CGNM: Cluster Gauss-Newton Method

An R package to find multiple approximate minimizers of a nonlinear least squares problem

argmin_x || f(x) - y* ||

without assuming the minimizer is unique. In the context of model fitting, f is the model, x is the parameter to estimate, and y* is the observed data. Because CGNM searches from a range of initial iterates rather than a single starting point, and returns many approximate minimizers at once, it can also reveal when a model's parameters are not practically identifiable (the returned minimizers won't converge to a single point).

See the papers below for the algorithm and comparisons with conventional multi-start optimization:

• Aoki, Y., Hayami, K., Toshimoto, K., & Sugiyama, Y. (2020). Cluster Gauss–Newton method. Optimization and Engineering, 1-31. doi:10.1007/s11081-020-09571-2
• Aoki, Y. & Sugiyama, Y. (2024). Cluster Gauss-Newton method for a quick approximation of profile likelihood: With application to physiologically-based pharmacokinetic models. CPT Pharmacometrics Syst Pharmacol. 13(1):54-67. doi:10.1002/psp4.13055

If you use CGNM in your research, please cite the relevant paper(s) above.

When to use CGNM

• You want to fit a relatively complex parameterized model to data but are unsure of a good initial guess — CGNM takes an initial range instead of a point.
• You want to know whether the fit is practically identifiable (one best-fit parameter set, or many) — CGNM returns multiple approximate minimizers so non-identifiability shows up directly.
• Your model is a black box (not necessarily continuous or differentiable in its parameters) — CGNM makes no assumptions about the model's internal form.

When not to use CGNM

• You already know roughly where the best-fit parameter is and just need one answer — plain nls() will be faster.
• The model is cheap to evaluate and computational cost isn't a concern — CGNM's main advantage is reducing the number of model evaluations needed compared to repeated multi-start nls().

Installation

# from a local checkout of this repository
install.packages("path/to/CGNM", repos = NULL, type = "source")

# or, from within the checked-out directory
# R CMD INSTALL .

Quick example

library(CGNM)

# flip-flop kinetics: a model known to have two distinct best-fit solutions
model_function <- function(x) {
  observation_time <- c(0.1, 0.2, 0.4, 0.6, 1, 2, 3, 6, 12)
  Dose <- 1000; F <- 1
  ka <- x[1]; V1 <- x[2]; CL_2 <- x[3]
  t <- observation_time
  Cp <- ka * F * Dose / (V1 * (ka - CL_2 / V1)) * (exp(-CL_2 / V1 * t) - exp(-ka * t))
  log10(Cp)
}

observation <- log10(c(4.91, 8.65, 12.4, 18.7, 24.3, 24.5, 18.4, 4.66, 0.238))

CGNM_result <- Cluster_Gauss_Newton_method(
  nonlinearFunction  = model_function,
  targetVector       = observation,
  initial_lowerRange = rep(0.01, 3),
  initial_upperRange = rep(100, 3),
  saveLog            = FALSE
)

acceptedApproximateMinimizers(CGNM_result)

Then inspect the fit visually:

library(ggplot2)
plot_Rank_SSR(CGNM_result)
plot_goodnessOfFit(CGNM_result, plotType = 1,
                    independentVariableVector = c(0.1,0.2,0.4,0.6,1,2,3,6,12),
                    plotRank = seq(1, 50))

For the full workflow — fitting, residual-resampling bootstrap, accepted minimizers, summary tables, and profile likelihood — see vignette("CGNM-vignette", package = "CGNM") or vignettes/CGNM-vignette.Rmd.

Documentation

• ?CGNM — package overview and workflow map.
• ?Cluster_Gauss_Newton_method — the main fitting function.

By default, Cluster_Gauss_Newton_method() and the bootstrap/EBE variants return a plain list. Pass outputS4 = TRUE to get a CGNM_result S4 object instead (see ?"CGNM_result-class") — every field is still reachable via $/[[ exactly as on the list, so this is a drop-in, fully backward- compatible opt-in.

License

MIT. See LICENSE.