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


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