(i−1)(N−i+k)
Qi,i(x) = −(Qi,i−1(x) + Qi,i+1(x)),0<k< β + 1
Q(x) only depends on βand a new pa ame e k.
Fo example: N= 3 phases, β= 1,k= 3/2:
Q(x) =
−1
1−x
1
1−x0
5x
2(1 −x)
−3−5x
2(1 −x)
3
2(1 −x)
03x
1−x−3x
1−x
,x∈(0,1)
The W igh -Fishe model A a ian o he W igh -Fishe model
S ochas ic ep esen a ion
Example: N= 3 phases, α= 0, β = 1,k= 3/2, allowing 4 changes
The W igh -Fishe model A a ian o he W igh -Fishe model
S ochas ic ep esen a ion
Example: N= 3 phases, α= 0, β = 1,k= 3/2, allowing 4 changes
Phase 1: σ2(x) = 2x(1 −x), τ1(x) = 3 −5x,Q1,1(x) = −1
1−x
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.2
0.4
0.6
0.8
1
1
The W igh -Fishe model A a ian o he W igh -Fishe model
S ochas ic ep esen a ion
Example: N= 3 phases, α= 0, β = 1,k= 3/2, allowing 4 changes
Phase 2: σ2(x) = 2x(1 −x), τ2(x) = 2 −4x,Q2,2(x) = −3−5x
2(1−x)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.2
0.4
0.6
0.8
1
1
2
The W igh -Fishe model A a ian o he W igh -Fishe model
S ochas ic ep esen a ion
Example: N= 3 phases, α= 0, β = 1,k= 3/2, allowing 4 changes
Phase 3: σ2(x) = 2x(1 −x), τ3(x) = 1 −3x,Q3,3(x) = −3x
1−x
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.2
0.4
0.6
0.8
1
1
23
The W igh -Fishe model A a ian o he W igh -Fishe model
S ochas ic ep esen a ion
Example: N= 3 phases, α= 0, β = 1,k= 3/2, allowing 4 changes
Phase 2: σ2(x) = 2x(1 −x), τ2(x) = 2 −4x,Q2,2(x) = −3−5x
2(1−x)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.2
0.4
0.6
0.8
1
1
23
2
The W igh -Fishe model A a ian o he W igh -Fishe model
Bounda y beha io
The bounda y 0 (o 1) is e lec ing i α≥0 (o β≥0) and
abso bing i −1< α < 0AND phase N(o −1< β < 0).
The W igh -Fishe model A a ian o he W igh -Fishe model
Wai ing imes and endency
Wai ing imes
We ha e o ake a look o he diagonal en ies o Q(x):
Qii (x) = −1
1−x[(N−i)(i+β−k) + x(i−1)(N−i+k)]
I x→1−⇒all phases a e ins an aneous.
I x→0+o k→0+⇒phase Nis abso bing.
I k→β+ 1 ⇒phase 1 is abso bing.
Tendency
I k→β+ 1 ⇒Backwa d endency
Meaning: The pa ame e khelps he popula ion o A’s o su i e
agains he popula ion o B’s.
I k→0+⇒Fo wa d endency
Meaning: Bo h popula ions Aand B’ igh ’ in he same condi ions.
The W igh -Fishe model A a ian o he W igh -Fishe model
Spec al me hods
The in ini esimal ope a o Ais now ma ix- alued
A=1
2A(x)d2
dx2+B(x)d
dx +Q(x)d0
dx0
A(x) = 2x(1 −x)I,Bii (x) = τi(x)
We al eady know (G ¨unbaum-Pacha oni-Ti ao, 2002) a amily o
ma ix- alued o hono mal eigen unc ions Φn(x) o A:
AΦn(x) = Φn(x)Γn,Γndiagonal
They a e called he ma ix- alued sphe ical unc ions associa ed wi h he
complex p ojec i e space.
The co esponding weigh ma ix W(x) is diagonal wi h en ies
Wii (x) = xα(1 −x)ββ−k+i−1
i−1N+k−i−1
N−ixN−i
Spec al ep esen a ion o he p obabili y densi y
P( ;x,y) =
∞
X
n=0
Φn(x)eΓn Φ∗
n(y)W(y)
The W igh -Fishe model A a ian o he W igh -Fishe model
Spec al me hods
The in ini esimal ope a o Ais now ma ix- alued
A=1
2A(x)d2
dx2+B(x)d
dx +Q(x)d0
dx0
A(x) = 2x(1 −x)I,Bii (x) = τi(x)
We al eady know (G ¨unbaum-Pacha oni-Ti ao, 2002) a amily o
ma ix- alued o hono mal eigen unc ions Φn(x) o A:
AΦn(x) = Φn(x)Γn,Γndiagonal
They a e called he ma ix- alued sphe ical unc ions associa ed wi h he
complex p ojec i e space.
The co esponding weigh ma ix W(x) is diagonal wi h en ies
Wii (x) = xα(1 −x)ββ−k+i−1
i−1N+k−i−1
N−ixN−i
Spec al ep esen a ion o he p obabili y densi y
P( ;x,y) =
∞
X
n=0
Φn(x)eΓn Φ∗
n(y)W(y)
The W igh -Fishe model A a ian o he W igh -Fishe model
Spec al me hods
The in ini esimal ope a o Ais now ma ix- alued
A=1
2A(x)d2
dx2+B(x)d
dx +Q(x)d0
dx0
A(x) = 2x(1 −x)I,Bii (x) = τi(x)
We al eady know (G ¨unbaum-Pacha oni-Ti ao, 2002) a amily o
ma ix- alued o hono mal eigen unc ions Φn(x) o A:
AΦn(x) = Φn(x)Γn,Γndiagonal
They a e called he ma ix- alued sphe ical unc ions associa ed wi h he
complex p ojec i e space.
The co esponding weigh ma ix W(x) is diagonal wi h en ies
Wii (x) = xα(1 −x)ββ−k+i−1
i−1N+k−i−1
N−ixN−i
Spec al ep esen a ion o he p obabili y densi y
P( ;x,y) =
∞
X
n=0
Φn(x)eΓn Φ∗
n(y)W(y)
The W igh -Fishe model A a ian o he W igh -Fishe model
In a ian dis ibu ion
The in a ian dis ibu ion ψ(y) (α, β ≥0) comes om he s udy o
lim
→∞
P( ;x,y)
This should be independen o he ini ial s a e and phase.
The e o e we should expec a ow ec o in a ian dis ibu ion
ψ(y) = (ψ1(y), ψ2(y), . . . , ψN(y))
wi h 0 ≤ψj(y)≤1 and
N
X
j=1 Z1
0
ψj(y)dy = 1
Explici o mula (MdI, 2012)
⇒ψ(y) = Z1
0
eT
NW(x)eNdx−1
eT
NW(y)
whe e eT= (1,1,...,1). In pa icula
ψj(y) = yα+N−j(1 −y)βN−1
j−1α+β+N
α(β+N)(k)N−j(β−k+1)j−1
(α+β−k+2)N−1
The W igh -Fishe model A a ian o he W igh -Fishe model
In a ian dis ibu ion
The in a ian dis ibu ion ψ(y) (α, β ≥0) comes om he s udy o
lim
→∞
P( ;x,y)
This should be independen o he ini ial s a e and phase.
The e o e we should expec a ow ec o in a ian dis ibu ion
ψ(y) = (ψ1(y), ψ2(y), . . . , ψN(y))
wi h 0 ≤ψj(y)≤1 and
N
X
j=1 Z1
0
ψj(y)dy = 1
Explici o mula (MdI, 2012)
⇒ψ(y) = Z1
0
eT
NW(x)eNdx−1
eT
NW(y)
whe e eT= (1,1,...,1). In pa icula
ψj(y) = yα+N−j(1 −y)βN−1
j−1α+β+N
α(β+N)(k)N−j(β−k+1)j−1
(α+β−k+2)N−1
The W igh -Fishe model A a ian o he W igh -Fishe model
S udy o he in a ian dis ibu ion