Post on 17-Apr-2015
EE-240/2009
Proportional Hazards Model EE-240/2009Proportional Hazards
Model
EE-240/2009
Proportional Hazards Model
Proportional Hazards Model
EE-240/2009
Proportional Hazards Model
Informações da População
t0
p (t)
t
0
p (t)
t
t
=1
=2
=n-1
=n
=1
=2
=n-1
=n
T
EE-240/2009
Proportional Hazards Model
Informações da População
t0 0
t
=1
=2
=n-1
=n
=1
=2
=n-1
=n
x1 = T
x2 = fON-OFF
xgtt 0x t0
EE-240/2009
Proportional Hazards Model
2
1
20
10
x
x
xg
xg
xgt
xgt
t
t
2
1
xgtt 0x
Hazard em Proporção Constante para Qualquer t
EE-240/2009
Proportional Hazards Model
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}
EE-240/2009
Proportional Hazards Model
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) =
t
xxtT|ttTtPt j
x j
jx xxtT|ttTtPttj
j
iT
jT
j i Ri
xj0
xj0
Ri jix
jx
et
et
tt
tt
j
iT
jT
Ri
x
x
e
e
j
iT
jT
Ri
x
xk
1j e
eL
EE-240/2009
Proportional Hazards Model
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
140
1 eeeeee
eL
509075
75
2 eee
eL
5090
90
3 ee
eL
321 LLLL
Lmaxargˆ
j
iT
jT
Ri
x
xk
1j e
eL
EE-240/2009
Proportional Hazards Model
-0.05 0 0.05 0.1 0.15 0.20
0.05
0.1
0.15
0.2
0.25
Beta
Pa
rtia
l Lik
elih
oo
d
= 0.068
EE-240/2009
Proportional Hazards Model
Relação com Tempo de Falha Acelerada
EE-240/2009
Proportional Hazards Model
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 00
0xx
xT
e)x(a tdt
dRt x
x
xatxaxatxadt
dRt 0
0x
EE-240/2009
Proportional Hazards Model
Exemplo de Aplicação
Modelo de Riscos Proporcionais
EE-240/2009
Proportional Hazards Model
Exemplo
t0
=1
=2
=n-1
=n
t0 nn11 x...x0x ett
T0x
1ett
0t
=1
=2
=n-1
=n
x1 = T
290
830
580
560
EE-240/2009
Proportional Hazards Model
nn11 x...x0x ett
T0x
1ett
0t
=1
=2
=n-1
=n
x1 = T
290
830
580
560ti tfalha [dias] xi=T
t1 290 140
- >360 100
- >410 105
t2 560 75
t3 580 90
t4 830 50
Ordenado
Exemplo
EE-240/2009
Proportional Hazards Model
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
140
1 eeeeee
eL
509075
75
2 eee
eL
5090
90
3 ee
eL
j
iT
jT
Ri
x
xk
1j e
eL
Dados Censurados
Conjunto sob Risco
EE-240/2009
Proportional Hazards Model
-0.05 0 0.05 0.1 0.15 0.20
0.05
0.1
0.15
0.2
0.25
Beta
Pa
rtia
l Lik
elih
oo
d
= 0.068
Maximização de L(): Método Gráfico
EE-240/2009
Proportional Hazards Model
Exemplo: Componentes sujeitos a Ciclos de Temperatura
EE-240/2009
Proportional Hazards Model
t
Temp
Como contar ciclos?
EE-240/2009
Proportional Hazards Model
Rainflow Counting
t
y
EE-240/2009
Proportional Hazards Model
Rainflow Counting
t
y
EE-240/2009
Proportional Hazards Model
Rainflow Counting
10
EE-240/2009
Proportional Hazards Model
Rainflow Counting
10 5
EE-240/2009
Proportional Hazards Model
Rainflow Counting
10 5 5
EE-240/2009
Proportional Hazards Model
Rainflow Counting
10 5 5
10
EE-240/2009
Proportional Hazards Model
Rainflow Counting
10 5 5
10 7
EE-240/2009
Proportional Hazards Model
Rainflow Counting
10 5 5
10 7 6
EE-240/2009
Proportional Hazards Model
Rainflow Counting
10 5 5
10 7 6 9
EE-240/2009
Proportional Hazards Model
Rainflow Counting
10 5 5
10 7 6 9 9
EE-240/2009
Proportional Hazards Model
Rainflow Counting
10 5 5
10 7 6 9 9
Num ciclos
Amplitudedos Ciclos
1095
ni ciclos de amplitude i observadosNi ciclos de amplitude i até falha
k
i i
i
N
ndegradação
1
Falha esperada se degradação = 1
Miner's Rule
EE-240/2009
Proportional Hazards Model
Exemplo: Transistor sujeito a Ciclos de Temperatura
0t
=1
=2
=n-1
=n
x2 = T
x1 = fciclos
nn11 x...x0x ett
1 = 0.080
2 = 0.002
0(t) = 1.000
EE-240/2009
Proportional Hazards Model
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
EE-240/2009
Proportional Hazards Model
k
1jRi
xx
xx
j
i,22i,11
j,22j,11
e
eL
k
i,22i,11
2
i,22i,11
1
i,22i,11
Ri
xxk,22k,12
Ri
xx2,222,12
Ri
xx1,221,11
elogxx
...
elogxx
elogxx)(Llog
>> x=fminsearch(@neglogpartlikelihood,[0.09 ; 0.002])
x =
0.0892 0.0033
Método da Maximização da Verossimilhança (log)
EE-240/2009
Proportional Hazards Model
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)));
EE-240/2009
Proportional Hazards Model
Método Gráfico
2.9 9.4
2.0 8.0
log L()
beta1beta2 [ x 10-3 ] [ x10-2 ]
EE-240/2009
Proportional Hazards Model
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 }
EE-240/2009
Proportional Hazards Model
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
Proportional Hazards Model
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
Proportional Hazards Model
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
Proportional Hazards Model
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
Proportional Hazards Model
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
Proportional Hazards Model
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
Proportional Hazards Model
4097.0ˆ
Visualização Gráfica
EE-240/2009
Proportional Hazards Model
Muito Obrigado!