weibulltools (version 1.0.1)

r_squared_profiling: \(R<U+00B2>\)-Profile Function for Log-Location-Scale Distributions with Threshold

Description

This function evaluates the coefficient of determination with respect to a given threshold parameter of a three-parametric lifetime distribution. In terms of Median Rank Regression this function can be optimized (optim) to estimate the threshold parameter.

Usage

r_squared_profiling(x, y, thres, distribution = c("weibull3",
  "lognormal3", "loglogistic3"))

Arguments

x

a numeric vector which consists of lifetime data. Lifetime data could be every characteristic influencing the reliability of a product, e.g. operating time (days/months in service), mileage (km, miles), load cycles.

y

a numeric vector which consists of estimated failure probabilities regarding the lifetime data in x.

thres

a numeric value of the threshold parameter.

distribution

supposed distribution of the random variable. The value can be "weibull3", "lognormal3" or "loglogistic3".

Value

Returns the coefficient of determination for a specified threshold value.

References

Meeker, William Q; Escobar, Luis A., Statistical methods for reliability data, New York: Wiley series in probability and statistics, 1998

Examples

# NOT RUN {
# Alloy T7987 dataset taken from Meeker and Escobar(1998, p. 131)
cycles   <- c(300, 300, 300, 300, 300, 291, 274, 271, 269, 257, 256, 227, 226,
              224, 213, 211, 205, 203, 197, 196, 190, 189, 188, 187, 184, 180,
              180, 177, 176, 173, 172, 171, 170, 170, 169, 168, 168, 162, 159,
              159, 159, 159, 152, 152, 149, 149, 144, 143, 141, 141, 140, 139,
              139, 136, 135, 133, 131, 129, 123, 121, 121, 118, 117, 117, 114,
              112, 108, 104, 99, 99, 96, 94)
state <- c(rep(0, 5), rep(1, 67))

df_john <- johnson_method(x = cycles, event = state)

# Determining threshold parameter for which the coefficient of determination is
# maximized subject to the condition that the threshold parameter must be smaller
# as the first failure cycle, i.e 94:
threshold <- seq(0, min(cycles[state == 1]) - 0.1, length.out = 100)
profile_r2 <- sapply(threshold, r_squared_profiling,
                     x = df_john$characteristic[df_john$status == 1],
                     y = df_john$prob[df_john$status == 1],
                     distribution = "weibull3")
threshold[which.max(profile_r2)]

# plot:
# plot(threshold, profile_r2, type = "l")
# abline(v = threshold[which.max(profile_r2)], h = max(profile_r2), col = "red")
# }