Example: Multi-start fitting

For models with ridged or multi-modal OFV surfaces — Michaelis-Menten elimination is the classic case — a single fit from one set of starting values often lands on a local minimum. Multi-start runs n_starts independent fits from perturbed initial values in parallel and returns the one with the lowest OFV.

When to use multi-start

  • Michaelis-Menten (Vmax/Km) models: Vmax and Km are weakly identifiable from sparse data and the OFV surface has a narrow ridge
  • Full block-omega models with many correlated ETAs
  • Models with many covariate parameters where the optimiser can stall

How it works

  • All starts apply a log-space perturbation of size start_sigma to the initial theta values. The default start_sigma = 0.3 is a ±30% multiplicative perturbation on the natural scale
  • All starts run in parallel via Rayon; on an 8-core machine n_starts = 8 costs approximately the same wall-clock time as a single run
  • The fit with the lowest OFV is returned

Model file

The mm_multistart example is a Michaelis-Menten one-compartment oral model with starting values intentionally set above the true values to demonstrate local-minimum trapping:

[parameters]
  # Initial values are intentionally high — a single run often stays here.
  # Multi-start explores the neighbourhood and finds the true optimum.
  theta TVVMAX(12.0, 0.1, 50.0)    # maximum elimination rate (mg/h); true ≈ 3.5
  theta TVKM(20.0,  0.1, 100.0)    # Michaelis constant (mg/L);       true ≈ 5.5
  theta TVV(11.0,   1.0, 200.0)    # volume of distribution (L);       true ≈ 11
  theta TVKA(1.3,   0.05, 20.0)    # absorption rate constant (1/h);   true ≈ 1.3

  omega ETA_VMAX ~ 0.25
  omega ETA_V    ~ 0.20

  sigma PROP_ERR ~ 0.04 (sd)

[individual_parameters]
  VMAX = TVVMAX * exp(ETA_VMAX)
  KM   = TVKM
  V    = TVV * exp(ETA_V)
  KA   = TVKA

[structural_model]
  ode(obs_cmt=central, states=[depot, central])

[odes]
  d/dt(depot)   = -KA * depot
  d/dt(central) = KA * depot / V - VMAX * central / (KM + central)

[error_model]
  DV ~ proportional(PROP_ERR)

[fit_options]
  method           = focei
  maxiter          = 500
  covariance       = true
  n_starts         = 8
  start_sigma      = 0.5
  multi_start_seed = 42

Running

library(ferx)
ex <- ferx_example("mm_multistart")

# Single-start — override the model-file default of n_starts = 8
fit_single <- ferx_fit(ex$model, ex$data,
                       settings = list(n_starts = 1L))

# Multi-start — 8 parallel runs with wider perturbation
fit_multi  <- ferx_fit(ex$model, ex$data)

cat("Single-start OFV:", fit_single$ofv, "\n")
cat("Multi-start OFV: ", fit_multi$ofv,  "\n")

print(fit_multi)

The single-start run from the inflated starting values often converges to a local minimum with TVVMAX and TVKM both higher than the true values. The multi-start run returns estimates close to the simulation truth (TVVMAX ≈ 3.5, TVKM ≈ 5.5).

Fit options reference

Key Default Description
n_starts 1 Number of independent runs. 1 = single run, no overhead
start_sigma 0.3 Log-space perturbation size for starts 1..n
multi_start_seed — RNG seed for reproducible perturbations

See also