Weibull Distribution

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The survival function -- the probability that a unit survives for at least time t:

t λ β S.W e t λ / (β^-^:

The cumulative failure function -- the probability that a unit fails at a time T<t:

t λ β F.W 1 t λ β S.W - :

The probability density function of failure a time t:

t λ β f.W t λ β F.W t diff :

Given a population of n units observed for a total time of tt with failures observed at times t the log likelihood function is:

λ β n tt t.f ll.W t.f λ β f.W ln vectorize sum (n t.f rows - (tt λ β S.W ln * + :

T 20307180116246311379384421458495497499114 mat transpose :

The life cycle analysis starts with n=130 units under test for time_test=499 days. Several units fail. What is the probability that the ith unit will fail at time T hours. Use a Weibull probability destiny distribution function. The family of rate distribution functions is given by T: Time to failure in days.

time.test T max eval : 499

n 130 :

k 1 T length range :

Y k el k :

LM Algorithm

Diagonal matrix for Numerical Procedures A\032E\ N\032E\ nDiag D\0324\ N\032E\ N\032E\ matrix : k\0324\ 1 N\032E\ range D\0324\ k\0324\ k\0324\ el A\032E\ k\0324\ k\0324\ el : for D\0324\ eval 31 line :

Diagonal matrix for Numerical Procedures A\032E\ nDiag A\032E\ A\032E\ rows nDiag 11 line :

Levenberg–Marquardt 1 f\032E\ 1 b\032E\ a\032E\ νo\032E\ εx\032E\ εy\032E\ M\032E\ nLM x\0324\ a\032E\ : y\0324\ a\032E\ f\032E\ eval : R\0324\ y\0324\ length eval : C\0324\ a\032E\ length eval : 14 mat εx\0324\ εx\032E\ a\032E\ 1 el UnitsOf * eval : εo\0324\ C\0324\ identity εx\0324\ * eval : 12 mat εy\0324\ εy\032E\ y\0324\ 1 el UnitsOf * : 31 line y\0324\ Co\0324\ 12 / y\0324\ k\0324\ el (2^k\0324\ 1 R\0324\ sum * 11 line : x\0324\ Y\0324\ Grd\0324\ G\0324\ C\0324\ 1 matrix : c\0324\ 1 C\0324\ range G\0324\ c\0324\ el x\0324\ εo\0324\ c\0324\ col + f\032E\ Co\0324\ Y\0324\ - εx\0324\ / eval : for G\0324\ 2^sqrt eval 31 line : x\0324\ y\0324\ Jac J\0324\ R\0324\ C\0324\ matrix : r\0324\ 1 R\0324\ range c\0324\ 1 C\0324\ range J\0324\ r\0324\ c\0324\ el x\0324\ εo\0324\ c\0324\ col + f\032E\ r\0324\ el y\0324\ r\0324\ el - εx\0324\ / eval : for for J\0324\ eval 31 line : y\0324\ x\0324\ f\032E\ eval : Y\0324\ y\0324\ Co\0324\ : J\0324\ x\0324\ y\0324\ Jac : H\0324\ J\0324\ transpose J\0324\ * eval : 14 mat λo\0324\ H\0324\ C\0324\ nDiag max eval : xm\0324\ x\0324\ : Gr\0324\ x\0324\ Y\0324\ Grd\0324\ : 13 mat k\0324\ 1 M\032E\ range d\0324\ H\0324\ λo\0324\ H\0324\ C\0324\ nDiag * + (1 -^J\0324\ transpose y\0324\ * (* eval : k\0324\ 1 M\032E\ + : break 21 line try B\0324\ x\0324\ d\0324\ - b\032E\ eval : xo\0324\ x\0324\ d\0324\ B\0324\ * vectorize - : yo\0324\ xo\0324\ f\032E\ eval : Yo\0324\ yo\0324\ Co\0324\ : 12 mat Yo\0324\ Y\0324\ < x\0324\ xo\0324\ : y\0324\ yo\0324\ : Y\0324\ Yo\0324\ : λo\0324\ λo\0324\ νo\032E\ / eval : 14 mat Go x\0324\ Y\0324\ Grd\0324\ : (Gr\0324\ < xm\0324\ x\0324\ : Gr\0324\ Go : 12 mat 0 if Go εy\0324\ > y\0324\ x\0324\ f\032E\ eval : Y\0324\ y\0324\ Co\0324\ : J\0324\ x\0324\ y\0324\ Jac : H\0324\ J\0324\ transpose J\0324\ * eval : 14 mat break if 31 line λo\0324\ λo\0324\ νo\032E\ * eval : if 51 line for k\0324\ M\032E\ ≤ x\0324\ eval xm\0324\ eval if 81 line :

1 f\032E\ a\032E\ nLM B\0324\ a\032E\ length identity nDiag : x\032E\ b\032E\# B\0324\ : x\032E\# f\032E\ x\032E\# b\032E\ a\032E\ nLM 31 line :

1 f\032E\ 1 b\032E\ a\032E\ nLM x\032E\# f\032E\ x\032E\# b\032E\ a\032E\ νo εx εy MaxIters nLM 11 line :

MaxIters 300 :

εx 105 -^:

εy 103 -^:

νo 2 :

A maximum likelihood estimate of the parameters, as a function of the data and guess values for the parameters, is then

λ β n tt t.f 1 b ml.W x f x 1 el x 2 el n tt t.f ll.W 11 mat 11 line : x# f x# b λ β 21 mat nLM eval 21 line :

We need guess values for the parameters. Here is a very simplistic estimate, taking b as just 1:

n tt t.f guess.W tt n t.f rows / ln * 112 mat :

λ.g β.g 12 mat n time.test T guess.W : 1112.01 112 mat

λ.g β.g n time.test T ll.W 153.8518 -

β 1 <

x b 0 x 1 el < 0 x 2 el < (x 2 el 1 < (& 12 mat transpose : λ.1 β.1 21 mat λ.g 0.5 n time.test T x# b ml.W : 21 sys

n x λ.1 β.1 F.W * T Y o 5 red augment 2 1 sys

λ.1 β.1 12 mat 5608.5324 0.9288 12 mat

x β.1 n time.test T ll.W λ.1 β.1 n time.test T ll.W 2 1 sys λ.1 x n time.test T ll.W λ.1 β.1 n time.test T ll.W 2 1 sys

β 1 >

x b 0 x 1 el < 1 x 2 el < 12 mat transpose : λ.2 β.2 21 mat λ.g 2 n time.test T x# b ml.W : 21 sys

n x λ.2 β.2 F.W * T Y o 5 red augment 2 1 sys

λ.2 β.2 12 mat 3450.4506 1.1179 12 mat

x β.2 n time.test T ll.W λ.2 β.2 n time.test T ll.W 2 1 sys λ.2 x n time.test T ll.W λ.2 β.2 n time.test T ll.W 2 1 sys

f.W refuses to plot if isn't epxlicity defined.

f.W t λ β f.W eval :

t λ β f.W f.W :

β 1 <

β 1 >

x λ.1 β.1 f.W 1 1 sys x λ.2 β.2 f.W 1 1 sys

Alvaro