-
Markov Chain
article version
180705 2
. Cyclic Decomposition Theorem .1
Cyclic Decomposition Theorem 2 ()
. motivation.2
[ I, 8.58.7] Cyclic Decomposition Theorem , Jordan canonical
form ,
Markov chain 1
.
article Markov chain subject
. Markov chain
application . article Markov chain
,
() (genotype frequency),
() Google ranking system
. , Markov chain Perron-
Frobenius Theory () .
article Jordan canonical form([I, 238 (j) t ] [ I, 239 i ]) . ,
Cyclic Decomposition The-orem . , Jordan canonical form ,
article () F =C.1, , [3] Cyclic Decomposition Theorem .
1 1- , 2 2-
.2 [II, 12] .
1
-
1 section version
180705 componentwise .
1.1 Ck =(c(k)ij
)Mm,n(C), i, j lim
kc(k)ij
, {Ck}k=0 ,
limk
Ck =(limk
c(k)ij
) . Mm,n(C) = Cmn identify
.
2 (1.2,
1.3 1.6).
1.2 (Exponential map) exp : Mn,n(C) Mn,n(C)
exp(A) =k=0
1
k !Ak, (AMn,n(C))
.
.3
, , M. Artin [1, p. 119].
1.3 (Eigen-vector motivation) () A=
(2 1
3 4
)Mn,n(R)
( ) eigen-vector
. Ae1 = (2, 3)t Ae2 = (1, 4)
t, LA 1
S ={ae1+ be2 | a, b 0
} cone AS =
{a(2, 3)
t+ b(1, 4)
t | a, b 0}
. A2e1 = (7, 18)t A2e2 = (6, 19)
t, LA AS cone
A2S ={a(7, 18)
t+ b(6, 19)
t | a, b 0}. ,
cone sequence
S AS A2S A3S A4S
.
3Exponential map Lie group Lie algebra.
2
-
() cone half-line .4 , =
m=0 AmS
1 () half-line. ,
A = A(
m=0 AmS)=
m=0 Am+1S =
, LA half-line . X (non-
zero) vector, AX = X A eigen-value R. > 0.
() = 5, (1, 3)t half-line
([I, 7.2.1 ). (A eigen-value 1.)
Perron-Frobenius Theory( 7).
() eigen-value = 5 dominant eigen-
value.
1.4 A Mn,n(C) , A eigen-value(in C) maximum ab-solute value A
dominant eigen-value. ( A
dominant eigen-value.) Dominant eigen-value
eigen-vector dominant eigen-vector . (
principal eigen-vector.5)
Numerical Linear Algebra Matrix Computation
, , size eigen-value( ) eigen-
vector( ) . n 10( 230)
characteristic polynomial 10 root
.
eigen-vector dominant eigen-vector
. Matrix Computation ,
(!), dominant eigen-vector.6
4 , A Am
. (, eigen-vector , eigen-
vector .)5 Principal Component Analysis(PCA) principal.6Matrix
Computation Golub-Van Loan[4] Meyer[5]
. .
3
-
A Mn,n(C) dominant eigen-vector , 1.3 . , X Cn , lim
mAmX ( )
.7 , , 1.3 , m Am
AmX , .
limm
AmXAmX ?
A,A2, A3, . . . step n2-
. AX, A(AX), A(A2X), . . . step n-
. limm
Am limm
AmX .
n, n2 n significant.
1.5 Numerical Linear Algebra, ,
Linear Algebra.
() A Mn,n(C) . det(A) = 0, A() 230
.
, .
() A Mn,n(C) diagonalizable . A(t) ([I, 7.2.18] ).
AMn,n(C) unique dominant eigen-value.
1.6 A Mn,n(C), X Cn , lim
mAmX
AmX A dominant eigen-vector.
() m AmX = 0 ( A invertible, X = 0 OK). A diagonalizable. , AXi
= iXi i C Xi Cn,
U1AU = D = diag(1, . . . , n), AmX = UDmU1X
. U = (X1, . . . , Xn)([I, 7.2.13] ).
A Mn,n(C) unique dominant eigen-value 1 , 1 > 0. j = 1 1 = j
, . U = (uij), U
1X = (yj), y1 = 0.71.3 , 1 ( ) half-line ,
limm Am.
4
-
() limm
AmXAmX k- , 1 > 0 unique domi-
nant eigen-value 1 = j(, j = 1),j ukj
mj yj
i |
j uijmj yj |2
=m1
j ukj(j1
)myj
m1
i
j uij
(j1
)myj2 y1
i |ui1y1|2uk1
. limm
AmXAmX [U ]
1 =X1(
i |ui1y1|2 = 0
?). , limm
AmXAmX A dominant eigen-vector.
() . ()
. , 1.5
1.7 (), .
() () essential assumption 1 > 0 . (
() 1 > 0 .)
.
positive real number zero.
() A diagonalizable , A Jordan canonical form
. Detail.8
() () () L2-norm L1-
norm . A Markov matrix
X provability vector( 2.2 4.5() ),
m AmX L1-norm 1, limm
AmX , A
dominant eigen-vector( 4.9 ).9
1.7 1.6(), y1 = 0 X / X2, . . . , Xn.10
, limm
AmXAmX
? Google Ranking System( n 10) A52X
( 6).
8() . 3.2 .9 [6] L1-norm.
10 Hint: Uei =Xi.
5
-
terminology .
1.8 A Mn,n(C), X Cn , AX, A(AX), A(A2X), . . . lim
mAmX
AmX () , A dominant eigen-vector()
power method .
1.9 () , power method.
() , n 230 , A X 1.6()
. power method
, limm
AmXAmX . , A X
230 . ( idea 800 Google Ranking System. 6.)
() Power method
.11
A unique dominant eigen-value positive real number
, . , , unique dominanteigen-value positive real number square
matrix
. Positive Markov matrix( 2.1 2.2 )
. ( , non-negative matrix( 2.1 )
.12 non-negative matrix dominant eigen-vector
Perron-Frobenius Theory.13)
, () Cn- norm 180705
. X = (x1, . . . , xn)t Cn, X norm()
X=|x1|2 + + |xn|2
.14 X,Y Cn X Y .
11. = .127 Meyer[5] . Non-negative case.13, power method
computer , power method
Perron-Frobenius Theory . Perron-Frobenius Theorem
20 . non-negative matrix
, .14 Cn R2n naturally identify.
6
-
2 A Toy Example
section version
180102 Markov matrix, toy example.
2.1 C = (cij) Mm,n(C) , [cij > 0 for all i, j ], C > 0 C
positive .15 [cij 0 for all i, j ],C 0 C non-negative . , X Cn ,X
> 0 X 0.
2.2 () X Cn , X 0 X 1, X probability vector . A = (aij) Mm,n(C)
, A column probability vector(, A column sum 1),
A (left) Markov matrix( A 0).16
() At Markov matrix, A right Markov matrix.
Markov matrix. :
2.3 (Right) Markov matrix 1 eigen-value .
: A(t) = At(t)([I, 7.6.24]), right
. . A
right Markov, A row sum 1,
A (1, . . . , 1)t = 1(1, . . . , 1)t
. , A Markov, (AI) row zero vector, (A I) row, det(1A I) =
0.
(2 2)- example , Markov matrix (2 2)- . toy example [3] ( )
. ( toy example idea 5 6(Random Surfer
Model).)
15 positive definite matrix (Linear Algebra)
. [I, , 15.3] .16Markov matrix stochastic matrix, probability
matrix, transition matrix
. Probability vector stochastic vector.
7
-
2.4 () (city) (suburbs)
(probability vector) X = (c0, s0)t (, , c0, s0 0
c0+s0 = 1).17 ,
A=
(0.90 0.02
0.10 0.98
)=
(city city suburbs city
city suburbs suburbs suburbs
)
. , , 1 city 90%
city , city 10% suburbs .
, 1 (c1, s1)t(
c1
s1
)=
(0.90c0+0.02s0
0.10c0+0.98s0
)=
(0.90 0.02
0.10 0.98
)(c0
s0
)= AX
. , 2 (c2, s2)t(
c2
s2
)= A
(c1
s1
)= A2X
, m- AmX.18
() A,
U1AU =
(16
16
56
16
)1(0.90 0.02
0.10 0.98
)(16
16
56
16
)=
(1 0
0 0.88
)=D
. ,
Am =UDmU1 =
(1+5(0.88)m
61(0.88)m
655(0.88)m
65+(0.88)m
6
), lim
mAm =
(16
16
56
56
)
,
AmX =
(1+5(0.88)m
6 c0+1(0.88)m
6 s055(0.88)m
6 c0+5+(0.88)m
6 s0
), lim
mAmX =
(1656
)
().
17c for city, s for suburbs.18 , mathematical model A
constant,() () ( ) . ,
, 1.
8
-
() A , suburbs city
. (16 ,
56
)t equilibrium()
. city 0 . ( suburbs
2% city, suburbs .)
() A positive Markov matrix. 1 A unique dominant
eigen-value, dominant eigen-vector U column(16 ,
56
)t. (Dominant eigen-vector probability vector
(16 ,
56
)t.)
, () limm
AmX A dominant eigen-vector
1.6.
() , limm
Am column A dominant eigen-vector
.
() equilibrium(16 ,
56
)t
X = (c0, s0)t .
c0 = 0 s0 = 0 equilibrium.
() () Am AmX, 0.88m 0
, .
toy example [ I, 7 ].
toy example ( 4.8 4.9
).
9
-
3 Preliminary Resultssection version
180102, F = C, A Mn,n(C) . A
characteristic polynomial
A(t) = (t1)e1(t2)e2 (tk)ek
(, ei 1, i distinct). ei eigen-value i multiplicity, ei =
multA(i). A -eigen-space
EA = ker (AI) = {X Cn |AX = X}
([I, 7.6.20] ).
, article !
3.1 Prove or disprove : dimEA = multA().19
Jordan canonical form .20 A similar Jordan
canonical form J i-Jordan block Ji Mei,ei(C) block diag-onal
matrix. Ji i-Jordan -block Jij Mrij ,rij (C) block diagonal matrix.
,
A J = diag(J1, J2, . . . , Jk), Ji = diag(Ji1, Ji2, . . . ,
Jihi)
.21 -Jordan -block Jij
K = (),
( 1
0
), . . . ,
1
1 0
0 1
. , A diagonalizable, A Jordan canonical form
diag(1Ie1 , 2Ie2 , . . . , kIek).
19 A eigen-value, dimEA =0= multA().20, Cyclic Decomposition
Theorem OK.21Jordan block, Jordan -block.
10
-
Jordan canonical form (upper-)triangular matrix, triangular
matrix diagonal matrix .
Jordan canonical form
. , m Jm
. , J A Jordan canonical form , Jm ,
{Am}m=0.
3.2 K = I+N Mr,r(F ) , N2, N3, . . . . , m, m r1,
(K)m =
m mm1(m2
)m2
(m3
)m3
(m
r1)mr+1
m mm1(m2
)m2
(m
r2)mr+2
. . .. . .
. . ....
0. . .
. . ....
m mm1
m
.22 (, , m < i binomial coefficient
(mi
)= 0
. , = 0 m < r 1 . = 0, (K)
m =Nm.23)
, , .
3.3 Ak Mn,n(C), B Mm,n(C), C Mn,r(C) U Mn,n(C), .
() {Ak}k=0 , limk
Ak = L . , {BAk}k=0{AkC}k=0. lim
k(BAk) =BL lim
k(AkC) =LC.
() {U1AkU}k=0 {Ak}k=0 . lim
kAk =L, lim
k(U1AkU) =U
1LU .
22Hint: I N commute , (I)N =N(I) (I+N)m 2-
.23 N nilpotent.
11
-
.
3.4 D={C
|| < 1 or = 1}.24 . , {m}m=0
D. {Am}m=0.
3.5 AMn,n(C) , {Am}m=0
(i) C A eigen-value, D,(ii) 1 A eigen-value, dimEA1 = multA(1)(,
A 1-Jordan
block ).
: J A Jordan canonical form , 3.3()
, {Am}m=0 {Jm}m=0 . 3.2 (K)
m. (.)
3.1 ,
.
3.6 A eigen-value,
(1) dimEA = multA().
(2) A -Jordan block I.
.
Linear Algebra A eigen-value 1, . . . , k
. , Numerical Linear Algebra 3.5
eigen-value () . Eigen-value
.
3.7 A= (aij)Mn,n(C),
i(A) =n
j=1 |aij |, j(A) =n
i=1 |aij |
. A row sum column sum
(A) = max {i(A) | 1 i n}, (A) = max {j(A) | 1 j n}
.
24D disk.
12
-
simple, .
3.8 (Gershgorins Disk Theorem, 1931) A = (aij)
Mn,n(C)eigen-value eigen-vector X = (xi) Cn . |xk|= max
{|xi|
1 i n} ,25 |akk| k(A)|akk|
.26
: AX = X,n
j=1 akjxj = xk, j =k akjxj = xkakkxk
. , xk = 0,
|akk| =j =k akj xjxk j =k |akj | = k(A)|akk|
.27
3.7 row sum column sum.
3.9 AMn,n(C) eigen-value, || min{(A), (A)}.
: Gershgorins Disk Theorem,
|| |akk|+ |akk| (k(A)|akk|
)+ |akk| = k(A) (A)
. At eigen-value( ?),
|| (At) = (A)
.
Gershgorins Disk Theorem 3.9
. , A eigen-value A similarity class invariant
, (A), (A) similarity class invariant . , , A eigen-value A
n2-.
25 k.26 akk k(A)|akk| disk Gershgorins disk.27 1931 . (
A. Markov(18561922) .) historical comment
Meyer[5, p. 497] .
13
-
Positive matrix eigen-value a technical lemma
. motivation .28
3.10 A Mn,n(C) , A > 0 . A || = (A)eigen-value .29 , .
() = (A)> 0.
() dimEA = 1. EA =
(1, 1, . . . , 1)
t. : 3.8(Gershgorins Disk Theorem),
|||xk|=
j akjxj j |akjxj | j |akj ||xk|= k(A)|xk| (A)|xk|
. , ||= (A), ()(=). ,
(i)
j akjxj= j |akjxj |,
(ii)
j |akjxj |=
j |akj ||xk|,
(iii) k(A) = (A)
. (i),
akjxj = cj z, (1 j n), |z|= 1
real number c1, . . . , cn 0 z C ( 3.11 ). , A> 0, (ii)
|xj | = |xk|, (1 j n)
. , A> 0,
akj |xk| = akj |xj | = |cj z| = cj , (1 j n)
. ,
xj =cjakj
z = |xk|z, (1 j n)
. , X = |xk|z(1, 1, . . . , 1)t. eigen-vector 1 = (1, 1, . . . ,
1)t . 1 eigen-
vector. A1> 0, A1= 1, > 0.30 28 3.10 [3].29, 3.9, A
dominant eigen-value.30(iii).
14
-
3.11 1, . . . , r C
|1 + + r| = |1|+ + |r|
,
i = ci, (i = 1, . . . , r)
C 0 c1, . . . , cr R . ||= 1 OK.31
3.12 A Mn,n(C) , A > 0 . A || = (A)eigen-value . = (A)> 0,
dimEA = 1
.32
3.13 ,
(2 00 1
)
(0 1
1 0
) , 3.10 A
positive . .33
. :
3.14 A= (aij)Mn,n(C), A matrix norm
A = max{|aij |
1 i, j n}.34
. .
3.15 A,B Mn,n(C), .
() A = 0, A> 0.
() cC, cA= |c|A.
() A+B A+B.
() AB nAB.
31Hint: r. [I, , 7.5] .32 [I, 7.6.24()] . ( (1, 1, . . . , 1)t
eigen-vector.)33 Markov matrix.34 matrix norm . max norm .
Mn,n(C) =Cn2 identify, max norm L-norm. Matrix norm exp(A)
(1.2 ).
15
-
4 Markov Matrix
version
150909 2 toy example .
3 Markov matrix.
4.1 AMn,n(C) Markov matrix,
() j(A) = (A) = 1 for all j = 1, . . . , n.
() C A eigen-vector, || 1.() 1 is a dominant eigen-value of
A.
: () Markov matrix . () 3.9
(Gershgorins Disk Theorem ) () . () 2.3
().
positive Markov matrix.
4.2 AMn,n(C) positive Markov matrix,
() C A eigen-vector , = 1, ||< 1. , 1 Aunique dominant
eigen-value.
() dimEA1 = 1.
: 3.12( 3.10 transpose version) 4.1 direct con-
sequence.
4.3 A Mn,n(C) right Markov matrix , 4.1 4.2.
4.2, A positive Markov, dimEA1 = 1
, dimEA1 = multA(1) .
Markov chain essence !
easy exercise. ( 1= (1, 1, . . . , 1)t.)
4.4 .
() 0AMn,n(C) Markov matrix At 1= 1.() 0X Cn probability vector
Xt 1= (1)11.
16
-
4.5 .
() A,B Mn,n(C) Markov matrix, AB Markov matrix.
() A Mn,n(C) Markov matrix, X Cn probability vector,AX
probability vector.
4.6 A Mn,n(C) right Markov matrix, 4.4 4.5.35
Final touch matrix norm.36
4.7 AMn,n(C) (right) Markov matrix, dimEA1 = multA(1)(,A
1-Jordan block ).
: (i) m , Am Markov( 4.4()
), Am 1.
(ii) U1AU = J A Jordan canonical form,
Jm = U1AmU n2U1Am|U n2U1U
. ,{Jm
m 0} bounded above.(iii) J 1-Jordan -block (1 1)- ( 3.2 ). , J
1-Jordan block( 3.6 ).
(iv) A right Markov [ I, 7.6.24] .
article main theorem state.
4.8 AMn,n(C) positive (right) Markov matrix,
() dimEA1 = multA(1) = 1.37 ( A unique dominant eigen-value
1
dominant eigen-vector (up to scalar) unique.)
() limm
Am.
: () 4.1, 4.2 4.7.
() 3.5. 35X probability vector, AX probability vector.36, 4.7
4.1, first touch OK.37 multA(1)= 1, dimE
A1 =1. ?
17
-
4.9 A Mn,n(C) positive Markov matrix , limm
Am =L
. A unique dominant eigen-value 1 dominant
eigen-vector unique probability vector P ,
() AL=LA=L.
() L Markov matrix, L= (P, P, . . . , P ).
() probability vector X , limm
AmX =LX = P .38
() P > 0. ( L> 0.)
: () AL=A limm
Am = limm
Am+1 =L. (LA=L.)
() Am Markov( 4.5),
1t L = 1t limm
Am = limm
1t Am = limm
1t = 1t
( L 0), L Markov( 4.4). , L columnprobability vector. , AL = L,
L column eigen-
value 1 A eigen-vector. , L column P .
() P = (p1, . . . , pn)t, X = (x1, . . . , xn)
t . , L = (P, P, . . . , P )
, LX i-
pix1+ +pixn = pi(x1+ +xn) = pi
. , LX = P .
() . AP =P , A> 0, P 0, P > 0.
, positive Markov matrix A , probability
vector X AX, A(AX), A(A2X), . . . limm
AmX
, A dominant eigen-vector P .
( limm
Am = L. ?)
power method(1.8), , X depend
. , P .
4.9 dominant eigen-vector P Perron-Frobenius
vector, stochastic vector, stationary vector, fixed probability
vector
.39
38, , LP =P .39 6 PageRank vector.
18
-
.
4.10 Ai Mn,n(C) positive Markov matrix , block diagonalmatrix A=
diag(A1, . . . , Ak).
() limm
Am.
() Eigen-space EA1 basis.
Right Markov matrix. (.)
4.11 A Mn,n(C) positive right Markov , limm
Am = L
. At unique dominant eigen-value 1 dominant
eigen-vector unique probability vector P ,40
() AL=LA=L.
() L right Markov matrix, L= (P, P, . . . , P )t.
() probability vector X , Xt L=P t,41 LX = (P t X) 1.
comment().
4.12 () , A diagonalizable
, article . Jordan
canonical form.
() A Markov matrix , As > 0 s , A
regular Markov matrix( primitive Markov matrix) . Regular
Markov matrix 4.2 4.8 4.9
4.11 .42 [ ] : A eigen-value
||= 1, s As > 0 eigen-value, s =1( 4.2). , As+1 > 0, s+1 =
1. = 1. dimEA
s
1 = 1
( 4.8), EA1 =EAs
1 .
, article Markov chain( Markov process)
chain .
, . .43
40A dominant eigen-vector 1( 2.3).41 Right Markov right right
notation( ). , L
act.42, n, A regular.43Wikipedia . stochastic process.
19
-
5 Hardy-Weinberg Equilibrium
version
180102 Population Genetics ( ) .44
19081909 Hardy-Weinberg equilibrium(principle, law)
.45
5.A.
, .
allele() (T, t) . genotype TT Tt
phenotype, genotype tt phenotype.46
, m 0, m-(m-) genotype frequency
pm = Probm-(TT ), qm = Probm-(Tt), rm = Probm-(tt)
, m- genotype frequency vector Pm = (pm, qm, rm)t
.47 (0-) genotype frequency vector
P0 = (p0, q0, r0)t= (p, q, r)
t
. m- allele frequency
am = Probm-(T ), bm = Probm-(t)
, m- allele frequency vector Qm= (am, bm)t .
0- allele frequency vector
Q0 = (a0, b0)t= (a, b)
t
. , Pm, Qm 0,
pm+ qm+ rm = 1, am+ bm = 1
.
44 population , . . .
45G. H. Hardy(18771947) Hardys Theorem, Hardys Inequality,
Hardy-Littlewood
Theorem (). 20 neo-Darwinism(modern
synthesis) Hardy. (W. Weinberg(18621937) ().)46Mendel. Dominant
trait recessive trait .47Frequency vector = probability vector.
20
-
, m- allele frequency
am = pm+12 qm, bm =
12 qm+ rm
( ?).48 0-
a = p+ 12 q, b =12 q+ r
.
, 1- genotype frequency. TT -type
genotype table(Markov matrix)
ATT = [ TT -type]
TT Tt tt
TT 1 12 0
Tt 0 12 1
tt 0 0 0
. Tt-type tt-type
genotype Markov matrix
ATt = [ Tt-type] Att = [ tt-type]
TT Tt tt
TT 1214 0
Tt 1212
12
tt 0 1412
TT Tt tt
TT 0 0 0
Tt 1 12 0
tt 0 12 1
. ,
A = pATT + qATt+ rAtt
,
A =
p+ 12 q
12 p+
14 q 0
12 q+ r
12 p+
12 q+
12 r p+
12 q
0 14 q+12 r
12 q+ r
=a 12 a 0
b 12 a
0 12 b b
. A Markov matrix.
48, , allele frequency genotype frequency.
21
-
Markov matrix A 2 toy example,
A =
a 12 a 0
b 12 a
0 12 b b
=TT TT Tt TT tt TTTT Tt T t Tt tt TtTT tt T t tt tt tt
. , , TT Tt () TT() Tt . , 2 toy example
,49 1- genotype frequency vector
P1 =
p1
q1
r1
= AP0 =a 12 a 0
b 12 a
0 12 b b
p
q
r
=
ap+ 12 aq
bp+ 12 q+ar12 bq+ br
=
a2
2ab
b2
().50 , 1- allele frequency vector
Q1 =
(a1
b1
)=
(a2+ 12 2ab12 2ab+ b
2
)=
(a
b
)= Q0
. , allele frequency !
, 2- genotype frequency vector
P2 =
p2
q2
r2
=a1
12 a1 0
b112 a1
0 12 b1 b1
p1
q1
r1
=a 12 a 0
b 12 a
0 12 b b
a2
2ab
b2
=
a3+a2b
a2b+ab+ab2
ab2+ b3
=
a2
2ab
b2
= P1, allele frequency vector Q2 = (a, b)
t=Q0. ,
Pm =
pm
qm
rm
= AmP0 =
a2
2ab
b2
= P1, Qm =(a
b
)= Q0, (m 1)
.
49= .50 , , mathematical model random mating, sex
independent genotype frequency . (
) . , (, m- m-).
22
-
,51 allele frequency vector (a, b)t constant
, genotype frequency vector equilibrium(Hardy-
Weinberg equilibrium) (a2, 2ab, b2)t . (20 neo-Darwinism
(modern synthesis).)
equilibrium , ,
A regular Markov matrix.52 , P1 = (a2, 2ab, b2)t
A dominant eigen-vector.53 , limm
AmP0 = P1.
5.B. Sex Linked Gene
, () X- .
allele (X,x). genotype
XX, Xx, xx xx, genotype XY, xY xY
.54
, m- genotype frequency
pm = Probm-(XX), qm = Probm-(Xx), rm = Probm-(xx)
, m- genotype frequency vector Pm = (pm, qm, rm)t
. ( Pm 0, pm + qm + rm = 1.) m- allelefrequency
am = Probm- (X), bm = Probm- (x)
( am, bm 0, am+ bm =1), 5.A,
am = pm+12 qm, bm =
12 qm+ rm, (m 0)
. m- genotype frequency
cm = Probm-(XY ) = Probm- (X),
dm = Probm-(xY ) = Probm- (x)
( allele frequency genotype frequency ), m-
allele frequency vector Rm = (cm, dm)t.
51 (!) [ : = 3 : 1]. ?52a, b =0, A2> 0. 4.12().53A
eigen-value 1, 1
2, 0. , , A diagonalizable, det(A) = 0.
54.
23
-
, (m+1)- genotype frequency. 5.A, genotype Markov matrix
AXY = [ XY -type] AxY = [ xY -type]
XX Xx xx
XX 1 12 0
Xx 0 12 1
xx 0 0 0
XX Xx xx
XX 0 0 0
Xx 1 12 0
xx 0 12 1
. ,
Am = cmAXY + dmAxY =
cm
12 cm 0
dm12 cm
0 12 dm dm
, (m 0)(Am Markov matrix), (m+1)- genotype frequency
vector
Pm+1 =
pm+1
qm+1
rm+1
= AmPm =cm
12 cm 0
dm12 cm
0 12 dm dm
pm
qm
rm
=
amcm
dmpm+12 qm+ cmrm
bmdm
=
amcm
1amcm bmdmbmdm
.55
5.1 () (m+1)- genotype(allele) frequency
table(Markov matrix) BXX , BXx, Bxx
.
() Bm = pmBXX + qmBXx+ rmBxx,
Rm+1 =
(cm+1
dm+1
)= BmRm =
(am am
bm bm
)(cm
dm
)=
(am
bm
)
.
55 vector probability vector( 4.5), .
24
-
() , , genotype
( ?), . , dm+1 = bm. ,
allele frequency bm+1
bm+1 =12 (1amcm bmdm)+ bmdm
= 12(1 (1 bm)(1dm) bmdm
)+ bmdm
= 12 (bm+ dm)
. M =
(12
12
1 0
), Xm =
(bm
dm
),
Xm+1 =
(bm+1
dm+1
)=
(12
12
1 0
)(bm
dm
)= MXm
. ,
X1 = MX0, X2 = MX1 = M2X0, . . . , Xm = M
mX0
.
, M2 > 0, M regular right Markov matrix.
, 4.11 ( 4.12() ). M t
dominant eigen-vector(23 ,
13
)t(),
limm
(bm
dm
)= lim
mMm
(b0
d0
)=
(23
13
23
13
)(b0
d0
)=
(23 b0+
13 d0
23 b0+
13 d0
)
. , , allele frequency vector
allele frequency vector . , ,
allele frequency vector,
b = limm
bm =23 b0+
13 d0, a = limm
am = 1 b
, Hardy-Weinberg equilibrium
limm
pm
qm
rm
= limmam1cm1
bm1dm1
= limmam1am2
bm1bm2
=
(a)2
2ab
(b)2
( 5.1).56
56.
25
-
, , 112 = 0.0833
, 1200 = 0.0050 . ,
limm
dm = b,
limm
rm = (b)2.
(112
)2= 0.0069,
() .57
5.C. Generalization
. :
() r- T, S, . . . ( genotype
3r-),
() Allele pair (T, t), s-tuple (T1, . . . , Ts)( genotype
(s+12
)-),
() ,
() ()() genotype frequency,
() ()() sex linked gene,
() mutation, sexual selection,
mathematical model
.
(. (?).)
5.2 20 neo-Darwinism(modern synthe-
sis) G. H. Hardy Hardy ,
, . , Hardy I have never done anything
useful. No discovery of mine has made, or is likely to make the
least differ-
ence to the amenity of the world ,
. Hardy cricket team geneticist
, () . Hardy thus became the
somewhat unwitting founder of a branch of applied
mathematics.58
57 (0.019) . . ( limm
rm = (b)2
initial condition( founder effect).)58Wikipedia.
26
-
6 Google Ranking System
version
180102 Markov chain PageRank algorithm(Google ranking
system) . Internet search engine (
web page ). ranking system
: search result(rank) ?
1995(?) Stanford University Computer Science
L. Page S. Brin G. H. Golub[4] Matrix Computation
(Golub dominant eigen-vector power method). Page Brin
,59 dominant eigen-
vector Marcov chain.
PageRank idea :
. ,
.
6.1 i ri(, 0 ri R) ? follower , i
. , follower , ri
j i rj
. j i j i.,
ri =j i
1
Njrj , , Nj =
{k | j k}.60
, ? ?
(Golub (?)) ? .
59 (1998) [6] eigen-value
. [6] T. Winograd Page .Google ,
honor system . [6]
, , .60, 0 follower 0.
27
-
6.1, rank(, reputation, , ) ri
. , rank.
800 !61
6.2 web page i rank ri
ri =j i
1
Njrj , , Nj =
{k | j k} (, 0 ri R).62 j i web page j webpage i link.
, ri ? ri ?
ri 1- ? ,
aij =
1
Nj(if j i and i = j)
0 (otherwise)
, 1-
ri =
j aij rj
. ( aii = 0. , self-link.) , , A= (aij)Mn,n(R), rank vector R=
(rj)Rn,
AR = R
. , R eigen-value 1 A eigen-vector. ,
A matrix size n 230.
6.3 A zero column, A Markov matrix
. (, A dominant eigen-value 1.)
, A zero column , outlink web
page A eigen-value 1 . (
R eigen-value 1 , .)
61 2017 Google 800.62 web page.
28
-
R , R (up to scalar multiple) ? , dimEA1
1 ? multA(1) ? A 0 A very very sparse matrix.63 , 0 .
, A (after renumbering) block diagonal matrix
. (Page-Brin[6] there is a small problem , two web pagesthat
point to each other but to no other page . , A
diagonal block
(0 1
1 0
) . , (1)
A eigen-value, 1 unique dominant eigen-value.) ,
multA(1) diagonal block (
4.10).64
dominant eigen-vector Marcov chain
, .
, 1.5 , A 230
.65
1- : A Mn,n(C) zero column , zerocolumn 1n 1 (, 1 = (1, 1, . . .
, 1)
t). A
, A Markov matrix( 6.3 ).
, outlink page(
) page outlink . ,
( ) (?).
1n , A A . (
.)
, Markov matrix A very very sparse.
6.2 , .
63Sparse matrix non-zero entry.64 diagonal block ,
, .65 PageRank algorithm , 1.5, 230
.
29
-
Page-Brin 800 () :
theory positive Markov matrix, A
positive Markov matrix ! ,
A 230 .
2- : Google matrix GMn,n(R)
G = dA+1dn
1n
(, 1n Mn,n(C) 1).
d = 0.85
(d damping factor). G positive Markov matrix
().
Google matrix G = (gij) . Nj = 0
(, j -page outlink , A j -th column 0,
A j -th column 1n 1), G j -th column1n 1. ,
Nj = 0, gij =1n for all i. , Nj = 0
gij =
0.15
n+
0.85
Nj(if j i and i = j)
0.15
n(otherwise)
. (, G positive Markov.)
, 1n0.15n
, 1Nj0.85Nj 0.15Nj . ,
A non-zero component 1Nj . ,
damping factor, , 0.5 0.88, 0.85
? damping factor 0.9999 1 1230 , 6.2 ?66
, damp,
/ . , limm
GmX
(, X Rn probability vector).66 0 < d < 1, G positive
Markov matrix.
30
-
G positive Markov matrix, Perron-Frobenius vector
limm
GmX = P theory( 4.8 4.9)
(, X Rn probability vector). dominant eigen-vector P PageRank
vector (P i- i-page
PageRank).67 P power method. Page-Brin[6]
, G52X.68 69
damping factor d = 0.85 Page-Brin[6]
.70 Random Surfer Model.
Random Surfer Model G (i, j)- gij j -page random
surfer i-page ( ) . , X web
page ,71 - lim
mGmX . ( 2 toy example .) ,
backlink (, ) page surfer
. , (? !) , page surfer
page outlink click 85%, outlink
15% .72 (, j i , j -page surfer i-page outlink click i-page
0.15n , (j -page)0.15n .)
PageRank algorithm implement,
() .73 , ()
, .
676.2 rank vector.68 [2], can be computed in a few hours on a
medium size workstation.69, X PageRank vector, .70 [6] 0.85, [2].71
(random) surfer.72Random surfer 85% randomly . ,
random imaginary virtual.73 0.85().
31
-
7 Perron-Frobenius Theory
version
160102 Perron-Frobenius Theory
. 4 ,
4 . ,
, article
. Meyer[5].
A> 0. A 0, .74
7.1 C,D Mm,n(C) , .
() C = (cij), |C| =(|cij |
)Mm,n(C) .75 X Cn
, |X|.() C >D if and only if CD> 0.() C D if and only if
CD 0.
main theorem, .
7.2 (Perron-Frobenius Theorem : Positive Case) A Mn,n(C)A> 0
, A dominant eigen-value . :
() = ||> 0.() AX = X( 0 =X Cn), A |X|= |X|, |X|> 0.()
dimEA = multA() = 1.
, (), A> 0 unique dominant eigen-value > 0
. , (), positive eigen-vector
. , () , positive eigen-vector
probability vector P , P EA = P . P A Perron-Frobenius
vector.
7.3 A Mn,n(C) , m(A) = max{|| A eigen-value}
. (, A dominant eigen-value, ||=m(A).)
74 A > 0 Perron Theory , A 0 Perron-FrobeniusTheory.
Meyer[5]
75, |C| det(C).
32
-
AMn,n(C), A> 0.
7.4 .
() m(A)> 0.
() 1m(A) A> 0, m(
1m(A) A
)=1.
7.5 0 = cC, .
() EA =EcAc .
() A(t) = (t1) (tn), cA(t) = (t c1) (t cn).76 ,multA() =
multcA(c).
7.2 , , 3.10 4.7 modify
. technical .
7.6 () m(A) A eigen-value.
() AX = X(, ||=m(A), 0 =X Cn), A |X|=m(A) |X|, |X|> 0.
: (i) (notational convenience) A 1m(A) A normalize,
m(A) = 1().
(ii) ||=m(A) = 1, AX = X(, 0 =X Cn). ,
|X| = || |X| = |X| = |AX| |A| |X| = A |X|
. A |X|= |X|, A |X| = |X|, A
(A|X||X|
)> 0( ?). , Z =A|X|
, Z > 0. A(Z|X|
)> Z > 0.
AZZ > Z, 11+AZ >Z, B =1
1+A
, BZ >Z. ,
B2Z = B(BZ) > BZ > Z, . . . , BmZ > Z, (m 1)
. , m(A) = 1, m(B) = 11+ < 1, limmBm = 0
( 3.2 3.5 ). , BmZ >Z limit
, 0 Z. ! , A |X|= |X|=m(A) |X|. , A |X|> 0, A |X|= |X|>
0.
76Hint: .
33
-
7.7 A dominant eigen-value(, ||=m(A)),
() = ||> 0.
() dimEA = multA() = 1.
: (i) A normalize, m(A) = 1 = || ().
(ii) 0 =X = (x1, . . . , xn)t Cn A eigev-vector(, AX = X). , 7.6
, A |X| = |X|> 0 . ,
A |X| = |X| = || |X| = |X| = |AX|
, i () (, 1 i n), i-,j aij |xj | =
j aijxj
. , 3.11,
aijxj = cj z, , xj = zcjaij
(1 j n)
c1, . . . , cn 0 z C . eigen-vector Y =
(c1ai1
, . . . , cnain
)t.77 ( dimEA =1
. Y X depend.78)
(iii) (ii) eigen-value A eigev-vector Y 0 . , , EA vector 0
0 . , dimEA = 1
EA probability vector
( ?). P,Q EA probability vector .
0 = P Q EA , P Q 0, P Q . . dimEA =1.
(iv) (ii), |X|> 0, c1, . . . , cn = 0, Y > 0. ,
Y = AY = |AY | = |Y | = || Y = Y
, = 1= ||.
77 i.78 gap.
34
-
(v) dimEA = multA() , dimEA = multA()
. J = U1AU A Jordan canonical form
, limm
Jm=( 3.2 3.14 ). ,
Jm = U1AmU n2U1UAm
( 3.15()), limm
Am=. Am =(a(m)ij
)
, Am = a(m)imjm . , Y = (y1, . . . , yn)t
, AY = Y ,
yim =
j a(m)imj
yj (
j a(m)imj
)min
k{yk} Ammin
k{yk}
. .
7.2 ( 7.6 7.7) . , A > 0
, A positive dominant eigen-vector probability vector P
, P . P A Perron-Frobenius vector. ,
At > 0, At Perron-Frobenius vector Q(At
dominant eigen-value ).
Perron-Frobenius Theorem.
7.8 X 0 A eigen-value eigen-vector,=m(A). ( X Perron-Frobenius
vector P positive scalar multiple.)
: Q At Perron-Frobenius vector. ,
At Q=m(A)Q, Qt X > 0. , AX = X,
Qt X = Qt (X) = (Qt A)X = m(A)Qt X
, =m(A).
7.9 (Collatz-Wielandt Formula) N = {X Cn |X 0 and X = 0}, f : N
R
f(X) = f((x1, . . . , xn)
t)= min
{AX i-
xi
1 i n, xi =0}, (X N ), m(A) = max{f(X) |X N}.
35
-
version
180102 [1] M. Artin, Algebra, Prentice-Hall, 1991.
[2] S. Brin and L. Page, The anatomy of a large-scale
hypertextual Web search
engine, Computer Networks and ISDN Systems, 30, 107117,
1998.
[3] S. H. Friedberg, A. J. Insel and L. E. Spence, Linear
Algebra, 4th ed.,
Pearson, 2002.
[4] G. H. Golub and C. F. Van Loan, Matrix Computation, 4th ed.,
JHU
Press, 2012.
[5] C. D. Meyer, Matrix Analysis and Applied Linear Algebra,
SIAM, 2000.
[6] L. Page, S. Brin, R. Motwani and T. Winograd, The PageRank
citation
ranking : bringing order to the web, Technical Report, Stanford
InfoLab,
1999.
36