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EE-240/2009 Proportional Hazards Model EE-240/2009 Proportional Hazards Model.

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Apresentação em tema: "EE-240/2009 Proportional Hazards Model EE-240/2009 Proportional Hazards Model."— Transcrição da apresentação:

1 EE-240/2009 Proportional Hazards Model EE-240/2009 Proportional Hazards Model

2 EE-240/2009 Proportional Hazards Model

3 EE-240/2009 Proportional Hazards Model Informações da População t 0 p (t) t 0 t t =1 =2 =n-1 =n =1 =2 =n-1 =n T

4 EE-240/2009 Proportional Hazards Model Informações da População t 0 0 t =1 =2 =n-1 =n =1 =2 =n-1 =n x 1 = T x 2 = f ON-OFF

5 EE-240/2009 Proportional Hazards Model Hazard em Proporção Constante para Qualquer t

6 EE-240/2009 Proportional Hazards Model Risk Set: Seja t 1 < t 2 <... < t k R i = conjunto de componentes sobreviventes até t i - Compt falha [dias]Temp [ o ] 183050 256075 358090 4>360100 5>410105 6290140 Exemplo: R 1 = {1,2,3,4,5,6} R 2 = {2,3,4} R 3 = {3,4} R 4 = {4} Compt falha [dias]titi xixi RiRi 6290t1t1 140{1,2,3,4,5,6} 4>360 100 5>410 105 2560t2t2 75{2,3,1} 3580t3t3 90{3,1} 1830t4t4 50{1}

7 EE-240/2009 Proportional Hazards Model P(componente j falhar em t j | algum componente de R j falhar em t j ) = ? falhar em t j havia sobrevivido até t P(componente x = x j falhar em t j | algum componente de R j falhar em t j ) = P(componente x = x j falhar em t j ) P(algum componente de R j falhar em t j ) =

8 EE-240/2009 Proportional Hazards Model Compt falha [dias]titi xixi RiRi 6290t1t1 140{1,2,3,4,5,6} 4>360 100 5>410 105 2560t2t2 75{2,3,1} 3580t3t3 90{3,1} 1830t4t4 50{1}

9 EE-240/2009 Proportional Hazards Model = 0.068

10 EE-240/2009 Proportional Hazards Model Relação com Tempo de Falha Acelerada

11 EE-240/2009 Proportional Hazards Model Em modelos com tempo de falha acelerada os covariates x atuam diretamente sobre a escala de tempo: Tempo de falha t i associado com x i xa t xaxatxa dt dR t 0 0 x

12 EE-240/2009 Proportional Hazards Model Exemplo de Aplicação Modelo de Riscos Proporcionais

13 EE-240/2009 Proportional Hazards Model Exemplo t 0 =1 =2 =n-1 =n 0 t =1 =2 =n-1 =n x 1 = T 290 830 580 560

14 EE-240/2009 Proportional Hazards Model 0 t =1 =2 =n-1 =n x 1 = T 290 830 580 560 titi t falha [dias]x i =T t1t1 290140 ->360100 ->410105 t2t2 56075 t3t3 58090 t4t4 83050 Ordenado Exemplo

15 EE-240/2009 Proportional Hazards Model titi t falha [dias]xixi RiRi t1t1 290140{1,2,3,4,5,6} ->360100- ->410105- t2t2 56075{2,3,1} t3t3 58090{3,1} t4t4 83050{1} Dados Censurados Conjunto sob Risco

16 EE-240/2009 Proportional Hazards Model = 0.068 Maximização de L( ): Método Gráfico

17 EE-240/2009 Proportional Hazards Model Exemplo: Componentes sujeitos a Ciclos de Temperatura

18 EE-240/2009 Proportional Hazards Model t Temp Como contar ciclos?

19 EE-240/2009 Proportional Hazards Model Rainflow Counting t y

20 EE-240/2009 Proportional Hazards Model Rainflow Counting t y

21 EE-240/2009 Proportional Hazards Model Rainflow Counting 10

22 EE-240/2009 Proportional Hazards Model Rainflow Counting 10 5

23 EE-240/2009 Proportional Hazards Model Rainflow Counting 10 5 5

24 EE-240/2009 Proportional Hazards Model Rainflow Counting 10 5 5

25 EE-240/2009 Proportional Hazards Model Rainflow Counting 10 5 5 7

26 EE-240/2009 Proportional Hazards Model Rainflow Counting 10 5 5 7 6

27 EE-240/2009 Proportional Hazards Model Rainflow Counting 10 5 5 7 6 9

28 EE-240/2009 Proportional Hazards Model Rainflow Counting 10 5 5 7 6 9 9

29 EE-240/2009 Proportional Hazards Model Rainflow Counting 10 5 5 7 6 9 9 Num ciclos Amplitude dos Ciclos 10 9 5 n i ciclos de amplitude i observados N i ciclos de amplitude i até falha Falha esperada se degradação = 1 Miner's Rule

30 EE-240/2009 Proportional Hazards Model Exemplo: Transistor sujeito a Ciclos de Temperatura 0 t =1 =2 =n-1 =n x 2 = T x 1 = f ciclos 1 = 0.080 2 = 0.002 0 (t) = 1.000

31 EE-240/2009 Proportional Hazards Model x 1 x 2 t f 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 x 1 x 2 t f 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

32 EE-240/2009 Proportional Hazards Model >> x=fminsearch(@neglogpartlikelihood,[0.09 ; 0.002]) x = 0.0892 0.0033 Método da Maximização da Verossimilhança (log)

33 EE-240/2009 Proportional Hazards Model function [L]=neglogpartlikelihood(beta) % Número de componentes N = 30; table = [2.2974 49.5063 2.6412 6.4795 40.0443 0.1454... 3.0467 81.7767 0.1391]; % Ordenar por instantes de falha tables=sortrows(table,3); % Calcula Somatoria de beta*x bex=0; for kk=1:N bex=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:-1:1 somat(kk)=somat(kk+1)+ exp(beta(1)*tables(kk,1)+beta(2)*tables(kk,2)); end L = -bex + sum(log(somat(1:N)));

34 EE-240/2009 Proportional Hazards Model Método Gráfico 2.9 9.4 2.0 8.0 log L( ) beta1 beta2 [ x 10 -3 ] [ x10 -2 ]

35 EE-240/2009 Proportional Hazards Model Exemplo: Rolamento 20 rolamentos em ambiente limpo x = 0: t f = { 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: t f = { 1 1 2 2 3 4 5 8 8 9 11 12 14 16 18 21 27+ 31 38+ 44 }

36 EE-240/2009 Proportional Hazards Model tftf xtiti didi 1011t1t1 3e 2 /(20+20e ) 3 211t2t2 2e 2 /(19+18e ) 2 3001t3t3 3e /(19+16e ) 3 41t4t4 1e /(17+15e ) 51t5t5 1e /(17+14e ) 60t6t6 11/(17+13e ) 700t7t7 21/(16+13e ) 2 811t8t8 2e 2 /(14+13e ) 2 91t9t9 1e /(14+11e ) 100t 10 11/(14+10e ) 111t 11 1e /(13+10e ) 1210t 12 2e /(13+9e ) 2 tftf xtiti didi 1410t 13 2e /(12+8e ) 2 150t 14 11/(11+7e ) 161t 15 1e /(10+7e ) 1810t 16 2e /(10+6e ) 2 190t 17 11/(9+5e ) 211t 18 1e /(8+5e ) 220t 19 11/(8+4e ) 260t 20 11/(7+4e ) 2900 + 1 + t 21 11/(5+3e ) 311t 22 1e /(4+3e ) 3401 + t 23 11/(4+e ) 400t 24 11/(3+e ) x = 0: t f = { 1 3 3 6 7 7 10 12 14 15 18 19 22 26 28+ 29 34 40 48+ 49+ } x = 1: t f = { 1 1 2 2 3 4 5 8 8 9 11 12 14 16 18 21 27+ 31 38+ 44 }

37 EE-240/2009 Proportional Hazards Model tftf xtiti didi 1011t1t1 3e 2 /(20+20e ) 3 211t2t2 2e 2 /(19+18e ) 2 3001t3t3 3e /(19+16e ) 3 41t4t4 1e /(17+15e ) 51t5t5 1e /(17+14e ) 60t6t6 11/(17+13e ) 700t7t7 21/(16+13e ) 2 811t8t8 2e 2 /(14+13e ) 2 91t9t9 1e /(14+11e ) 100t 10 11/(14+10e ) 111t 11 1e /(13+10e ) 1210t 12 2e /(13+9e ) 2 tftf xtiti didi 1410t 13 2e /(12+8e ) 2 150t 14 11/(11+7e ) 161t 15 1e /(10+7e ) 1810t 16 2e /(10+6e ) 2 190t 17 11/(9+5e ) 211t 18 1e /(8+5e ) 220t 19 11/(8+4e ) 260t 20 11/(7+4e ) 2900 + 1 + t 21 11/(5+3e ) 311t 22 1e /(4+3e ) 3401 + t 23 11/(4+e ) 400t 24 11/(3+e ) x = 0: t f = { 1 3 3 6 7 7 10 12 14 15 18 19 22 26 28+ 29 34 40 48+ 49+ } x = 1: t f = { 1 1 2 2 3 4 5 8 8 9 11 12 14 16 18 21 27+ 31 38+ 44 }

38 EE-240/2009 Proportional Hazards Model tftf xtiti didi 1011t1t1 3e 2 /(20+20e ) 3 211t2t2 2e 2 /(19+18e ) 2 3001t3t3 3e /(19+16e ) 3 41t4t4 1e /(17+15e ) 51t5t5 1e /(17+14e ) 60t6t6 11/(17+13e ) 700t7t7 21/(16+13e ) 2 811t8t8 2e 2 /(14+13e ) 2 91t9t9 1e /(14+11e ) 100t 10 11/(14+10e ) 111t 11 1e /(13+10e ) 1210t 12 2e /(13+9e ) 2 tftf xtiti didi 1410t 13 2e /(12+8e ) 2 150t 14 11/(11+7e ) 161t 15 1e /(10+7e ) 1810t 16 2e /(10+6e ) 2 190t 17 11/(9+5e ) 211t 18 1e /(8+5e ) 220t 19 11/(8+4e ) 260t 20 11/(7+4e ) 2900 + 1 + t 21 11/(5+3e ) 311t 22 1e /(4+3e ) 3401 + t 23 11/(4+e ) 400t 24 11/(3+e ) x = 0: t f = { 1 3 3 6 7 7 10 12 14 15 18 19 22 26 28+ 29 34 40 48+ 49+ } x = 1: t f = { 1 1 2 2 3 4 5 8 8 9 11 12 14 16 18 21 27+ 31 38+ 44 }

39 EE-240/2009 Proportional Hazards Model tftf xtiti didi 1011t1t1 3e 2 /(20+20e ) 3 211t2t2 2e 2 /(19+18e ) 2 3001t3t3 3e /(19+16e ) 3 41t4t4 1e /(17+15e ) 51t5t5 1e /(17+14e ) 60t6t6 11/(17+13e ) 700t7t7 21/(16+13e ) 2 811t8t8 2e 2 /(14+13e ) 2 91t9t9 1e /(14+11e ) 100t 10 11/(14+10e ) 111t 11 1e /(13+10e ) 1210t 12 2e /(13+9e ) 2 tftf xtiti didi 1410t 13 2e /(12+8e ) 2 150t 14 11/(11+7e ) 161t 15 1e /(10+7e ) 1810t 16 2e /(10+6e ) 2 190t 17 11/(9+5e ) 211t 18 1e /(8+5e ) 220t 19 11/(8+4e ) 260t 20 11/(7+4e ) 2900 + 1 + t 21 11/(5+3e ) 311t 22 1e /(4+3e ) 3401 + t 23 11/(4+e ) 400t 24 11/(3+e ) x = 0: t f = { 1 3 3 6 7 7 10 12 14 15 18 19 22 26 28+ 29 34 40 48+ 49+ } x = 1: t f = { 1 1 2 2 3 4 5 8 8 9 11 12 14 16 18 21 27+ 31 38+ 44 }

40 EE-240/2009 Proportional Hazards Model tftf xtiti didi 1011t1t1 3e 2 /(20+20e ) 3 211t2t2 2e 2 /(19+18e ) 2 3001t3t3 3e /(19+16e ) 3 41t4t4 1e /(17+15e ) 51t5t5 1e /(17+14e ) 60t6t6 11/(17+13e ) 700t7t7 21/(16+13e ) 2 811t8t8 2e 2 /(14+13e ) 2 91t9t9 1e /(14+11e ) 100t 10 11/(14+10e ) 111t 11 1e /(13+10e ) 1210t 12 2e /(13+9e ) 2 tftf xtiti didi 1410t 13 2e /(12+8e ) 2 150t 14 11/(11+7e ) 161t 15 1e /(10+7e ) 1810t 16 2e /(10+6e ) 2 190t 17 11/(9+5e ) 211t 18 1e /(8+5e ) 220t 19 11/(8+4e ) 260t 20 11/(7+4e ) 2900 + 1 + t 21 11/(5+3e ) 311t 22 1e /(4+3e ) 3401 + t 23 11/(4+e ) 400t 24 11/(3+e ) x = 0: t f = { 1 3 3 6 7 7 10 12 14 15 18 19 22 26 28+ 29 34 40 48+ 49+ } x = 1: t f = { 1 1 2 2 3 4 5 8 8 9 11 12 14 16 18 21 27+ 31 38+ 44 }

41 EE-240/2009 Proportional Hazards Model tftf xtiti didi 1011t1t1 3e 2 /(20+20e ) 3 211t2t2 2e 2 /(19+18e ) 2 3001t3t3 3e /(19+16e ) 3 41t4t4 1e /(17+15e ) 51t5t5 1e /(17+14e ) 60t6t6 11/(17+13e ) 700t7t7 21/(16+13e ) 2 811t8t8 2e 2 /(14+13e ) 2 91t9t9 1e /(14+11e ) 100t 10 11/(14+10e ) 111t 11 1e /(13+10e ) 1210t 12 2e /(13+9e ) 2 tftf xtiti didi 1410t 13 2e /(12+8e ) 2 150t 14 11/(11+7e ) 161t 15 1e /(10+7e ) 1810t 16 2e /(10+6e ) 2 190t 17 11/(9+5e ) 211t 18 1e /(8+5e ) 220t 19 11/(8+4e ) 260t 20 11/(7+4e ) 2900 + 1 + t 21 11/(5+3e ) 311t 22 1e /(4+3e ) 3401 + t 23 11/(4+e ) 400t 24 11/(3+e )

42 EE-240/2009 Proportional Hazards Model Visualização Gráfica

43 EE-240/2009 Proportional Hazards Model Muito Obrigado!


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