littlewood.verall {Reliability} | R Documentation |
littlewood.verall
computes the Maximum Likelihood estimates for the parameters
theta0
, theta1
and rho
of the mean value function for the
Littlewood-Verall model.
littlewood.verall(t, linear = T, init = c(1, 1, 1), method = "Nelder-Mead", maxit = 10000, ...)
t |
time between failure data |
linear |
logical. Should the linear or the quadratic form of the mean value
function for the Littlewood-Verrall model be used of computation?
If TRUE , which is the default, the linear form of the mean
value function is used. |
init |
initial values for Maximum Likelihood fit of the mean value function for the Littlewood-Verall model. |
method |
the method to be used for optimization, see optim for details. |
maxit |
the maximum number of iterations, see optim for details. |
... |
control parameters and plot parameters optionally passed to the
optimization and/or plot function. Parameters for the optimization
function are passed to components of the control argument of
optim .
|
This function estimates the parameters theta0
, theta1
and rho
of
the mean value function in the linear or the quadratic form for the Littlewood-Verall
model.
First, the computation with the mean value function in the linear form is explained. With Maximum Likelihood estimation one gets the following equations, which have to be minimized. This is
equation_1 := frac{n}{rho} + sum_{i = 1}^{n} log(theta_0 + theta_1 i) - sum_{i = 1}^{n} log(theta_0 + theta_1 i + t_i) = 0,
equation_2 := rho sum_{i = 1}^{n} frac{1}{theta_0 + theta_1 i} - rho + 1 sum_{i = 1}^{n} frac{1}{theta_0 + theta_1 i + t_i} = 0
and
equation_3 := rho sum_{i = 1}^{n} frac{i}{theta_0 + theta_1 i} - rho + 1 sum_{i = 1}^{n} frac{i}{theta_0 + theta_1 i + t_i} = 0.
Second, the computation with the mean value function in the quadratic form is explained. With Maximum Likelihood estimation one gets the following equations, which have to be minimized. This is
equation_1 := frac{n}{rho} + sum_{i = 1}^{n} log(theta_0 + theta_1 i^2) - sum_{i = 1}^{n} log(theta_0 + theta_1 i^2 + t_i) = 0,
equation_2 := rho sum_{i = 1}^{n} frac{1}{theta_0 + theta_1 i^2} - rho + 1 sum_{i = 1}^{n} frac{1}{theta_0 + theta_1 i^2 + t_i} = 0
and
equation_3 := rho sum_{i = 1}^{n} frac{i^2}{theta_0 + theta_1 i^2} - rho + 1 sum_{i = 1}^{n} frac{i^2}{theta_0 + theta_1 i^2 + t_i} = 0.
Where t is the time between failure data and n is the length or in other words the size of the time between failure data. So the simultaneous minimization of these equations happens by minimization of the equation
equation_1^2 + equation_2^2 + equation_3^2 = 0.
A list containing following components:
theta0 |
Maximum Likelihood estimate for theta0 |
theta1 |
Maximum Likelihood estimate for theta1 |
rho |
Maximum Likelihood estimate for rho |
Andreas Wittmann andreas_wittmann@gmx.de
J.D. Musa, A. Iannino, and K. Okumoto. Software Reliability: Measurement, Prediction, Application. McGraw-Hill, 1987.
Michael R. Lyu. Handbook of Software Realibility Engineering. IEEE Computer Society Press, 1996. http://www.cse.cuhk.edu.hk/~lyu/book/reliability/
littlewood.verall.plot
, mvf.ver.lin
,
mvf.ver.quad
# time between-failure-data from DACS Software Reliability Dataset # homepage, see system code 1. Number of failures is 136. t <- c(3, 30, 113, 81, 115, 9, 2, 20, 20, 15, 138, 50, 77, 24, 108, 88, 670, 120, 26, 114, 325, 55, 242, 68, 422, 180, 10, 1146, 600, 15, 36, 4, 0, 8, 227, 65, 176, 58, 457, 300, 97, 263, 452, 255, 197, 193, 6, 79, 816, 1351, 148, 21, 233, 134, 357, 193, 236, 31, 369, 748, 0, 232, 330, 365, 1222, 543, 10, 16, 529, 379, 44, 129, 810, 290, 300, 529, 281, 160, 828, 1011, 445, 296, 1755, 1064, 1783, 860, 983, 707, 33, 868, 724, 2323, 2930, 1461, 843, 12, 261, 1800, 865, 1435, 30, 143, 108, 0, 3110, 1247, 943, 700, 875, 245, 729, 1897, 447, 386, 446, 122, 990, 948, 1082, 22, 75, 482, 5509, 100, 10, 1071, 371, 790, 6150, 3321, 1045, 648, 5485, 1160, 1864, 4116) littlewood.verall(t, linear = TRUE) littlewood.verall(t, linear = FALSE)