SAR Models

sars implements all 20 models from the R sars package. Each model has a dedicated fitting function (e.g., sars.sar_power()) that returns a SARFit object.

Non-asymptotic models

These models increase without bound as area increases.

Function	Formula	Parameters
sar_power	S = c · A^z	c, z
sar_powerR	S = f + c · A^z	f, c, z
sar_loga	S = c + z · log(A)	c, z
sar_linear	S = c + m · A	c, m
sar_epm1	S = c · A^z · exp(d · (log A)^2)	c, z, d
sar_epm2	S = c · A^(z1 · A^z2)	c, z1, z2
sar_p1	S = c · A^z · exp(-d · A)	c, z, d
sar_p2	S = c · A^z · exp(-d / A)	c, z, d

Asymptotic convex models

These models approach a finite asymptote with a convex (decelerating) curve.

Function	Formula	Parameters
sar_koba	S = c · log(1 + A/z)	c, z
sar_monod	S = d · A / (c + A)	d, c
sar_negexpo	S = d · (1 - exp(-z · A))	d, z
sar_asymp	S = d - c · exp(-z · A)	d, c, z
sar_ratio	S = (c + z · A) / (1 + d · A)	c, z, d

Asymptotic sigmoid models

These models have an S-shaped curve, approaching an asymptote via an inflection point.

Function	Formula	Parameters
sar_mmf	S = d / (1 + c · A^(-z))	d, c, z
sar_gompertz	S = d · exp(-exp(-z · (A - c)))	d, z, c
sar_weibull3	S = d · (1 - exp(-c · A^z))	d, c, z
sar_weibull4	S = d · (1 - exp(-c · A^z))^f	d, c, z, f
sar_chapman	S = d · (1 - exp(-z · A))^c	d, z, c
sar_betap	S = d · (1 - (1 + (A/c)^z)^(-f))	d, c, z, f
sar_heleg	S = d / (1 + slope^log(c / A))	d, slope, c

Fitting details

All models are fitted using nonlinear least squares (scipy least_squares) with a multi-start grid of initial values to avoid local minima:

• 2-parameter models: ≥36 starting points
• 3-parameter models: ≥100 starting points
• 4-parameter models: ≥200 starting points

If fitting fails for all starting values, a SARFit with converged=False is returned rather than raising an exception.

Information criteria (AIC, AICc, BIC) are computed using the normal log-likelihood convention, consistent with the R sars package.