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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 ] > > 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> > t2t2 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> > t2t2 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

14 EE-240/2009 Proportional Hazards Model 0 t =1 =2 =n-1 =n x 1 = T titi t falha [dias]x i =T t1t > > t2t t3t t4t Ordenado Exemplo

15 EE-240/2009 Proportional Hazards Model titi t falha [dias]xixi RiRi t1t {1,2,3,4,5,6} -> > t2t {2,3,1} t3t {3,1} t4t {1} Dados Censurados Conjunto sob Risco

16 EE-240/2009 Proportional Hazards Model = 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

24 EE-240/2009 Proportional Hazards Model Rainflow Counting

25 EE-240/2009 Proportional Hazards Model Rainflow Counting

26 EE-240/2009 Proportional Hazards Model Rainflow Counting

27 EE-240/2009 Proportional Hazards Model Rainflow Counting

28 EE-240/2009 Proportional Hazards Model Rainflow Counting

29 EE-240/2009 Proportional Hazards Model Rainflow Counting Num ciclos Amplitude dos Ciclos 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 = = (t) = 1.000

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

32 EE-240/2009 Proportional Hazards Model >> ; 0.002]) x = Método da Maximização da Verossimilhança (log)

33 EE-240/2009 Proportional Hazards Model function [L]=neglogpartlikelihood(beta) % Número de componentes N = 30; table = [ ]; % 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 log L( ) beta1 beta2 [ x ] [ x10 -2 ]

35 EE-240/2009 Proportional Hazards Model Exemplo: Rolamento 20 rolamentos em ambiente limpo x = 0: t f = { } 20 rolamentos em ambiente com partículas abrasivas x = 1: t f = { }

36 EE-240/2009 Proportional Hazards Model tftf xtiti didi 1011t1t1 3e 2 /(20+20e ) 3 211t2t2 2e 2 /(19+18e ) t3t3 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 ) t 21 11/(5+3e ) 311t 22 1e /(4+3e ) t 23 11/(4+e ) 400t 24 11/(3+e ) x = 0: t f = { } x = 1: t f = { }

37 EE-240/2009 Proportional Hazards Model tftf xtiti didi 1011t1t1 3e 2 /(20+20e ) 3 211t2t2 2e 2 /(19+18e ) t3t3 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 ) t 21 11/(5+3e ) 311t 22 1e /(4+3e ) t 23 11/(4+e ) 400t 24 11/(3+e ) x = 0: t f = { } x = 1: t f = { }

38 EE-240/2009 Proportional Hazards Model tftf xtiti didi 1011t1t1 3e 2 /(20+20e ) 3 211t2t2 2e 2 /(19+18e ) t3t3 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 ) t 21 11/(5+3e ) 311t 22 1e /(4+3e ) t 23 11/(4+e ) 400t 24 11/(3+e ) x = 0: t f = { } x = 1: t f = { }

39 EE-240/2009 Proportional Hazards Model tftf xtiti didi 1011t1t1 3e 2 /(20+20e ) 3 211t2t2 2e 2 /(19+18e ) t3t3 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 ) t 21 11/(5+3e ) 311t 22 1e /(4+3e ) t 23 11/(4+e ) 400t 24 11/(3+e ) x = 0: t f = { } x = 1: t f = { }

40 EE-240/2009 Proportional Hazards Model tftf xtiti didi 1011t1t1 3e 2 /(20+20e ) 3 211t2t2 2e 2 /(19+18e ) t3t3 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 ) t 21 11/(5+3e ) 311t 22 1e /(4+3e ) t 23 11/(4+e ) 400t 24 11/(3+e ) x = 0: t f = { } x = 1: t f = { }

41 EE-240/2009 Proportional Hazards Model tftf xtiti didi 1011t1t1 3e 2 /(20+20e ) 3 211t2t2 2e 2 /(19+18e ) t3t3 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 ) t 21 11/(5+3e ) 311t 22 1e /(4+3e ) 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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