數值方法, Applied Mathematics NDHU 1

Linear Recursive Relations:•Data generation•Reconstruction of LRR•Predictable by LRR•Unpredictable by LRR

數值方法, Applied Mathematics NDHU 2

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mb

b

b

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bAx

m1,...,ifor bi xaTi

數值方法, Applied Mathematics NDHU 3

m > n

The number of unknowns is less than the number of linear equations.

m > n: over-determined linear system

數值方法, Applied Mathematics NDHU 4

數值方法, Applied Mathematics NDHU 5

Solving linear systems with m>n

Input paired data Form matrix A and

vector b Set x1 to pinv(A)*b Set x2 to

數值方法, Applied Mathematics NDHU 6

N=30;A=rand(N,3);b=A*[1 0.5 -1]'+rand(N,1)*0.1-0.05;x1=pinv(A)*b;x2=inv(A'*A)*(A'*b);

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Applications

Linear convolution (auto regression)Linear recursive relations

數值方法, Applied Mathematics NDHU 8

Fibonacci

Linear combination of predecessors

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Recurrent Relation

nf

F

2nf

1nf

Given and

the recurrent relation can generate an infinite sequence

0f 1f

n}{ nf

數值方法, Applied Mathematics NDHU 10

Linear recursive relation

nf

F

2nf

1nf

F is linear

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Forward problem : data generation

function F=Fibonacci(N)F(0+1)=0;F(1+1)=1;for i=2:N F(i+1)=F(i-1+1)+F(i-2+1);endplot(1:length(F),F,'o')

Fibonacci.m

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Inverse problem

F[n]=a1F[n-1]+ a2F[n-2]+e, n=2..N

e denotes noiseGiven F[n], n=0,…,N, find a1 and a2

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Linear recursion

F[n]=a1F[n-1]+ a2F[n-2]+ e ,n=2..N

Given F[n], n=0,…,N, find a1 and a2

Linear system:

aa11 F[1]+ aa22 F[0]= F[2]aa11 F[2]+ aa22 F[1]= F[3]aa11 F[3]+ aa22 F[2]= F[4]aa11 F[4]+ aa22 F[3]= F[5]aa11 F[5]+ aa22 F[4]= F[6]...aa11 F[9]+ aa22 F[8]=F[10]

數值方法, Applied Mathematics NDHU 14

Linear system

function [A,b]=formAb_Fib(F)N=length(F);b=F(2+1:N+1)';A=[F(1+1:N-1+1)' F(0+1:N-2+1)'];

formAb_Fib.m

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Linear recursion

f=Fibonacci(30);[A,b]=formAb_Fib(f)x =pinv(A)*b

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Linear recursive relation

Linear combination of predecessorsf[t]=a1f[t-1]+ a2f[t-2]+…+ af[t-]+ e[t],

t= ,…,N

數值方法, Applied Mathematics NDHU 17

Linear recursive relation: delays

nf

F2nf

1nf

F is linear

nf

...

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nf

2nf

1nf

nf

.

.

.

nnnn fafafaf ...2211

Data generation by linear recursive relation

Fgen.m

L=10; N=80;a=pdf('norm', linspace(pi,-pi,L),0,1)-0.2;F=Fgen(a,N);

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Construction of Linear recursive relation

demo_FG.m

Form A and b x =pinv(A)*b

Blue: a1 … a

Red: Estimation

數值方法, Applied Mathematics NDHU 20

Construction of Linear recursive relation

Form A and b x =pinv(A)*b

Blue: a1 … a

Red: Estimation

nf

2nf

1nf

nf

.

.

.

nnnn fafafaf ...2211

數值方法, Applied Mathematics NDHU 21

Reconstruction of linear recursive relation

L=10; N=80;a=pdf('norm', linspace(pi,-pi,L),0,1)-0.2;F=Fgen(a,N);

[A b]=formAb(F,L);a_hat=pinv(A)*b;

Prediction of time series after N

time series before N

1:N

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Prediction of time series after N

Fprediction.m

ini_F=F(N-L+1:N);New_F=Fprediction(a_hat,ini_F,N);plot((1:length(New_F))+N,New_F);

Time series after N

N:2N

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Nonlinear recursive relation

z[t]=tanh(a1z[t-1]+ a2z[t-2]+…+ az[t-])+ e[t],

t= ,…,N

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Nonlinear recursiondemo_FG2.m

Form A and b x =pinv(A)*b

Blue: a1

… a

Red: x1 … x

Estimation

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Unpredictable by linear recursive relation

Data generation by nonlinear recursive relation

Linear recursive relation can not predict time series that are created by nonlinear recursive relation

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Prediction

Use initial -1 instances to generate the full time series (red) based on estimated linear parameters (red)