lecture 5ii

軟體實作與計算實驗 1

Markov ChainRoots of nonlinear functionsNested FOR loops

Lecture 5II


State

1S 3S2S


Initial population

in denotes the initial population of being state iS

ni

i n


State transition

A person has a current state, e.g. Si Its next state may be any one of possible

statesState transition from Si to Sj


Transition probabilities

ijp denotes the probability of transition from toiS

1p

conditionUnitary

j

ij

jS


State

1S 3S2S

31p

12p


State transition

Populations after one time of state transition

Pv

mmmmm

m

m

n

nn

ppp

pppppp

2

1

21

22221

11211

, vP


State transition

Populations after one time of state transition

Populations after k times of state transition

vP'

vPP ''...k


Exercise I

Draw a flow chart to calculate populations after k times of state transitions using for-loops

Write a matlab function to implement the flow chart

Find the result forP =

0.1000 0.4500 0.4500 0.3333 0 0.6667 0.5000 0 0.5000

v =

100 600 300

k=1k=10k=20k=100


Convergence

Change

Norm

new

new

vvvPv

'

)( newnorm vv


Convergence

The answer converges if

8

6

10)(

10)(

e.g.number positive small a than less is )(

new

new

new

norm

norm

norm

vv

vv

vv


for i=1:k

v_new=P'*v

norm(v-v_new)<10^-6

Yes:converge

break v=v_new


for i=1:kv_new=P' *v;

if norm(v_new-v)<10^-6 break else v=v_new

endend

vP k)'(lim k


Exercise II

Draw a flow chart to illustrate finding

by a for-loopImplement your flow chart by a matlab

function

vP k)'(lim k


for i=1:k

v_new=P'*v

norm(v-v_new)<10^-6

Yes:converge

break v=v_new

change>10^-6

change=1

v_new=P'*v

change=norm(v-v_new)v=v_new


while-loop

change>10^-6

change=1

v_new=P'*v

change=norm(v-v_new)v=v_new


change = 1while change < 10^-6

v_new=P' *v;change = norm(v-v_new);v=v_new

end


Exercise III

Draw a flow chart to illustrate finding

by a while-loopImplement your flow chart by a matlab

function

vP k)'(lim k


Nonlinear system

004

21

22

21

xxxx


A set of nonlinear functions

21212

22

21211

211

21121

),(4),(

),(),(

),F(

xxxxfxxxxf

xxfxxf

xx


function F = myfun(x) F(1) = x(1).^2 + x(2)^2-4; F(2) = x(1) - x(2);return

Function evaluation


x=linspace(-2,2);plot(x,x); hold on; plot(x,sqrt(4-x.^2))plot(x,-sqrt(4-x.^2))


Least square of nonlinear functions

i

ixxxxf 2

21,),(min

21

0),(04),(

SOLVE

21212

22

21211

xxxxfxxxxf


x0= ones(1,2)*2;x = lsqnonlin(@myfun,x0)

Find a zero

y=myfun(x);sum(y.^2)

CHECKING


Multiple roots

x0= ones(1,2)*2;x = lsqnonlin(@myfun,x0)v=[v;x];plot(x(1),x(2),'o');

v=[]

for i=1:n

demo_lsq.m


012),(

0242),(

SOLVE

22212

22

21211

21

xxxxf

xxxxf


function F = myfun2(x) F(1) = 2*x(1).^2 + x(2)^2-24; F(2) = x(1).^2 - x(2).^2+12;return

Function evaluation

012),(

0242),(22

212

22

21211

21

xxxxf

xxxxf


x0= ones(1,2)*2;x = lsqnonlin(@myfun2,x0)

Find a zero

y=myfun2(x);sum(y.^2)

CHECKING


Multiple roots

x0= ones(1,2)*2;x = lsqnonlin(@myfun2,x0)v=[v;x];plot(x(1),x(2),'o');

v=[]

for i=1:n


x=linspace(-5,5);plot(x,sqrt(24-2*x.^2),'r');hold on; plot(x,-sqrt(24-2*x.^2),'r')plot(x,sqrt(12+x.^2),'b')plot(x,-sqrt(12+x.^2),'b')


Demo_lsq_2c

source code


02),(022),(

SOLVE

21212

22

21211

xxxxfxxxxf


function F = myfun3(x) F(1) = x(1).^2 -2* x(2)^2-2; F(2) = x(1).*x(2) -2;return

Function evaluation

02),(022),(

SOLVE

21212

22

21211

xxxxfxxxxf


x=linspace(-3,3);x=x(find(abs(x) > 0.1));plot(x,sqrt((x.^2-2)/2),'r');hold on; plot(x,-sqrt((x.^2-2)/),'r')x=linspace(0.5,3);plot(x,2./x,'b');x=linspace(-0.5,-3);plot(x,2./x,'b')


x0= ones(1,2)*2;x = lsqnonlin(@myfun3,x0)

Find a zero

y=myfun3(x);sum(y.^2)

CHECKING


Multiple roots

x0= ones(1,2)*2;x = lsqnonlin(@myfun3,x0)v=[v;x];plot(x(1),x(2),'o');

v=[]

for i=1:n


demo_lsq_hyperbola.m

two_hyperbola.m


Matrix Multiplication

C=A*BC(i,j)=A(i,:) *B(:,j) for all i,j


Nested FOR Loops

C(i,j)=A(i,:) *B(:,j)

[n,d1]=size(A);[d2,m]=size(B)

for i=1:n

for j=1:m


[n,d1]=size(A);[d2,m]=size(B);for i=1:nfor j=1:m C(i,j)=A(i, :) *B(:,j);endend


Constrained optimization

13

100102

22

21

12

2

1

xx

xxxx

Inequality constraint

Nonlinear equality


12

2

1

100102

xxxx

222

21,

)13(min21

xxxx

Constrained optimization


Constraints

Lower bound and upper boundEquality constraintInequality constraint


Constraints

Lower bound

Upper bound

],[]0,2[ 21 xx

]10,10[],[ 21 xx

Inequality constraints

0112

1

12

xx

xx


Demo_conop2

function demo_conop2()lb =[-2 0];ub =[10,10];Aeq=[-1 1];beq=[0];A=[];b=[];x = fmincon(@fun,[1 1],A,b,Aeq,beq,lb,ub)x(1).^2+x(2).^2returnfunction y = fun(x)y = (x(1).^2+x(2).^2-13).^2;return

demo_conop2.m


function y = fun(x)y = (x(1).^2+x(2).^2-13).^2;return

Objective function


Calling fmincon

x = fmincon(@fun,[1 1],A,b,Aeq,beq,lb,ub)

objective function initial

guesslinearequality

inequalityLower and upperbound


Random variable

Sample space S={A,T,C,G}X is a discrete random variable with

sample space S.

apAX )Pr(

cpCX )Pr(tpTX )Pr(

gpGX )Pr(


Generative model

pa

pt pc

pg

A T C G

Flip-flop1 gcta pppp

gatcacaggt


Partition

ap0 1cta

ta

a

pppnppn

pn

3

2

1

1n 2n 3n

Knots partition [0,1] into four intervals respectively with lengths,

321 and , nnn

gcta pppp and ,,

tp cp gp


r=rand;Tag= (r<=c(1))+ 2*(r<=c(2) & r > c(1))+3*(r<=c(3) & r > c(2))+4*(r<=c(4) & r > c(3))switch Tagcase 1

s=[s 'A'];case 2

s=[s 'T'];case 3

s=[s 'C'];case 4

s=[s 'G'];end


Sequence generation

Emulation of a generative model for creation of an atcg sequence


FOR Loops

r=rand;switch r…end

input p,mn=length(p)p_sum=0;s=[];

for i=1:n for j=1:m

p_sum=p_sum+p(i);c(i)=p_sum;


Mixture model

pa

pt pc

pg

A T C G

Flip-flop

pa

pt pc

pg

A T C G

Flip-flop1 2


Mixture model

According to probabilities, and each time one of two joined models is

selected to generate a characterCreated characters are collected to form

an atcg sequence

1 2


Hidden Markov model

1 2

pa

pt pc

pg

A T C G

pa

pt pc

pg

A T C G

11 21

22

12


Transition probability

2221

1211

lecture 5ii

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