Helper functions for analytical solutions. More...

Collaboration diagram for Helpers for Analytical Solutions:

Functions
int	check_for_uniform_terms (int dist_id, int primary_id)

int	check_for_analytical (int dist_id, int primary_id)

real	primarycensored_uniform_lcdf_from_terms (vector terms_d, vector terms_q, data real pwindow)

vector	primarycensored_gamma_uniform_terms (real t, array[] real params)

vector	primarycensored_lognormal_uniform_terms (real t, array[] real params)

real	log_weibull_g (real t, real shape, real scale)

vector	primarycensored_weibull_uniform_terms (real t, array[] real params)

vector	primarycensored_gengamma_uniform_terms (real t, array[] real params)

vector	primarycensored_uniform_terms (real t, data int dist_id, array[] real params)

int	check_for_analytical_vectorized (int dist_id, int primary_id, data real pwindow)

Description

Helper functions for analytical solutions.

Function Documentation

◆ check_for_analytical()

int check_for_analytical	(	int	dist_id,
		int	primary_id )

Check if an analytical solution exists for the given distribution combination

The non-parametric step (26) and discrete-hazard (27, 28) delays are analytic for every primary primary_lcdf currently supports, the uniform (1) and exponential growth (2). That list is repeated by hand below, so adding a primary to primary_lcdf does not extend the analytic path on its own: without a matching update here the new primary silently falls back to numerical integration.

Parameters

dist_id	Distribution identifier for the delay distribution
primary_id	Distribution identifier for the primary distribution

Returns: 1 if an analytical solution exists, 0 otherwise

Definition at line 37 of file primarycensored_analytical_cdf.stan.

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◆ check_for_analytical_vectorized()

int check_for_analytical_vectorized	(	int	dist_id,
		int	primary_id,
		data real	pwindow )

Check if the analytical solution can be vectorised over integer delays

The analytical uniform primary CDF at d combines terms at d and at q = max(d - pwindow, 0). With an integer pwindow q is an integer delay too, so primarycensored_analytical_lcdf_vectorized() can compute the terms once per delay and share them. This needs the analytical solutions built from primarycensored_uniform_terms(), see check_for_uniform_terms(). The non-parametric delays in check_for_analytical() have no such terms.

Parameters

dist_id	Distribution identifier for the delay distribution
primary_id	Distribution identifier for the primary distribution
pwindow	Primary event window

Returns: 1 if the vectorised analytical solution applies, 0 otherwise

Definition at line 455 of file primarycensored_analytical_cdf.stan.

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◆ check_for_uniform_terms()

int check_for_uniform_terms	(	int	dist_id,
		int	primary_id )

Check if the analytical solution is built from uniform primary terms

These are the delays whose censored CDF with a uniform primary is primarycensored_uniform_lcdf_from_terms() applied to primarycensored_uniform_terms() at d and q.

Parameters

dist_id	Distribution identifier for the delay distribution
primary_id	Distribution identifier for the primary distribution

Returns: 1 if the solution is built from uniform primary terms, 0 otherwise

Definition at line 15 of file primarycensored_analytical_cdf.stan.

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◆ log_weibull_g()

real log_weibull_g	(	real	t,
		real	shape,
		real	scale )

Compute the log of the lower incomplete gamma function

This function is used in the analytical solution for the primary censored Weibull distribution with uniform primary censoring. It corresponds to the g(t; λ, k) function described in the analytic solutions document.

Parameters

t	Upper bound of integration
shape	Shape parameter (k) of the Weibull distribution
scale	Scale parameter (λ) of the Weibull distribution

Returns: Log of g(t; λ, k) = γ(1 + 1/k, (t/λ)^k)

Definition at line 153 of file primarycensored_analytical_cdf.stan.

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◆ primarycensored_gamma_uniform_terms()

vector primarycensored_gamma_uniform_terms	(	real	t,
		array[]real	params )

Compute the uniform primary terms at t for a Gamma delay

Parameters

t	Time (d or q)
params	Array of Gamma distribution parameters [shape, rate]

Returns: Vector [log(t * F_T(t; k)), log(E * F_T(t; k + 1))], both -inf for t <= 0

Definition at line 91 of file primarycensored_analytical_cdf.stan.

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◆ primarycensored_gengamma_uniform_terms()

vector primarycensored_gengamma_uniform_terms	(	real	t,
		array[]real	params )

Compute the uniform primary terms at t for a generalised gamma delay

Uses the Stacy parameterisation of flexsurv::pgengamma.orig(), see gengamma_lcdf. The mean is E = scale * Gamma(k + 1/shape) / Gamma(k) and the partial expectation distribution is the generalised gamma with k replaced by k + 1/shape, so this generalises the Gamma (shape = 1) and Weibull (k = 1) solutions.

Parameters

t	Time (d or q)
params	Array of generalised gamma distribution parameters [shape, scale, k]

Returns: Vector [log(t * F_T(t)), log(E * tilde F_T(t))], both -inf for t <= 0

Definition at line 201 of file primarycensored_analytical_cdf.stan.

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◆ primarycensored_lognormal_uniform_terms()

vector primarycensored_lognormal_uniform_terms	(	real	t,
		array[]real	params )

Compute the uniform primary terms at t for a Lognormal delay

Each term is formed whole and dropped whole. Adding a -inf log CDF to the parameter-dependent log(t) or log_E first would leave an edge back to the parameters that log_sum_exp differentiates to exp(-inf - -inf). t <= 0 underflows on the same test.

Parameters

t	Time (d or q)
params	Array of Lognormal distribution parameters [mu, sigma]

Returns: Vector [log(t * F_T(t)), log(E * tilde F_T(t))]

Definition at line 123 of file primarycensored_analytical_cdf.stan.

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◆ primarycensored_uniform_lcdf_from_terms()

real primarycensored_uniform_lcdf_from_terms	(	vector	terms_d,
		vector	terms_q,
		data real	pwindow )

Combine the uniform primary terms at d and q into the censored log CDF

For a delay T with mean E and a uniform primary over a window of width w_P, the primary event censored CDF at d is F_{S+}(d) = (A - B) / w_P, with A = d * F_T(d) + E * tilde F_T(q), B = q * F_T(q) + E * tilde F_T(d), where q = max(d - w_P, 0) and tilde F_T is the CDF of the partial expectation distribution. Each of A and B is a sum of one term at d and one at q, and those terms depend on d or q alone (see primarycensored_uniform_terms()). Ordering A >= B is guaranteed by F_{S+}(d) >= 0.

Parameters

terms_d	Terms at d from primarycensored_uniform_terms()
terms_q	Terms at q from primarycensored_uniform_terms()
pwindow	Primary event window

Returns: Log of the primary event censored CDF at d

Definition at line 68 of file primarycensored_analytical_cdf.stan.

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◆ primarycensored_uniform_terms()

vector primarycensored_uniform_terms	(	real	t,
		data int	dist_id,
		array[]real	params )

Compute the uniform primary terms at t for a delay distribution

Parameters

t	Time (d or q)
dist_id	Distribution identifier (1: Lognormal, 2: Gamma, 3: Weibull, 5: Generalised gamma), see check_for_uniform_terms()
params	Array of distribution parameters