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kofn

Component Lifetime Estimation from k-out-of-n System Data

The Problem

You observe a system (an engine, a power grid, a redundant server cluster) and you record when it fails. But you don’t care about the system. You care about the components inside it. How long do the individual bearings, transformers, or nodes last?

This is fundamentally a censoring problem. When a k-out-of-n system fails (meaning k components have failed), the component lifetimes are only partially observed:

  • 1 component is observed exactly (the one that triggered system failure)
  • k − 1 components are right-censored (still functioning at system failure)
  • m − k components are left-censored (already failed, but we don’t know when)

The kofn package provides maximum likelihood estimation of component lifetime parameters from this structured censoring, supporting exponential and Weibull component distributions under multiple observation schemes.

Why This Matters

Parallel systems (k = n) are the hardest case: all components have failed by the time the system fails, but we only observe the system failure time, not the individual failure times. The last-to-fail component is observed exactly; all others are deeply left-censored.

This creates an information asymmetry: fast components (those that typically fail early) contribute almost no information to the likelihood, because their CDF is near 1 at typical system failure times. The kofn package quantifies this asymmetry and provides tools to resolve it through periodic inspection.

Installation 

# Install from GitHub
remotes::install_github("queelius/kofn")

Quick Start 

library(kofn)

# Create a parallel system model (k=m) with 3 exponential components
model <- kofn(k = 3, m = 3, family = "exponential")

# Generate data: true rates = (0.5, 0.3, 0.2)
gen <- rdata(model)
set.seed(42)
df <- gen(theta = c(0.5, 0.3, 0.2), n = 300)

# Fit via maximum likelihood
fitter <- fit(model)
result <- fitter(df, n_starts = 5)

# Full base R stats interface
coef(result)       # parameter estimates
confint(result)    # confidence intervals
summary(result)    # coefficient table with SEs
AIC(result)        # model comparison

Weibull Components with EM

When components follow a Weibull distribution (capturing increasing or decreasing failure rates), the estimation problem is substantially harder. The kofn package provides an EM algorithm whose E-step computes truncated Weibull moments via incomplete gamma functions.

# Weibull parallel system with EM estimation
model_wei <- kofn(k = 2, m = 2, family = "weibull", method = "em")
gen <- rdata(model_wei)
set.seed(42)
df <- gen(theta = c(1.5, 2.0, 2.0, 3.0), n = 300)
#                   ^shape ^scale ^shape ^scale

fitter <- fit(model_wei)
result <- fitter(df, n_starts = 3)
summary(result)

Periodic Inspection (Scheme 1)

Under Scheme 0 (system-level observation only), fast components are poorly estimated. Periodic inspection at interval δ\delta localizes each component’s failure time to an inspection interval, resolving the information asymmetry:

# Generate data with periodic inspection every 0.5 time units
s1_gen <- rdata_scheme1(model_wei)
df_s1 <- s1_gen(theta = c(1.5, 2.0, 2.0, 3.0), n = 300, delta = 0.5)

# Fit using the Scheme 1 likelihood
s1_fitter <- fit_scheme1(model_wei)
result_s1 <- s1_fitter(df_s1, n_starts = 3)

Shape RMSE drops by an order of magnitude: even coarse inspection intervals provide dramatic improvement for the worst-estimated components.

Topology and DGP

As of v0.3.0, kofn delegates topology and data-generating process queries to the dist.structure package. For non-k-of-n topologies (bridges, arbitrary coherent systems), use dist.structure directly:

library(dist.structure)

# Standard systems with arbitrary component distributions
sys_parallel <- parallel_dist(replicate(5, algebraic.dist::exponential(1),
                                        simplify = FALSE))
sys_2of5     <- exp_kofn(k = 4, rates = c(1, 1, 1, 1, 1))  # dist.structure :G
sys_bridge   <- bridge_dist(replicate(5, algebraic.dist::exponential(1),
                                       simplify = FALSE))

# System properties
system_signature(sys_2of5)
phi(sys_bridge, c(1L, 0L, 1L, 1L, 0L))

Note the convention: kofn uses :F (k = number of failures triggering system failure), dist.structure uses :G (k = number of functioning components required). They convert via k_dist = m - k_kofn + 1.

Key Features

Feature Description
Exponential MLE Closed-form likelihood via inclusion-exclusion expansion for parallel; dist.structure delegation for general k
Weibull EM EM algorithm with incomplete gamma E-step and profile M-step
Direct MLE L-BFGS-B with Nelder-Mead fallback, multi-start
Observation schemes Scheme 0 (black box), Scheme 1 (periodic inspection), Scheme 2 (complete), masked cause-of-failure
Fisher information Compare information across observation schemes
dist.structure integration Full topology, signature, importance measures, compositional operations via the distribution protocol
Base R integration coef(), vcov(), logLik(), AIC(), confint(), summary()

Ecosystem

kofn is part of a family of packages for likelihood-based reliability inference:

Vignettes