Derivatives of phase cumulative and instantaneous hazard w.r.t. shape params

Computes \(\Phi_j(t)\), \(\phi_j(t)\), and their derivatives with respect to log_t_half, nu, and m using finite differences on hzr_decompos(): central in log_t_half and nu, and in m central except near m = 0, where the stencil keeps the sign of m. hzr_decompos() changes formula at m = 0 and the two sides meet in a cusp, so for m >= 0 a stencil that would reach 0 becomes one-sided forward (second order), and for m < 0 the step is capped at 1% of |m| (floored at 1e-10) and becomes one-sided backward if it would still reach 0. The log_t_half derivative is a central difference in log_t_half itself, so its step is proportional to t_half at every scale. It is NaN where the two points of the difference cannot both be used: t_half too small or too large to step, or a point at which hzr_decompos() fails. A "hazard" phase's value is -log_surv from hzr_decompos(), which keeps its accuracy far past saturation, so the derivative does too.

Usage

.hzr_phase_derivatives(
  time,
  t_half,
  nu,
  m,
  type = c("cdf", "hazard", "constant")
)

Arguments

time

Numeric vector of positive times.

t_half

Positive scalar half-life.

nu

Numeric scalar time exponent.

m

Numeric scalar shape exponent.

type

Phase type: "cdf", "hazard", or "constant".

Value

Named list:

Phi

Cumulative hazard contribution \(\Phi(t)\).

phi

Instantaneous hazard contribution \(\phi(t) = d\Phi/dt\).

dPhi_dlog_thalf

\(d\Phi / d(\log t_{1/2})\).

dPhi_dnu

\(d\Phi / d\nu\).

dPhi_dm

\(d\Phi / dm\).

dphi_dlog_thalf

\(d\phi / d(\log t_{1/2})\).

dphi_dnu

\(d\phi / d\nu\).

dphi_dm

\(d\phi / dm\).