• A. J. Clark School of Engineering •Department of Civil and Environmental Engineering CHAPTER 4c CHAPMAN HALL/CRC Risk Analysis in Engineering and Economics Risk Analysis for Engineering Department of Civil and Environmental Engineering University of Maryland, College Park RELIABILITY ASSESSMENT CHAPTER 4c. RELIABILITY ASSESSMENT Slide No. 1 Bayesian Methods ̈ Estimating Binomial Distribution – The function p(t) is the time to failure cumulative distribution function, whereas (1 - p(t)) is the reliability or survivor function. – An estimate of the failure probability, p, is which is also the maximum likelihood estimate n r p = ˆ CHAPTER 4c. RELIABILITY ASSESSMENT Slide No. 2 Bayesian Methods ̈ Estimating Binomial Distribution (cont’d) – In order to obtain the Bayesian estimate for the probability p, a binomial test, in which the number of units n placed tested is fixed in advance, is considered. – The probability distribution of the number, r, of failed units during the test is given by the binomial distribution probability density function with parameters n and r as follows: ) p - ( p r r - n n = p) n, f(r; r-nr 1 !)!( ! (73) CHAPTER 4c. RELIABILITY ASSESSMENT Slide No. 3 Bayesian Methods ̈ Estimating Binomial Distribution (cont’d) – The corresponding likelihood function is given by where c is a constant which does not depend on the parameter of interest, p, and can be assigned a value of one since the constant c drops out from the posterior prediction equation. Where f = binomial probability mass function r = random variable, n and p =binomial distribution parameters. p - p c = r)|l(p rnr − )1( (74) CHAPTER 4c. RELIABILITY ASSESSMENT Slide No. 4 Bayesian Methods ̈ Estimating Binomial Distribution (cont’d) – For any continuous prior distribution of parameter p with probability density function h(p), the corresponding posterior probability density function can be written as dpphp - p php - p = r)|f(p rnr - rn r )()1( )()1( − ∞ ∞ − ∫ (75) CHAPTER 4c. RELIABILITY ASSESSMENT Slide No. 5 Bayesian Methods ̈ Estimating Binomial Distribution (cont’d) – In order to better understand the difference between statistical inference and Bayes’ estimation, the following case of the uniform prior distribution is discussed. – The prior distribution in this case is the standard uniform distribution, which is given by:    ≤≤ otherwise p = ph 0 101 )( (76) CHAPTER 4c. RELIABILITY ASSESSMENT Slide No. 6 Bayesian Methods ̈ Estimating Binomial Distribution (cont’d) – Based on Eq. 75, the respective posterior distribution can be written as – The posterior probability density function of 77 is the probability density function of the beta distribution. dp - pp - pp f(p|r) = rnr rnr ∫ −+−−+ −+−−+ 1 0 1)1(1)1( 1)1(1)1( )1( )1( (77) CHAPTER 4c. RELIABILITY ASSESSMENT Slide No. 7 Bayesian Methods ̈ Estimating Binomial Distribution (cont’d) – The mean value of this distribution, which is the Bayes’ estimate of interest p posterior is given by 2 1 + n + r = p posterior (78) CHAPTER 4c. RELIABILITY ASSESSMENT Slide No. 8 Bayesian Methods ̈ Parameter Estimation for the Exponential Distribution – A sample of n failure times from the exponential distribution, among which only r are distinct times to failure t 1 < t 2 < . . . < t r , and n - r times to censoring t c1 , t c2 , . . . , t c(n-r) , so that the so-called total time on test, T, is given by tt = T ci r - n 1=i i r 1=i + ∑∑ (88) CHAPTER 4c. RELIABILITY ASSESSMENT Slide No. 9 Bayesian Methods ̈ Parameter Estimation for the Exponential Distribution (cont’d) – Using the gamma distribution as the prior distribution of parameter , it is convenient to write the probability density of gamma distribution as a function of  in the following form: where the parameters e 1 = ,;h 1 ρλδ δ λ ρ δ ρδλ )( )( Γ (89) 0and00 , , > ≥≥ δ ρ λ CHAPTER 4c. RELIABILITY ASSESSMENT Slide No. 10 Bayesian Methods ̈ Parameter Estimation for the Exponential Distribution (cont’d) – These parameters can be interpreted as having δ fictitious failures in p total time leading to λ = δ /p. – For the time being these parameters are assumed known. – Also, it is assumed that the quadratic loss function of Eq. 70 is used. CHAPTER 4c. RELIABILITY ASSESSMENT Slide No. 11 Bayesian Methods ̈ Parameter Estimation for the Exponential Distribution (cont’d) – For the exponential time-to-failure data, the likelihood function can be written as )()()()()()()( ,2,1,21 rncccr tRtRtRtftftft|l − = LL λ (90a) Where f(t i ) = probability density function at time to failure t i R(t c,i ) = the reliability value at the time to censoring t c,i . CHAPTER 4c. RELIABILITY ASSESSMENT Slide No. 12 Bayesian Methods ̈ Parameter Estimation for the Exponential Distribution (cont’d) – Therefore, the following likelihood function can be obtained: where T is the total time on test as given by Eq. 88. Tr t - r-n 1=j t - r 1=i e ee = t|l cj i λ λ λ λ λλ − = ∏∏ )( (90b) CHAPTER 4c. RELIABILITY ASSESSMENT Slide No. 13 Bayesian Methods ̈ Parameter Estimation for the Exponential Distribution (cont’d) – Using the Bayes’ theorem with the prior distribution given by Eq. 89 and the likelihood function of Eq. 90, one can find the posterior density function of the parameter, , as: λ λ λ λ ρλ δ δρλ d e e = T|f +(T- -1+r 0 -1+r+(T- ) ) )( ∫ ∞ (91) CHAPTER 4c. RELIABILITY ASSESSMENT Slide No. 14 Bayesian Methods ̈ Parameter Estimation for the Exponential Distribution (cont’d) – Recalling the definition of the gamma function of Eq. 80, the integral in the denominator of Eq. 91 is or e r + T + = T|f +T 1+r r )( )( )( )( ρλδ δ λ δ ρ λ Γ + T + r + = d e r +(T- -1+r 0 + ∞ Γ ∫ δ ρλ δ ρ δ λ λ )( )( ) CHAPTER 4c. RELIABILITY ASSESSMENT Slide No. 15 Bayesian Methods ̈ Parameter Estimation for the Exponential Distribution (cont’d) – Finally, the posterior probability density function of  can be written as – Comparing the above function with the prior one of Eq. 89 reveals that the posterior distribution is also a gamma distribution with parameters e r + T + = T|f +T-1-+r r )( )( )( )( ρλδ δ λ δ ρ λ Γ + (92) ρ λ δ ρ + = ′ + = ′ Tr and CHAPTER 4c. RELIABILITY ASSESSMENT Slide No. 16 Bayesian Methods ̈ Parameter Estimation for the Exponential Distribution (cont’d) – Since a quadratic loss function is assumed, the point Bayesian estimate of  is the mean of the posterior gamma distribution with parameters . – Therefore, the point Bayesian estimate,  posterior , can be obtained as λ ρ ′ ′ and ρ δ λ ρ λ + T + r = = posterior ′ ′ (93) CHAPTER 4c. RELIABILITY ASSESSMENT Slide No. 17 Bayesian Methods ̈ Parameter Estimation for the Exponential Distribution (cont’d) – The corresponding probability intervals can be obtained using Eq. 72. For example, the 100(1 - ) level upper one-sided Bayes’ probability interval for  can be obtained from the following equation based on the posterior distribution Eq. 92: - = < Pr u α λ λ 1)( (94) CHAPTER 4c. RELIABILITY ASSESSMENT Slide No. 18 Bayesian Methods ̈ Parameter Estimation for the Exponential Distribution (cont’d) – The same upper one-sided probability interval for  can be expressed in a more convenient form similar to the classical confidence interval, i.e., in terms of the chi-square distribution, as follows: such that ( ) αδ χ ρλ α - = r < T + 2Pr - 1)(2)( 2 1 + (95) T + = 2 r + - u )(2 )(2,1 ρ χ λ δα (96) CHAPTER 4c. RELIABILITY ASSESSMENT Slide No. 19 Bayesian Methods ̈ Parameter Estimation for the Exponential Distribution (cont’d) – Contrary to classical estimation, the number of degrees of freedom, 2( + r), for the Bayes’ probability limits is not necessarily integer. – The chi-square value in Eq. 96 can be obtained from tables of the chi-square probability distribution available in probability and statistics textbooks, such as Ayyub and McCuen (2003). [...]... and Cumulative Hazard Rate Functions for a Series System of Three Identical Components of Example 21 Year Time to Failure, Years 1980 43 20 07 2008 2009 2010 70 71 72 73 Hazard Rate Function 0.005985 Cumulative Hazard Rate Function 0.0191 07 0.035955 0.0 370 65 0.038 175 0.039285 0.5852 97 0.6218 07 0.6594 27 0.6981 57 Slide No 32 CHAPTER 4c RELIABILITY ASSESSMENT Reliability Analysis of Systems ̈ Example 21 (cont’d)... RELIABILITY ASSESSMENT Reliability Analysis of Systems ̈ Example 22 (cont’d) Table 25 Data and Empirical Survivor Function, Sn(t), for Component 3 for Example 22 Year 19 37 TTF (Years) 0 Number of Failures 0 Survivor Function 1.000000 2001 2002 2003 2004 64 65 66 67 174 176 181 182 0. 871 750 0.862950 0.853900 0.844800 Slide No 38 CHAPTER 4c RELIABILITY ASSESSMENT Reliability Analysis of Systems ̈ Example... 45 50 55 60 65 70 75 Time to Failure (Years) Figure 17a Hazard (Failure) Rate Function (HRF) for a Series System of Three Identical Components of Example 21 Slide No 33 CHAPTER 4c RELIABILITY ASSESSMENT Reliability Analysis of Systems ̈ Example 21 (cont’d) Cumulative Hazard Rate 0.8 Component 0 .7 System 0.6 0.5 0.4 0.3 0.2 0.1 0 40 45 50 55 60 65 70 75 Time to Failure (Years) Figure 17b Cumulative Hazard... RELIABILITY ASSESSMENT Reliability Analysis of Systems ̈ Example 21 (cont’d) – Applying Equations 99a and 99b with n = 3, the following expressions can be obtained: Hs(t) = 0 .78 79 47 - 0.04 174 5t + 0.000555t2 and hs(t) = - 0.04 174 5 + 0.001110t – The resulting hazard functions are given in Table 23 and Figures 17a and 17b Slide No 31 CHAPTER 4c RELIABILITY ASSESSMENT Reliability Analysis of Systems Table 23... ASSESSMENT Reliability Analysis of Systems ̈ Example 23 (cont’d) Table 28 Hazard Rate Functions for Parallel System Composed of Three Identical Components of Example 23 Year 1 975 Time to Failure (Years) 38 Hazard Rate Function 3.41962E-10 Cumulative Hazard Rate Function 1.05647E-09 2001 64 0.000380821 0.0018 073 17 2002 2003 65 66 0.000449465 0.000526 673 0.002223232 0.00 271 2222 2004 67 0.000612993 0.003283142... Example 24 Year Time to Hazard Rate Failure, Years Function Cumulative Hazard Rate Function 1 975 38 2.50140E-12 5.20 173 E-12 2001 64 8.2 273 7E-05 3.39908E-04 2002 65 1.03504E-04 4.43412E-04 2003 66 1.28933E-04 5 .72 345E-04 2004 67 1.59129E-04 7. 31 474 E-04 Slide No 58 CHAPTER 4c RELIABILITY ASSESSMENT Reliability Analysis of Systems ̈ Example 24 (cont’d) Hazard Rate Component 1 0.014 Component 2 0.012 Component... of a Series-parallel System of Example 4-25 Year 1 975 Time to Failure (Years) 38 Hazard Rate Function 5.00289E-12 Cumulative Hazard Rate Function 1.040346E-11 2001 2002 2003 2004 64 65 66 67 1.64547E-04 2. 070 07E-04 2. 578 65E-04 3.18258E-04 6 .79 8169E-04 8.868240E-04 1.144689E-03 1.462947E-03 Slide No 69 CHAPTER 4c RELIABILITY ASSESSMENT Reliability Analysis of Systems ̈ Example 25 (cont’d) Hazard Rate... Function, Sn(t), for Component 4 for Example 22 Year TTF (Years) Number of Failures Survivor Function 19 37 0 0 1.000000 2001 2002 2003 2004 64 65 66 67 195 198 202 209 0.846850 0.836950 0.826850 0.816400 Slide No 39 CHAPTER 4c RELIABILITY ASSESSMENT Reliability Analysis of Systems ̈ Example 22 (cont’d) Table 27 Parameters of Hazard Rate Functions for Four Components and the Series System for Example 22... 99b as given in Table 27 Slide No 36 CHAPTER 4c RELIABILITY ASSESSMENT Reliability Analysis of Systems ̈ Example 22 (cont’d) Table 24 Data and Empirical Survivor Function, Sn(t), for Component 2 for Example 22 Year 19 37 TTF (Years) 0 Number of Failures 0 Survivor Function 1.000000 2001 2002 2003 2004 64 65 66 67 184 189 190 193 0.855350 0.845900 0.836400 0.82 675 0 Slide No 37 CHAPTER 4c RELIABILITY... are Hs(t) = 1.06 971 0 - 0.0 578 52t + 0.00 078 6t2 hs(t) = - 0.0 578 52 + 0.001 572 t – Figures 18a and 18b show the respective hazard rate functions Slide No 41 CHAPTER 4c RELIABILITY ASSESSMENT Reliability Analysis of Systems ̈ Example 22 (cont’d) Hazard Rate 0.05 Component 1 Component 2 Component 3 0.045 Component 4 0.04 System 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0 35 40 45 50 55 60 65 70 Time to Failure . Figures 17a and 17b H s (t) = 0 .78 79 47 - 0.04 174 5t + 0.000555t 2 h s (t) = - 0.04 174 5 + 0.001110t CHAPTER 4c. RELIABILITY ASSESSMENT Slide No. 31 Reliability Analysis of Systems 0.6981 570 .03928 573 2010 0.6594 270 .038 175 722009 0.6218 070 .0 370 6 571 2008 0.5852 970 .03595 570 20 07 0.0191 070 .005985431980 Cumulative. 31 Reliability Analysis of Systems 0.6981 570 .03928 573 2010 0.6594 270 .038 175 722009 0.6218 070 .0 370 6 571 2008 0.5852 970 .03595 570 20 07 0.0191 070 .005985431980 Cumulative Hazard Rate Function Hazard Rate. J. Clark School of Engineering •Department of Civil and Environmental Engineering CHAPTER 4c CHAPMAN HALL/CRC Risk Analysis in Engineering and Economics Risk Analysis for Engineering Department