Title:	Likelihood Models for Systems with Masked Component Cause of Failure
Version:	0.9.2
Description:	Maximum likelihood estimation for series systems where the component cause of failure is masked. Implements analytical log-likelihood, score, and Hessian functions for exponential, homogeneous Weibull, and heterogeneous Weibull component lifetimes under masked cause conditions (C1, C2, C3). Supports exact, right-censored, left-censored, and interval-censored observations via composable observation functors. Provides random data generation, model fitting, and Fisher information for asymptotic inference. See Lin, Loh, and Bai (1993) <doi:10.1109/24.257799> and Craiu and Reiser (2006) <doi:10.1111/j.1541-0420.2005.00498.x>.
License:	GPL (≥ 3)
Language:	en-US
Encoding:	UTF-8
RoxygenNote:	7.3.3
Suggests:	knitr, rmarkdown, testthat (≥ 3.0.0), algebraic.mle, algebraic.dist
Config/testthat/edition:	3
Imports:	stats, likelihood.model, generics, numDeriv
Depends:	R (≥ 4.1.0)
VignetteBuilder:	knitr
URL:	https://queelius.github.io/maskedcauses/
BugReports:	https://github.com/queelius/maskedcauses/issues
NeedsCompilation:	no
Packaged:	2026-03-04 04:52:58 UTC; spinoza
Author:	Alexander Towell [aut, cre]
Maintainer:	Alexander Towell <lex@metafunctor.com>
Repository:	CRAN
Date/Publication:	2026-03-08 10:50:02 UTC

Exponential series systems

Description

This file contains functions related to the Exponential series distribution. Functions include simulation, pdf, cdf, quantile, and other related functions for the Exponential series distribution.

Details

Exponential series system distribution

Density, distribution, quantile, random generation, hazard, survival, and mean functions for the exponential series system distribution. A series system of m independent exponential components with rates \lambda_1, \ldots, \lambda_m has an exponential system lifetime with rate \lambda = \sum \lambda_j.

Assumptions for exp_series_md_c1_c2_c3 model.

Description

Returns the assumptions made by this likelihood model.

Usage

## S3 method for class 'exp_series_md_c1_c2_c3'
assumptions(model, ...)

Arguments

Value

character vector of assumptions

Examples

assumptions(exp_series_md_c1_c2_c3())

Assumptions for wei_series_homogeneous_md_c1_c2_c3 model.

Description

Returns the assumptions made by this likelihood model.

Usage

## S3 method for class 'wei_series_homogeneous_md_c1_c2_c3'
assumptions(model, ...)

Arguments

Value

character vector of assumptions

Examples

assumptions(wei_series_homogeneous_md_c1_c2_c3())

Assumptions for wei_series_md_c1_c2_c3 model.

Description

Returns the assumptions made by this likelihood model.

Usage

## S3 method for class 'wei_series_md_c1_c2_c3'
assumptions(model, ...)

Arguments

Value

character vector of assumptions

Examples

assumptions(wei_series_md_c1_c2_c3())

Marginal cause-of-failure probability

Description

Returns a closure computing P(K=j | theta) for all components j, marginalized over the system failure time T. By Theorem 5 of the foundational paper, this equals E_T[P(K=j | T, theta)].

Usage

cause_probability(model, ...)

## S3 method for class 'series_md'
cause_probability(model, ...)

Arguments

Details

The default method uses Monte Carlo integration via rdata().

Value

a function with signature ⁠function(par, ...)⁠ returning an m-vector where element j gives P(K=j | theta)

Methods (by class)

• cause_probability(series_md): Default for series_md models via Monte Carlo integration (Theorem 5)

Examples

model <- exp_series_md_c1_c2_c3()
cp <- cause_probability(model)
cp(par = c(0.5, 0.3, 0.2))

Component hazard function

Description

Returns a closure computing the hazard function h_j(t; theta) for the j-th component. The returned function takes the full parameter vector par and extracts the relevant component parameters internally.

Usage

component_hazard(model, j, ...)

Arguments

Value

a function with signature ⁠function(t, par, ...)⁠ computing h_j(t)

Examples

model <- exp_series_md_c1_c2_c3()
h1 <- component_hazard(model, j = 1)
h1(t = 1.0, par = c(0.5, 0.3, 0.2))

Conditional cause-of-failure probability

Description

Returns a closure computing P(K=j | T=t, theta) for all components j, conditional on a specific failure time t. By Theorem 6 of the foundational paper, this equals h_j(t; theta) / sum_l h_l(t; theta).

Usage

conditional_cause_probability(model, ...)

## S3 method for class 'series_md'
conditional_cause_probability(model, ...)

Arguments

Value

a function with signature ⁠function(t, par, ...)⁠ returning an n x m matrix where n = length(t) and column j gives P(K=j | T=t, theta)

Methods (by class)

• conditional_cause_probability(series_md): Default for series_md models via component hazard ratios (Theorem 6)

Examples

model <- exp_series_md_c1_c2_c3()
ccp <- conditional_cause_probability(model)
ccp(t = c(1, 2), par = c(0.5, 0.3, 0.2))

Cumulative hazard function for a component hazard function

Description

Creates a cumulative hazard function from a hazard function by integrating.

Usage

cum_haz(haz)

Arguments

Value

A function that computes the cumulative hazard at time t

Examples

# Exponential hazard h(t) = lambda
haz <- function(t, ...) rep(0.5, length(t))
H <- cum_haz(haz)
H(2)  # Should be 1.0 (0.5 * 2)

pdf for exponential series.

Description

pdf for exponential series.

Usage

dexp_series(t, rates, log = FALSE)

Arguments

Value

Density values for the exponential series distribution.

Examples

rates <- c(0.5, 0.3, 0.2)
dexp_series(1.0, rates)
dexp_series(c(0.5, 1.0, 2.0), rates, log = TRUE)

Constructs a likelihood model for exp_series_md_c1_c2_c3.

Description

Likelihood model for exponential series systems with masked component cause of failure with candidate sets that satisfy conditions C1, C2, and C3, described below.

Usage

exp_series_md_c1_c2_c3(
  rates = NULL,
  lifetime = "t",
  lifetime_upper = "t_upper",
  omega = "omega",
  candset = "x"
)

Arguments

Details

This model satisfies the concept of a likelihood_model in the likelihood.model package by providing the following methods:

(1) loglik.exp_series_md_c1_c2_c3 (2) score.exp_series_md_c1_c2_c3 (3) hess_loglik.exp_series_md_c1_c2_c3

These are useful for doing maximum likelihood estimation, hypothesis testing (e.g., likelihood ratio test), estimation of asymptotic sampling distribution given data from the DGP according to the specified model, etc.

It is designed to work well with the likelihood_model R package. In particular, it is intended to be used with the likelihood_contr_model object, which is a likelihood_model object that allows likelihood contributions to be added for whatever data model you have in mind.

In this likelihood model, masked component data approximately satisfies the following conditions:

C1: ⁠Pr{K[i] in C[i]) = 1⁠ C2: ⁠Pr{C[i]=c[i] | K[i]=j, T[i]=t[i]) = Pr(C[i]=c[i] | K[i]=j', T[i]=t[i])⁠ for any ⁠j, j' in c[i]⁠. C3: masking probabilities are independent of theta

As a special case, this model also includes exact component cause of failure data where the candidate set is a singleton.

Value

likelihood model object with class c("exp_series_md_c1_c2_c3", "series_md", "likelihood_model")

Examples

model <- exp_series_md_c1_c2_c3()
# Generate data and evaluate log-likelihood
set.seed(123)
gen <- rdata(model)
df <- gen(theta = c(0.5, 0.3, 0.2), n = 50, tau = 10, p = 0.3)
ll <- loglik(model)
ll(df, par = c(0.5, 0.3, 0.2))

Extract and validate model data from a masked data frame

Description

Shared validation logic for all likelihood model methods. Checks that the data frame is non-empty, required columns exist, decodes the candidate set matrix, and validates observation types.

Usage

extract_model_data(df, lifetime, omega, candset, lifetime_upper = NULL)

Arguments

Value

list with components: t (lifetimes), omega (character vector of observation types), C (candidate set matrix), m (number of components), n (number of observations), t_upper (upper bounds or NULL)

Extract model column name defaults

Description

Helper function to extract default column names from a likelihood model object. Used by all model methods to avoid repeating the same pattern.

Usage

extract_model_defaults(model)

Arguments

Value

list with lifetime, lifetime_upper, omega, candset defaults

Generate masked series system data

Description

Shared data generation logic for all rdata methods. Takes pre-generated component lifetimes and applies an observation mechanism, then generates candidate sets satisfying conditions C1, C2, C3.

Usage

generate_masked_series_data(
  comp_lifetimes,
  n,
  m,
  tau,
  p,
  default_lifetime,
  default_omega,
  default_candset,
  default_lifetime_upper = paste0(default_lifetime, "_upper"),
  observe = NULL
)

Arguments

Value

data frame with system lifetime, observation type, and candidate sets

Hazard function for exponential series.

Description

Hazard function for exponential series.

Usage

hazard_exp_series(t, rates, log.p = FALSE)

Arguments

Value

The hazard function evaluated at the specified lifetimes.

Examples

rates <- c(0.5, 0.3, 0.2)
hazard_exp_series(1.0, rates)
hazard_exp_series(c(0.5, 1.0), rates, log.p = TRUE)

Hessian of log-likelihood method for exp_series_md_c1_c2_c3 model.

Description

Returns the Hessian (second derivative matrix) of the log-likelihood for an exponential series system with respect to parameter theta for masked data with candidate sets that satisfy conditions C1, C2, and C3.

Usage

## S3 method for class 'exp_series_md_c1_c2_c3'
hess_loglik(model, ...)

Arguments

Details

All four observation types have closed-form Hessian contributions.

Value

hessian function that takes the following arguments:

• df: masked data frame

• par: rate parameters of exponential component lifetime distributions

• lifetime: system lifetime column name (default from model)

• lifetime_upper: interval upper bound column name (default from model)

• omega: observation type column name (default from model)

• candset: prefix of Boolean matrix encoding candidate sets

Examples

model <- exp_series_md_c1_c2_c3()
set.seed(1)
df <- rdata(model)(theta = c(0.5, 0.3, 0.2), n = 30, tau = 10, p = 0.3)
H <- hess_loglik(model)
H(df, par = c(0.5, 0.3, 0.2))

Hessian of log-likelihood method for wei_series_homogeneous_md_c1_c2_c3.

Description

Returns the Hessian (second derivative matrix) of the log-likelihood for a Weibull series system with homogeneous shape. Computed numerically via the Jacobian of the score.

Usage

## S3 method for class 'wei_series_homogeneous_md_c1_c2_c3'
hess_loglik(model, ...)

Arguments

Value

hessian function that takes the following arguments:

• df: masked data frame

• par: parameter vector (shape, scale1, scale2, ...)

• lifetime: system lifetime column name (default from model)

• lifetime_upper: interval upper bound column name (default from model)

• omega: observation type column name (default from model)

• candset: prefix of Boolean matrix encoding candidate sets

Examples

model <- wei_series_homogeneous_md_c1_c2_c3()
set.seed(1)
theta <- c(1.2, 1000, 900, 850)
df <- rdata(model)(theta = theta, n = 30, tau = 500, p = 0.3)
H <- hess_loglik(model)
H(df, par = theta)

Hessian of log-likelihood method for wei_series_md_c1_c2_c3 model.

Description

Returns the Hessian (second derivative matrix) of the log-likelihood for a Weibull series system. Computed numerically via the Jacobian of the score.

Usage

## S3 method for class 'wei_series_md_c1_c2_c3'
hess_loglik(model, ...)

Arguments

Value

hessian function that takes the following arguments:

• df: masked data frame

• par: parameter vector (shape1, scale1, shape2, scale2, ...)

• lifetime: system lifetime column name (default from model)

• lifetime_upper: interval upper bound column name (default from model)

• omega: observation type column name (default from model)

• candset: prefix of Boolean matrix encoding candidate sets

Examples

model <- wei_series_md_c1_c2_c3()
set.seed(1)
theta <- c(1.2, 1000, 0.8, 900)
df <- rdata(model)(theta = theta, n = 30, tau = 500, p = 0.3)
H <- hess_loglik(model)
H(df, par = theta)

Log-likelihood method for exp_series_md_c1_c2_c3 model.

Description

Returns a log-likelihood function for an exponential series system with respect to rate parameters for masked data with candidate sets that satisfy conditions C1, C2, and C3.

Usage

## S3 method for class 'exp_series_md_c1_c2_c3'
loglik(model, ...)

Arguments

Details

Supports four observation types: exact failures, right-censored, left-censored, and interval-censored. All have closed-form likelihood contributions for the exponential model.

Value

log-likelihood function that takes the following arguments:

• df: masked data frame

• par: rate parameters of exponential component lifetime distributions

• lifetime: system lifetime column name (default from model)

• lifetime_upper: interval upper bound column name (default from model)

• omega: observation type column name (default from model)

• candset: prefix of Boolean matrix encoding candidate sets

Examples

model <- exp_series_md_c1_c2_c3()
set.seed(1)
df <- rdata(model)(theta = c(0.5, 0.3, 0.2), n = 30, tau = 10, p = 0.3)
ll <- loglik(model)
ll(df, par = c(0.5, 0.3, 0.2))

Log-likelihood method for wei_series_homogeneous_md_c1_c2_c3 model.

Description

Returns a log-likelihood function for a Weibull series system with homogeneous shape parameter. The parameter vector is (k, scale_1, ..., scale_m) for masked data with candidate sets that satisfy conditions C1, C2, and C3.

Usage

## S3 method for class 'wei_series_homogeneous_md_c1_c2_c3'
loglik(model, ...)

Arguments

Details

Supports four observation types. Left-censored and interval-censored have closed-form likelihood contributions because all shapes are equal.

Value

log-likelihood function that takes the following arguments:

• df: masked data frame

• par: parameter vector (shape, scale1, scale2, ...)

• lifetime: system lifetime column name (default from model)

• lifetime_upper: interval upper bound column name (default from model)

• omega: observation type column name (default from model)

• candset: prefix of Boolean matrix encoding candidate sets

Examples

model <- wei_series_homogeneous_md_c1_c2_c3()
set.seed(1)
# theta: (shape, scale1, scale2, scale3)
theta <- c(1.2, 1000, 900, 850)
df <- rdata(model)(theta = theta, n = 30, tau = 500, p = 0.3)
ll <- loglik(model)
ll(df, par = theta)

Log-likelihood method for wei_series_md_c1_c2_c3 model.

Description

Returns a log-likelihood function for a Weibull series system with respect to parameter vector (shape_1, scale_1, ..., shape_m, scale_m) for masked data with candidate sets that satisfy conditions C1, C2, and C3.

Usage

## S3 method for class 'wei_series_md_c1_c2_c3'
loglik(model, ...)

Arguments

Details

Supports four observation types. Left-censored and interval-censored observations require numerical integration (via stats::integrate) because heterogeneous shapes prevent a closed-form solution.

Value

log-likelihood function that takes the following arguments:

• df: masked data frame

• par: parameter vector (shape1, scale1, shape2, scale2, ...)

• lifetime: system lifetime column name (default from model)

• lifetime_upper: interval upper bound column name (default from model)

• omega: observation type column name (default from model)

• candset: prefix of Boolean matrix encoding candidate sets

Examples

model <- wei_series_md_c1_c2_c3()
set.seed(1)
theta <- c(1.2, 1000, 0.8, 900)
df <- rdata(model)(theta = theta, n = 30, tau = 500, p = 0.3)
ll <- loglik(model)
ll(df, par = theta)

Create numeric Hessian function from score function via Jacobian

Description

Factory function for creating Hessian functions using numerical differentiation of the score function. Used by Weibull models which compute score analytically but need numerical Hessian.

Usage

make_numeric_hessian(score_fn, model)

Arguments

Value

function(df, par, ...) that computes the Hessian matrix

Bernoulli candidate set model for systems with unobserved components.

Description

Bernoulli candidate set model is a particular type of uninformed model. Note that we do not generate candidate sets with this function. See md_cand_sampler for that.

Usage

md_bernoulli_cand_c1_c2_c3(
  df,
  p,
  prob = "q",
  comp = "t",
  right_censoring_indicator = "delta"
)

Arguments

Details

This model satisfies conditions C1, C2, and C3. The failed component will be in the corresponding candidate set with probability 1, and the remaining components will be in the candidate set with probability p (the same probability for each component). p may be different for each system, but it is assumed to be the same for each component within a system, so p can be a vector such that the length of p is the number of systems in the data set (with recycling if necessary).

Value

Data frame with added candidate set probability columns (e.g., q1, q2, ..., qm).

Examples

# Generate component lifetimes and system data
mat <- matrix(rexp(9, rate = rep(c(0.5, 0.3, 0.2), each = 3)),
              nrow = 3, ncol = 3)
df <- md_encode_matrix(mat, "t")
df$t <- apply(mat, 1, min)
df$delta <- TRUE
md_bernoulli_cand_c1_c2_c3(df, p = 0.5)

Convert Boolean candidate set columns to character set notation

Description

Replaces Boolean matrix columns (e.g., ⁠x1, x2, x3⁠) with a single character column showing set notation like ⁠{1, 3}⁠.

Usage

md_boolean_matrix_to_charsets(df, setvar = "x", cname = NULL, drop_set = FALSE)

Arguments

Value

data frame with character set column added

Examples

df <- data.frame(x1 = c(TRUE, FALSE, TRUE),
                 x2 = c(TRUE, TRUE, FALSE),
                 x3 = c(FALSE, TRUE, TRUE))
md_boolean_matrix_to_charsets(df)

Sample candidate sets for systems with unobserved components.

Description

Candidate set generator. Requires columns for component probabilities e.g., ⁠q1,...,qm⁠ where qj is the probability that the jth component will be in the corresponding candidate set generated for that observation in the md table.

Usage

md_cand_sampler(df, prob = "q", candset = "x")

Arguments

Value

Data frame with added Boolean candidate set columns (e.g., x1, x2, ..., xm).

Examples

# Generate component lifetimes
set.seed(42)
mat <- matrix(rexp(9, rate = rep(c(0.5, 0.3, 0.2), each = 3)),
              nrow = 3, ncol = 3)
df <- md_encode_matrix(mat, "t")
df$t <- apply(mat, 1, min)
df$delta <- TRUE
df <- md_bernoulli_cand_c1_c2_c3(df, p = 0.5)
md_cand_sampler(df)

Decode a matrix from prefixed columns in a data frame

Description

Extracts columns matching the pattern ⁠var1, var2, ...⁠ or ⁠var.1, var.2, ...⁠ from a data frame and returns them as a numeric matrix, ordered by index.

Usage

md_decode_matrix(df, var)

Arguments

Value

a matrix, or NULL if no matching columns are found

Encode a matrix as a data frame with prefixed column names

Description

Converts a matrix to a data frame with columns named ⁠var1, var2, ...⁠.

Usage

md_encode_matrix(mat, var)

Arguments

Value

a data frame with named columns

Examples

mat <- matrix(1:6, nrow = 2, ncol = 3)
md_encode_matrix(mat, "x")

Mark columns as latent in a masked data frame

Description

Sets the "latent" attribute on a data frame, recording which columns represent unobserved (latent) variables.

Usage

md_mark_latent(md, vars)

Arguments

Value

the data frame with updated latent attribute

Masked data generation for series system lifetime data

Description

Generates right-censored system failure times and right-censoring indicators for a series system with the given data frame of component lifetimes.

Usage

md_series_lifetime_right_censoring(
  df,
  tau = Inf,
  comp = "t",
  lifetime = "t",
  right_censoring_indicator = "delta"
)

Arguments

Value

(masked) data frame with masked data as described in the paper

Examples

mat <- matrix(rexp(9, rate = 0.5), nrow = 3, ncol = 3)
df <- md_encode_matrix(mat, "t")
md_series_lifetime_right_censoring(df, tau = 5)

Mean function for exponential series.

Description

Computes the expected value of a series system with exponentially distributed component lifetimes. For a series system with component rates \lambda_1, \ldots, \lambda_m, the system lifetime is exponential with rate \sum \lambda_j, so E[T] = 1 / \sum \lambda_j.

Usage

## S3 method for class 'exp_series'
mean(x, ...)

Arguments

Value

The mean of the exponential series distribution (1/sum of rates).

Examples

rates <- structure(c(0.5, 0.3, 0.2), class = "exp_series")
mean(rates)

Number of components in a series system model

Description

Returns the number of components m in the series system. If the model was constructed without parameter values, returns NULL.

Usage

ncomponents(model, ...)

Arguments

Value

integer number of components, or NULL if not determinable

Examples

model <- exp_series_md_c1_c2_c3(rates = c(0.5, 0.3, 0.2))
ncomponents(model)

Left-censoring observation scheme (single inspection)

Description

Creates an observation functor for a single-inspection design. If the system has already failed by inspection time tau, we know it failed before tau but not exactly when (left-censored). If it is still running, we know it survived past tau (right-censored).

Usage

observe_left_censor(tau)

Arguments

Value

A function with signature function(t_true) returning a list with components:

t

inspection time tau

omega

"left" if failed before tau, "right" otherwise

t_upper

NA (not used for this scheme)

Examples

obs <- observe_left_censor(tau = 100)
obs(50)   # left-censored: list(t = 100, omega = "left", t_upper = NA)
obs(150)  # right-censored: list(t = 100, omega = "right", t_upper = NA)

Mixture of observation schemes

Description

Creates an observation functor that randomly selects from a set of observation schemes for each observation. This models heterogeneous monitoring environments where different units are observed differently.

Usage

observe_mixture(..., weights = NULL)

Arguments

Value

A function with signature function(t_true) returning a list from one of the constituent schemes, selected randomly according to weights.

Examples

obs <- observe_mixture(
  observe_right_censor(tau = 100),
  observe_left_censor(tau = 50),
  weights = c(0.7, 0.3)
)
set.seed(42)
obs(30)  # randomly selects one of the two schemes

Periodic inspection observation scheme

Description

Creates an observation functor for periodic inspections at intervals of delta. Failures are bracketed between the last inspection before failure and the first inspection after failure (interval-censored). Systems surviving past tau are right-censored.

Usage

observe_periodic(delta, tau = Inf)

Arguments

Value

A function with signature function(t_true) returning a list with components:

t

lower bound of interval (or tau if right-censored)

omega

"interval" or "right"

t_upper

upper bound of interval (NA if right-censored)

Examples

obs <- observe_periodic(delta = 10, tau = 100)
obs(25)   # interval: list(t = 20, omega = "interval", t_upper = 30)
obs(150)  # right-censored: list(t = 100, omega = "right", t_upper = NA)

Right-censoring observation scheme

Description

Creates an observation functor that applies right-censoring at time tau. Systems that fail before tau are observed exactly; systems surviving past tau are right-censored.

Usage

observe_right_censor(tau)

Arguments

Value

A function with signature function(t_true) returning a list with components:

t

observed time

omega

"exact" or "right"

t_upper

NA (not used for this scheme)

Examples

obs <- observe_right_censor(tau = 100)
obs(50)   # exact: list(t = 50, omega = "exact", t_upper = NA)
obs(150)  # right-censored: list(t = 100, omega = "right", t_upper = NA)

Cumulative distribution function for exponential series.

Description

Cumulative distribution function for exponential series.

Usage

pexp_series(t, rates, lower.tail = TRUE, log.p = FALSE)

Arguments

Value

The cumulative probabilities evaluated at the specified lifetimes.

Examples

rates <- c(0.5, 0.3, 0.2)
pexp_series(1.0, rates)
pexp_series(c(0.5, 1.0, 2.0), rates)

Quantile function for a component with custom survival function

Description

Finds the time t such that S(t) = p using root finding. The survival function S(t) is assumed to be monotonically decreasing from S(0) = 1 to S(inf) = 0.

Usage

qcomp(
  p,
  surv,
  theta,
  t_lower = .Machine$double.eps,
  t_upper = .Machine$double.xmax^0.5,
  ...
)

Arguments

Value

time t such that S(t) = p

Examples

# Exponential survival function
surv_exp <- function(t, theta) exp(-theta * t)

# Median lifetime (50th percentile) for rate = 2
qcomp(0.5, surv = surv_exp, theta = 2.0)

Quantile function for exponential series.

Description

Quantile function for exponential series.

Usage

qexp_series(p, rates, lower.tail = TRUE, log.p = FALSE)

Arguments

Value

Quantiles corresponding to the given probabilities p.

Examples

rates <- c(0.5, 0.3, 0.2)
qexp_series(0.5, rates)
qexp_series(c(0.25, 0.5, 0.75), rates)

Random generation for a component with custom survival function

Description

Generates random samples using inverse transform sampling.

Usage

rcomp(n, surv, theta)

Arguments

Value

vector of n random samples

Examples

# Exponential survival function
surv_exp <- function(t, theta) exp(-theta * t)

# Generate 10 random samples with rate = 2
set.seed(123)
rcomp(10, surv = surv_exp, theta = 2.0)

Random data generation for exp_series_md_c1_c2_c3 model.

Description

Returns a function that generates random masked data from the exponential series system DGP at a given parameter value.

Usage

## S3 method for class 'exp_series_md_c1_c2_c3'
rdata(model, ...)

Arguments

Value

function that takes (theta, n, tau, p, observe, ...) and returns a data frame with columns: t, omega, x1, x2, ..., xm

Examples

model <- exp_series_md_c1_c2_c3()
gen <- rdata(model)
set.seed(42)
df <- gen(theta = c(0.5, 0.3, 0.2), n = 10, tau = 5, p = 0.5)
head(df)

Random data generation for wei_series_homogeneous_md_c1_c2_c3 model.

Description

Returns a function that generates random masked data from the homogeneous Weibull series system DGP at a given parameter value.

Usage

## S3 method for class 'wei_series_homogeneous_md_c1_c2_c3'
rdata(model, ...)

Arguments

Value

function that takes (theta, n, tau, p, observe, ...) and returns a data frame with columns: t, omega, x1, x2, ..., xm

Examples

model <- wei_series_homogeneous_md_c1_c2_c3()
gen <- rdata(model)
set.seed(42)
df <- gen(theta = c(1.2, 1000, 900, 850), n = 10, tau = 500, p = 0.5)
head(df)

Random data generation for wei_series_md_c1_c2_c3 model.

Description

Returns a function that generates random masked data from the Weibull series system DGP at a given parameter value.

Usage

## S3 method for class 'wei_series_md_c1_c2_c3'
rdata(model, ...)

Arguments

Value

function that takes (theta, n, tau, p, observe, ...) and returns a data frame with columns: t, omega, x1, x2, ..., xm

Examples

model <- wei_series_md_c1_c2_c3()
gen <- rdata(model)
set.seed(42)
# theta: (shape1, scale1, shape2, scale2)
df <- gen(theta = c(1.2, 1000, 0.8, 900), n = 10, tau = 500, p = 0.5)
head(df)

Objects exported from other packages

Description

These objects are imported from other packages. Follow the links below to see their documentation.

generics

fit

likelihood.model

assumptions, fim, hess_loglik, loglik, observed_info, rdata, score

Random number generation for exponential series.

Description

Generates random variates from an exponential series distribution.

Usage

rexp_series(n, rates, keep_latent = FALSE)

Arguments

Value

If keep_latent = FALSE, a vector of random variates from the exponential series distribution. If keep_latent = TRUE, a matrix with system lifetime in the first column and component lifetimes in columns 2 through m+1.

Examples

set.seed(123)
rexp_series(5, rates = c(0.5, 0.3, 0.2))
rexp_series(3, rates = c(0.5, 0.3, 0.2), keep_latent = TRUE)

Score method for exp_series_md_c1_c2_c3 model.

Description

Returns a score (gradient) function for an exponential series system with respect to parameter theta for masked component failure with candidate sets that satisfy conditions C1, C2, and C3.

Usage

## S3 method for class 'exp_series_md_c1_c2_c3'
score(model, ...)

Arguments

Details

All four observation types (exact, right, left, interval) have closed-form score contributions.

Value

score function that takes the following arguments:

• df: masked data frame

• par: rate parameters of exponential component lifetime distributions

• lifetime: system lifetime column name (default from model)

• lifetime_upper: interval upper bound column name (default from model)

• omega: observation type column name (default from model)

• candset: prefix of Boolean matrix encoding candidate sets

Examples

model <- exp_series_md_c1_c2_c3()
set.seed(1)
df <- rdata(model)(theta = c(0.5, 0.3, 0.2), n = 30, tau = 10, p = 0.3)
s <- score(model)
s(df, par = c(0.5, 0.3, 0.2))

Score method for wei_series_homogeneous_md_c1_c2_c3 model.

Description

Returns a score (gradient) function for a Weibull series system with homogeneous shape parameter. The parameter vector is (k, scale_1, ..., scale_m) for masked data with candidate sets that satisfy conditions C1, C2, and C3.

Usage

## S3 method for class 'wei_series_homogeneous_md_c1_c2_c3'
score(model, ...)

Arguments

Details

Uses a hybrid approach: analytical score for exact and right-censored observations, numerical gradient (via numDeriv) for left-censored and interval-censored observations.

Value

score function that takes the following arguments:

• df: masked data frame

• par: parameter vector (shape, scale1, scale2, ...)

• lifetime: system lifetime column name (default from model)

• lifetime_upper: interval upper bound column name (default from model)

• omega: observation type column name (default from model)

• candset: prefix of Boolean matrix encoding candidate sets

Examples

model <- wei_series_homogeneous_md_c1_c2_c3()
set.seed(1)
theta <- c(1.2, 1000, 900, 850)
df <- rdata(model)(theta = theta, n = 30, tau = 500, p = 0.3)
s <- score(model)
s(df, par = theta)

Score method for wei_series_md_c1_c2_c3 model.

Description

Returns a score (gradient) function for a Weibull series system with respect to parameter vector (shape_1, scale_1, ..., shape_m, scale_m) for masked data with candidate sets that satisfy conditions C1, C2, and C3.

Usage

## S3 method for class 'wei_series_md_c1_c2_c3'
score(model, ...)

Arguments

Details

Uses a hybrid approach: analytical score for exact and right-censored observations, numerical gradient (via numDeriv) for left-censored and interval-censored observations.

Value

score function that takes the following arguments:

• df: masked data frame

• par: parameter vector (shape1, scale1, shape2, scale2, ...)

• lifetime: system lifetime column name (default from model)

• lifetime_upper: interval upper bound column name (default from model)

• omega: observation type column name (default from model)

• candset: prefix of Boolean matrix encoding candidate sets

Examples

model <- wei_series_md_c1_c2_c3()
set.seed(1)
theta <- c(1.2, 1000, 0.8, 900)
df <- rdata(model)(theta = theta, n = 30, tau = 500, p = 0.3)
s <- score(model)
s(df, par = theta)

Survival function for exponential series.

Description

Survival function for exponential series.

Usage

surv.exp_series(t, rates, log.p = FALSE)

Arguments

Value

The survival function evaluated at the specified lifetimes.

Examples

rates <- c(0.5, 0.3, 0.2)
surv.exp_series(1.0, rates)
surv.exp_series(c(0.5, 1.0, 2.0), rates)

Constructs a likelihood model for wei_series_homogeneous_md_c1_c2_c3.

Description

Likelihood model for Weibull series systems with homogeneous shape parameter and masked component cause of failure with candidate sets that satisfy conditions C1, C2, and C3.

Usage

wei_series_homogeneous_md_c1_c2_c3(
  shape = NULL,
  scales = NULL,
  lifetime = "t",
  lifetime_upper = "t_upper",
  omega = "omega",
  candset = "x"
)

Arguments

Details

This is a reduced model where all components share a common shape parameter k, while retaining individual scale parameters. The parameter vector is (k, scale_1, ..., scale_m), giving m+1 parameters instead of 2m.

A key property of this model is that the series system lifetime is itself Weibull distributed with shape k and scale \lambda_s = (\sum \beta_j^{-k})^{-1/k}.

This model satisfies the concept of a likelihood_model in the likelihood.model package by providing the following methods:

(1) loglik.wei_series_homogeneous_md_c1_c2_c3 (2) score.wei_series_homogeneous_md_c1_c2_c3 (3) hess_loglik.wei_series_homogeneous_md_c1_c2_c3

In this likelihood model, masked component data approximately satisfies:

C1: ⁠Pr{K[i] in C[i]} = 1⁠ C2: ⁠Pr{C[i]=c[i] | K[i]=j, T[i]=t[i]} = Pr{C[i]=c[i] | K[i]=j', T[i]=t[i]}⁠ for any ⁠j, j' in c[i]⁠. C3: masking probabilities are independent of theta

Value

likelihood model object with class c("wei_series_homogeneous_md_c1_c2_c3", "series_md", "likelihood_model")

See Also

wei_series_md_c1_c2_c3() for the full model with heterogeneous shapes

Examples

# Create model and fit to data using generic dispatch
model <- wei_series_homogeneous_md_c1_c2_c3()
# solver <- fit(model)
# theta: (shape, scale1, scale2, ...)
# mle <- solver(data, par = c(1.2, 1000, 900, 850))

Integrand for numerical integration of Weibull series likelihood

Description

Computes h_c(t) S(t) where h_c = \sum_{j \in c} h_j(t) and S(t) = \exp(-\sum_l H_l(t)). Used for left-censored and interval-censored observations in the heterogeneous Weibull model.

Usage

wei_series_integrand(t, shapes, scales, cind)

Arguments

Value

vector of integrand values

Constructs a likelihood model for wei_series_md_c1_c2_c3.

Description

Likelihood model for Weibull series systems with masked component cause of failure with candidate sets that satisfy conditions C1, C2, and C3.

Usage

wei_series_md_c1_c2_c3(
  shapes = NULL,
  scales = NULL,
  lifetime = "t",
  lifetime_upper = "t_upper",
  omega = "omega",
  candset = "x"
)

Arguments

Details

This model satisfies the concept of a likelihood_model in the likelihood.model package by providing the following methods:

(1) loglik.wei_series_md_c1_c2_c3 (2) score.wei_series_md_c1_c2_c3 (3) hess_loglik.wei_series_md_c1_c2_c3

The Weibull series system has 2m parameters: (shape_1, scale_1, ..., shape_m, scale_m).

In this likelihood model, masked component data approximately satisfies:

C1: ⁠Pr{K[i] in C[i]} = 1⁠ C2: ⁠Pr{C[i]=c[i] | K[i]=j, T[i]=t[i]} = Pr{C[i]=c[i] | K[i]=j', T[i]=t[i]}⁠ for any ⁠j, j' in c[i]⁠. C3: masking probabilities are independent of theta

Value

likelihood model object with class c("wei_series_md_c1_c2_c3", "series_md", "likelihood_model")

Examples

# Create model and fit to data using generic dispatch
model <- wei_series_md_c1_c2_c3()
# solver <- fit(model)
# theta: (shape1, scale1, shape2, scale2, ...)
# mle <- solver(data, par = c(1, 1000, 1, 1000, 1, 1000))

System scale parameter for homogeneous Weibull series

Description

For a series system with Weibull components sharing shape k but with individual scales, the system lifetime is itself Weibull with shape k and this computed scale.

Usage

wei_series_system_scale(k, scales)

Arguments

Value

system scale parameter

Examples

# 3-component system with common shape 1.2
wei_series_system_scale(k = 1.2, scales = c(1000, 900, 850))