EE-240/2009 Proportional Hazards Model

EE-240/2009

Proportional Hazards Model EE-240/2009Proportional Hazards

EE-240/2009

Proportional Hazards Model

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Informações da População

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Informações da População

x1 = T

x2 = fON-OFF

xgtt 0x t0

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xgtxgt

xgtt 0x

Hazard em Proporção Constante para Qualquer t

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Risk Set:

Seja t1 < t2 < ... < tk

Ri = conjunto de componentes sobreviventes até ti-

Comp tfalha [dias] Temp [o]

1 830 50

2 560 75

3 580 90

4 >360 100

5 >410 105

6 290 140

Exemplo:

R1 = {1,2,3,4,5,6}R2 = {2,3,4}R3 = {3,4}R4 = {4}

Comp tfalha [dias] ti xi Ri

6 290 t1 140 {1,2,3,4,5,6}

4 >360 100

5 >410 105

2 560 t2 75 {2,3,1}

3 580 t3 90 {3,1}

1 830 t4 50 {1}

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P(componente j falhar em tj | algum componente de Rj falhar em tj) = ?

falhar em t j havia sobrevivido até t

P(componente x = xj falhar em tj | algum componente de Rj falhar em tj) =

P(componente x = xj falhar em tj ) P(algum componente de Rj falhar em tj) =

xxtT|ttTtPt j

jx xxtT|ttTtPttj

j i Rix

Ri jix

1j eeL

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Comp tfalha [dias] ti xi Ri

6 290 t1 140 {1,2,3,4,5,6}

4 >360 100

5 >410 105

2 560 t2 75 {2,3,1}

3 580 t3 90 {3,1}

1 830 t4 50 {1}

509075105100140

1 eeeeeeeL

509075

2 eeeeL

3 eeeL

321 LLLL

Lmaxargˆ

1j eeL

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-0.05 0 0.05 0.1 0.15 0.20

tial L

= 0.068

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Relação com Tempo de Falha Acelerada

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Em modelos com tempo de falha acelerada os covariates x atuamdiretamente sobre a escala de tempo:

0x t)x(at

Tempo de falha ti

associado com xi

t)x(aRt)x(aTPt)x(a

TP)tT(P)t(R 000

xTe)x(a t

dtdRt x

xatxaxatxadt

dRt 00

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Exemplo de AplicaçãoModelo de Riscos Proporcionais

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Exemplo

t0 nn11 x...x0x ett

x1 = T

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nn11 x...x0x ett

x1 = T

560ti tfalha [dias] xi=T

t1 290 140

- >360 100

- >410 105

t2 560 75

t3 580 90

t4 830 50

Ordenado

Exemplo

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ti tfalha [dias] xi Ri

t1 290 140 {1,2,3,4,5,6}

- >360 100 -

- >410 105 -

t2 560 75 {2,3,1}

t3 580 90 {3,1}

t4 830 50 {1}

321 LLLL

Lmaxargˆ

509075105100140

1 eeeeeeeL

509075

2 eeeeL

3 eeeL

1j eeL

Dados Censurados

Conjunto sob Risco

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-0.05 0 0.05 0.1 0.15 0.20

tial L

= 0.068

Maximização de L(): Método Gráfico

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Exemplo: Componentes sujeitos a Ciclos de Temperatura

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Como contar ciclos?

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Rainflow Counting

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Rainflow Counting

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Rainflow Counting

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Rainflow Counting

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Rainflow Counting

10 5 5

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Rainflow Counting

10 5 5

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Rainflow Counting

10 5 5

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Rainflow Counting

10 5 5

10 7 6

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Rainflow Counting

10 5 5

10 7 6 9

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Rainflow Counting

10 5 5

10 7 6 9 9

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Rainflow Counting

10 5 5

10 7 6 9 9

Num ciclos

Amplitudedos Ciclos

ni ciclos de amplitude i observadosNi ciclos de amplitude i até falha

degradação1

Falha esperada se degradação = 1

Miner's Rule

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Exemplo: Transistor sujeito a Ciclos de Temperatura

x2 = T

x1 = fciclos

nn11 x...x0x ett

1 = 0.080

2 = 0.002

0(t) = 1.000

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x1 x2 tf

0.7944 12.3874 0.5682

7.7666 10.4694 0.5849

6.8524 97.3821 0.6050

2.6404 99.4168 0.6327

6.4964 4.1755 0.6453

3.7506 64.4742 0.7492

3.6956 38.3876 0.7518

4.0088 34.0689 0.7793

0.9710 39.1862 0.8369

1.7397 92.1909 0.9740

7.8625 86.9611 1.1034

8.9069 23.8137 1.1754

1.9242 23.3173 1.2395

5.2769 34.1514 1.6436

0.2712 32.0848 2.7048

x1 x2 tf

4.2306 89.2317 0.0048

6.8886 72.0119 0.0315

2.6111 38.8399 0.0813

6.8938 76.0953 0.0956

7.9026 33.1328 0.1225

2.0328 6.8882 0.1869

5.1552 89.4266 0.2455

3.0149 18.0024 0.2530

4.3336 9.8591 0.2664

3.6429 61.5732 0.3403

5.9924 20.8155 0.3917

7.6678 24.4291 0.4065

4.8371 84.4551 0.4382

0.0822 58.1682 0.5185

1.5801 74.4378 0.5431

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i,22i,11

j,22j,11

i,22i,11

xxk,22k,12

xx2,222,12

xx1,221,11

elogxx

elogxx)(Llog

>> x=fminsearch(@neglogpartlikelihood,[0.09 ; 0.002])

0.0892 0.0033

Método da Maximização da Verossimilhança (log)

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function [L]=neglogpartlikelihood(beta)

% Número de componentesN = 30;

table = [2.2974 49.5063 2.6412 6.4795 40.0443 0.1454 ... 3.0467 81.7767 0.1391];

% Ordenar por instantes de falhatables=sortrows(table,3);

% Calcula Somatoria de beta’*xbex=0;for kk=1:Nbex=bex + beta(1)*tables(kk,1)+beta(2)*tables(kk,2);end

% Calcular Somatoria de beta’*x para R(kk)somat(N+1)=0.;for kk=N:-1:1somat(kk)=somat(kk+1)+ exp(beta(1)*tables(kk,1)+beta(2)*tables(kk,2));end

L = -bex + sum(log(somat(1:N)));

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Método Gráfico

2.9 9.42.0 8.0

log L()

beta1beta2 [ x 10-3 ] [ x10-2 ]

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Exemplo: Rolamento

20 rolamentos em ambiente limpo x = 0:

tf = { 1 3 3 6 7 7 10 12 14 15 18 19 22 26 28+ 29 34 40 48+ 49+ }

20 rolamentos em ambiente com partículas abrasivas x = 1:

tf = { 1 1 2 2 3 4 5 8 8 9 11 12 14 16 18 21 27+ 31 38+ 44 }

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tf x ti di

1 011 t1 3 e2/(20+20e)3

2 11 t2 2 e2/(19+18e)2

3 001 t3 3 e/(19+16e)3

4 1 t4 1 e/(17+15e)

5 1 t5 1 e/(17+14e)

6 0 t6 1 1/(17+13e)

7 00 t7 2 1/(16+13e)2

8 11 t8 2 e2/(14+13e)2

9 1 t9 1 e/(14+11e)

10 0 t10 1 1/(14+10e)

11 1 t11 1 e/(13+10e)

12 10 t12 2 e/(13+9e)2

tf x ti di

14 10 t13 2 e/(12+8e)2

15 0 t14 1 1/(11+7e)

16 1 t15 1 e/(10+7e)

18 10 t16 2 e/(10+6e)2

19 0 t17 1 1/(9+5e)

21 1 t18 1 e/(8+5e)

22 0 t19 1 1/(8+4e)

26 0 t20 1 1/(7+4e)

29 00+1+ t21 1 1/(5+3e)

31 1 t22 1 e/(4+3e)

34 01+ t23 1 1/(4+e)

40 0 t24 1 1/(3+e)

x = 0: tf = { 1 3 3 6 7 7 10 12 14 15 18 19 22 26 28+ 29 34 40 48+ 49+ }

x = 1: tf = { 1 1 2 2 3 4 5 8 8 9 11 12 14 16 18 21 27+ 31 38+ 44 }

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tf x ti di

1 011 t1 3 e2/(20+20e)3

2 11 t2 2 e2/(19+18e)2

3 001 t3 3 e/(19+16e)3

4 1 t4 1 e/(17+15e)

5 1 t5 1 e/(17+14e)

6 0 t6 1 1/(17+13e)

7 00 t7 2 1/(16+13e)2

8 11 t8 2 e2/(14+13e)2

9 1 t9 1 e/(14+11e)

10 0 t10 1 1/(14+10e)

11 1 t11 1 e/(13+10e)

12 10 t12 2 e/(13+9e)2

tf x ti di

14 10 t13 2 e/(12+8e)2

15 0 t14 1 1/(11+7e)

16 1 t15 1 e/(10+7e)

18 10 t16 2 e/(10+6e)2

19 0 t17 1 1/(9+5e)

21 1 t18 1 e/(8+5e)

22 0 t19 1 1/(8+4e)

26 0 t20 1 1/(7+4e)

29 00+1+ t21 1 1/(5+3e)

31 1 t22 1 e/(4+3e)

34 01+ t23 1 1/(4+e)

40 0 t24 1 1/(3+e)

x = 0: tf = { 1 3 3 6 7 7 10 12 14 15 18 19 22 26 28+ 29 34 40 48+ 49+ }

x = 1: tf = { 1 1 2 2 3 4 5 8 8 9 11 12 14 16 18 21 27+ 31 38+ 44 }

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tf x ti di

1 011 t1 3 e2/(20+20e)3

2 11 t2 2 e2/(19+18e)2

3 001 t3 3 e/(19+16e)3

4 1 t4 1 e/(17+15e)

5 1 t5 1 e/(17+14e)

6 0 t6 1 1/(17+13e)

7 00 t7 2 1/(16+13e)2

8 11 t8 2 e2/(14+13e)2

9 1 t9 1 e/(14+11e)

10 0 t10 1 1/(14+10e)

11 1 t11 1 e/(13+10e)

12 10 t12 2 e/(13+9e)2

tf x ti di

14 10 t13 2 e/(12+8e)2

15 0 t14 1 1/(11+7e)

16 1 t15 1 e/(10+7e)

18 10 t16 2 e/(10+6e)2

19 0 t17 1 1/(9+5e)

21 1 t18 1 e/(8+5e)

22 0 t19 1 1/(8+4e)

26 0 t20 1 1/(7+4e)

29 00+1+ t21 1 1/(5+3e)

31 1 t22 1 e/(4+3e)

34 01+ t23 1 1/(4+e)

40 0 t24 1 1/(3+e)

x = 0: tf = { 1 3 3 6 7 7 10 12 14 15 18 19 22 26 28+ 29 34 40 48+ 49+ }

x = 1: tf = { 1 1 2 2 3 4 5 8 8 9 11 12 14 16 18 21 27+ 31 38+ 44 }

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tf x ti di

1 011 t1 3 e2/(20+20e)3

2 11 t2 2 e2/(19+18e)2

3 001 t3 3 e/(19+16e)3

4 1 t4 1 e/(17+15e)

5 1 t5 1 e/(17+14e)

6 0 t6 1 1/(17+13e)

7 00 t7 2 1/(16+13e)2

8 11 t8 2 e2/(14+13e)2

9 1 t9 1 e/(14+11e)

10 0 t10 1 1/(14+10e)

11 1 t11 1 e/(13+10e)

12 10 t12 2 e/(13+9e)2

tf x ti di

14 10 t13 2 e/(12+8e)2

15 0 t14 1 1/(11+7e)

16 1 t15 1 e/(10+7e)

18 10 t16 2 e/(10+6e)2

19 0 t17 1 1/(9+5e)

21 1 t18 1 e/(8+5e)

22 0 t19 1 1/(8+4e)

26 0 t20 1 1/(7+4e)

29 00+1+ t21 1 1/(5+3e)

31 1 t22 1 e/(4+3e)

34 01+ t23 1 1/(4+e)

40 0 t24 1 1/(3+e)

x = 0: tf = { 1 3 3 6 7 7 10 12 14 15 18 19 22 26 28+ 29 34 40 48+ 49+ }

x = 1: tf = { 1 1 2 2 3 4 5 8 8 9 11 12 14 16 18 21 27+ 31 38+ 44 }

EE-240/2009

tf x ti di

1 011 t1 3 e2/(20+20e)3

2 11 t2 2 e2/(19+18e)2

3 001 t3 3 e/(19+16e)3

4 1 t4 1 e/(17+15e)

5 1 t5 1 e/(17+14e)

6 0 t6 1 1/(17+13e)

7 00 t7 2 1/(16+13e)2

8 11 t8 2 e2/(14+13e)2

9 1 t9 1 e/(14+11e)

10 0 t10 1 1/(14+10e)

11 1 t11 1 e/(13+10e)

12 10 t12 2 e/(13+9e)2

tf x ti di

14 10 t13 2 e/(12+8e)2

15 0 t14 1 1/(11+7e)

16 1 t15 1 e/(10+7e)

18 10 t16 2 e/(10+6e)2

19 0 t17 1 1/(9+5e)

21 1 t18 1 e/(8+5e)

22 0 t19 1 1/(8+4e)

26 0 t20 1 1/(7+4e)

29 00+1+ t21 1 1/(5+3e)

31 1 t22 1 e/(4+3e)

34 01+ t23 1 1/(4+e)

40 0 t24 1 1/(3+e)

x = 0: tf = { 1 3 3 6 7 7 10 12 14 15 18 19 22 26 28+ 29 34 40 48+ 49+ }

x = 1: tf = { 1 1 2 2 3 4 5 8 8 9 11 12 14 16 18 21 27+ 31 38+ 44 }

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tf x ti di

1 011 t1 3 e2/(20+20e)3

2 11 t2 2 e2/(19+18e)2

3 001 t3 3 e/(19+16e)3

4 1 t4 1 e/(17+15e)

5 1 t5 1 e/(17+14e)

6 0 t6 1 1/(17+13e)

7 00 t7 2 1/(16+13e)2

8 11 t8 2 e2/(14+13e)2

9 1 t9 1 e/(14+11e)

10 0 t10 1 1/(14+10e)

11 1 t11 1 e/(13+10e)

12 10 t12 2 e/(13+9e)2

tf x ti di

14 10 t13 2 e/(12+8e)2

15 0 t14 1 1/(11+7e)

16 1 t15 1 e/(10+7e)

18 10 t16 2 e/(10+6e)2

19 0 t17 1 1/(9+5e)

21 1 t18 1 e/(8+5e)

22 0 t19 1 1/(8+4e)

26 0 t20 1 1/(7+4e)

29 00+1+ t21 1 1/(5+3e)

31 1 t22 1 e/(4+3e)

34 01+ t23 1 1/(4+e)

40 0 t24 1 1/(3+e)

4097.0ˆ

EE-240/2009

4097.0ˆ

Visualização Gráfica

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EE-240/2009 Proportional Hazards Model

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