nonParBayesSystemInference {ReliabilityTheory}

Non-parametric Bayesian posterior predictive system survival inference

Description

Computes a non-parametric Bayesian posterior predictive survival probability given the survival signature of a system and test data on each of the components as described in Aslett et al (2015).

Usage

nonParBayesSystemInference(at.times, survival.signature, test.data, alpha=1, beta=1)

Arguments

at.times

a vector of times at which the posterior predictive estimate of survival probability should be computed.

survival.signature

the survival signature matrix of the system/network for which inference is performed. This should be in the same format as returned by computeSystemSurvivalSignature.

test.data

a list of vectors containing the component test data. The elements of the list should be named identically to the component columns in the survival.signature argument.

alpha, beta

the Beta prior shape parameters. Each must match in type and can be:

• a single scalar for a fixed prior across time and component types;

• a vector of parameters of the same length as the at.times argument, which indicates the time-varying prior parameter at the corresponding time in at.times. This is therefore time-varying, but indicates the same time-varying prior for all component types;

• a data frame where each column is named using the same names as for the survival.signature argument and each row corresponds to the time-varying prior parameter at the corresponding time in at.times.

By default the 'uninformative' prior with alpha=1 and beta=1 is used for all components at all times.

Details

This function implements the technique described in detail in Section 4 of Aslett et al (2015).

In brief, at any fixed time t, the functioning of a single component of type k is Bernoulli(p_t^k) distributed for suitable p_t^k, irrespective of the lifetime distribution of the component. Correspondingly, the distribution of the number of components still functioning at time t in a collection of n_k iid components of type k is Binomial(n_k, p_t^k).

Taking the priors p_t^k \sim Beta(\alpha_t^k, \beta_t^k), Aslett et al (2015) show that this leads to a posterior predictive survival distribution with a nice closed form (see equations 9 and 10 in Section 4).

Value

A vector of the same length as the at.times argument, where each element is the posterior predictive probability of a new system surviving to the corresponding time in at.times.

Note

Please feel free to email [e-mail] with any queries or if you encounter errors when running this function.

Author(s)

Louis J. M. Aslett [e-mail] (https://www.louisaslett.com/)

References

Aslett, L. J. M., Coolen, F. P. A. and Wilson, S. P. (2015), ‘Bayesian Inference for Reliability of Systems and Networks using the Survival Signature’, Risk Analysis 39(9), 1640–1651. Download paper

See Also

computeSystemSurvivalSignature

Examples

## Exactly reproduce the example in Section 4.1 of Aslett et al (2015), including Figure 5
# First specify the system layout, numbered as per Figure 4
sys <- createSystem(s -- 1 -- 2:4:5, 2 -- 3 -- t, 4:5 -- 6 -- t,
                    s -- 7 -- 8 -- t, s -- 9 -- 10 -- 11 -- t, 7 -- 10 -- 8,
                    types = list(T1 = c(1, 6, 11),
                                 T2 = c(2, 3, 9),
                                 T3 = c(4, 5, 10),
                                 T4 = c(7, 8)))

# Compute the survival signature table from Appendix
sig <- computeSystemSurvivalSignature(sys)

# Simulate the test data (same seed as used in the paper)
set.seed(233)
t1 <- rexp(100, rate=0.55)
t2 <- rweibull(100, scale=1.8, shape=2.2)
t3 <- rlnorm(100, 0.4, 0.9)
t4 <- rgamma(100, scale=0.9, shape=3.2)

# Compile into a list as required by this function
test.data <- list("T1"=t1, "T2"=t2, "T3"=t3, "T4"=t4)

# Create a vector of times at which to evaluate the posterior predictive
# survival probability and compute using this function
t <- seq(0, 5, length.out=300)
yS <- nonParBayesSystemInference(t, sig, test.data)

# Compute the survival curves for the individual components (just to match
# Figure 5)
y1 <- sapply(t, pexp, rate=0.55, lower.tail=FALSE)
y2 <- sapply(t, pweibull, scale=1.8, shape=2.2, lower.tail=FALSE)
y3 <- sapply(t, plnorm, meanlog=0.4, sdlog=0.9, lower.tail=FALSE)
y4 <- sapply(t, pgamma, scale=0.9, shape=3.2, lower.tail=FALSE)

# Plot

library(ggplot2)
p <- ggplot(data.frame(Time=rep(t,5), Probability=c(yS,y1,y2,y3,y4),
            Item=c(rep(c("System", "T1", "T2", "T3", "T4"), each=300))))
p <- p + geom_line(aes(x=Time, y=Probability, linetype=Item))
p <- p + xlab("Time") + ylab("Survival Probability")
p