Saturday, December 3, 2022

HP 15C and TI-84 Plus CE: Weibull Distribution Parameter Calculation

HP 15C and TI-84 Plus CE:  Weibull Distribution Parameter Calculation



Introduction


The Weibull probability density distribution function is:


f(x) = (b / Θ) * (x / Θ)^(b-1) * exp(-(x / Θ)^b)


with the lower tail cumulative distribution of (-∞ to x):


Area = 1 - exp(-(x / Θ)^b)


The area function tells us what is the probability a device lasts no more than x time units.  


Area = 1 - Survival


The survival function is the probability a device lasts more than x time units.


Survival = exp(-(x / Θ)^b)


Generally, the higher Θ is, the flatter the Weibull Distribution curve.  


In today's blog, we are estimating the parameters b and Θ given the number of data points, N, and data points (time periods to failure) x_i.   For the HP 15C program, which is modeled after the HP 55 program (see source below).   



The Process


Each x_i is sorted in ascending order.  Then transform the following data:


x' = ln x


α = (K - 0.3) ÷ (N + 0.4),  K = 1, 2, 3, ... , N


y' = ln( ln( 1 ÷ (1 - α)))


Enter each point (x', y'), and perform a linear regression analysis.


Then:  


b = slope


Θ = e^-(intercept ÷ slope)


 


HP 15C Program:  Weibull Distribution - Parameter Determination


Line #;  Key;  Code


001;  LBL A; 42, 21, 11

002;  1;  1

003;  STO 0;  44, 0

004;  CLΣ;  43, 32

005;  R/S;  31

006;  STO 1;  44, 1

007;  LBL 9;  42, 21, 9

008;  R/S;   31

009;  LN;  43, 12

010;  RCL 0;  45, 0

011;  . ;  48

012;  3 ;  3

013;  - ;  30

014; RCL 1;  45, 1

015; .  ;  48

016; 4 ;  4

017; +  ; 40

018; ÷ ; 10

019; 1 ;  1

020;  STO+ 0;  44, 40, 0

021;  x<>y  ; 34

022;  - ; 30

023;  1/x ;  15

024;  LN ; 43, 12

025;  LN ; 43, 12

026;  x<>y ; 34

027;  Σ+ ; 49

028;  GTO 9; 22, 9

029;  LBL B; 42, 21, 12

030;  L.R.;  42, 49

031;  x<>y ; 34

032;  R/S ; 31

033;  ÷ ; 10

034;  CHS;  16

035;  e^x;  12

036;  RTN;  43, 32


1.  Execute label A.   

2.  Enter N, the number of data points, then press the R/S key.

3.  Enter each x_i in ascending order, press R/S key in between each keys.  

4.  Execute label B.   The b parameter is displayed.  

5.  Press R/S.  The Θ parameter is displayed.



TI-84 Plus CE Program: WBFIT  


Weibull Distribution - Parameter Determination


"EWS 2022-10-09"

ClrHome

Disp "WEIBULL DIST.","FIT CALCULATION"

Input "DATA LIST: ",L1

SortA(L1)

dim(L1)→N

ln(L1)→L1

N→dim(L2)

For(K,1,N)

(K-0.3)/(N+0.4)→A

ln(ln(1/(1-A)))→L2(K)

End

LinReg(a+bx) L1,L2

b→B

e^(­(a/b))→θ

ClrHome

Disp "1-e^(­(X/B)^θ)"

Disp "B:",B,"θ:",θ


The x_i data are sorted in the WBIT program.  



Examples


Example 1:

Hours to failure:

{ 11000, 11056, 11379, 11821, 11956, 12403, 12526, 13000, 13380, 13663 }

N = 10


b ≈ 14.01123

Θ ≈ 12649.59071


Example 2:

Days to failure:

{ 1760, 1799, 1882, 1931, 1996, 2004, 2150 }

N = 7


b ≈ 15.22473

Θ ≈ 1993.22461


Sources:


HP55 Statistics Programs  Hewlett Packard Company.  Cupertino, CA.  1975


Ma, Dan.  "The Weibull distribution"  Topics in Actuarial Modeling.  September 28, 2016.   https://actuarialmodelingtopics.wordpress.com/2016/09/28/the-weibull-distribution/  Last Retrieved September 20, 2022.  



Eddie


All original content copyright, © 2011-2022.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 


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