Attempts to fit two columns of x-y data to a number of nonlinear equations, using least squares. Select a function name from the list. To see more functions, grab a function name and drag up and down to scroll.
The 95% confidence intervals are based on 1999 bootstrap replicates.
Fitting to a nonlinear function can be a bit tricky. For most of the functions, Past uses an educated guess for the parameters, followed by Levenberg-Marquardt optimization.
The Akaike Information Criterion (AIC) may aid in the selection of model. Lower values for the AIC imply a better fit, adjusted for the number of parameters.
Linear
y = ax + b
Included for comparison with the nonlinear functions. Fitting by ordinary least squares regression. The “Zero constant” option will set b = 0.
Quadratic
y = ax2 + bx + c
Included for reference. Fitting by least-squares and SVD (the equation is linear in its coefficients). The “Zero constant” option will set c = 0. See also the Polynomial Model module.
Power
y = axb + c
The usual power law equation. Initial guess by log-log transformation and linear regression (i.e. c = 0), followed by nonlinear optimization. The “Zero constant” option will set c=0.
Exponential
y = aebx + c
Initial guess by linearization (log-transforming y), followed by nonlinear optimization. The “Zero constant” option will set c = 0. See also the Generalized Linear Model module.
Von Bertalanffy
y = a(1-be-cx)
This equation is used for modelling growth of multi-celled animals (Brown & Rothery 1993). It is sometimes given in a slightly different form:
y = L∞(1-e-K(x-t0))
It is easy to see that L∞ = a, K = c and t0 = (ln b)/c.
The value of a is first estimated by the maximal value of y, and b and c using a straight-line fit to a linearized model. Finally nonlinear optimization.
Michaelis-Menten
y = ax / (b+x)
The Michaelis-Menten curve can make accurate fits to rarefaction curves, and may therefore (somewhat controversially) be used for extrapolating these curves to estimate biodiversity (Colwell & Coddington 1994). It is also an important model equation for chemical kinetics.
The algorithm uses maximum-likelihood estimators for the so-called Eadie-Hofstee transformation (Raaijmakers 1987; Colwell & Coddington 1994), followed by nonlinear optimization.
Logistic
y = a / (1 + be-cx)
A sigmoidal (S-shaped) curve. The logistic equation can model growth with saturation (Brown & Rothery 1993), and was used by Sepkoski (1984) to describe the proposed stabilization of marine diversity in the late Palaeozoic.
The value of a is first estimated by the maximal value of y, and b and c using a straight-line fit to a linearized model. Finally nonlinear optimization. See also the Generalized Linear Model module.
Gompertz
y = aebe^(cx)
Initial estimate is computed using regression on a linearized model, followed by nonlinear optimization.
Gaussian
y = ae-(x-b)^2 / (2c^2)
The ‘bell curve’ with mean b and standard deviation c.
Initial guess of a by maximal value of y, b by weighted mean, and c = 1, followed by nonlinear optimization.
Hill’s equation
y = d + (a-d) / (1 + (b/x)c)
This sigmoidal function is often used to model dosage-response data. d is the minimum and a the maximum asymptote. b is the dosage at which 50% of subjects show the response (the IC50 value), while c is the “Hill slope”. The “Zero constant” option will set d = 0.
References
Brown, D. & P. Rothery. 1993. Models in biology: mathematics, statistics and computing. John Wiley & Sons.
Colwell, R.K. & J.A. Coddington. 1994. Estimating terrestrial biodiversity through extrapolation. Philosophical Transactions of the Royal Society of London B 345:101-118.
Raaijmakers, J.G.W. 1987. Statistical analysis of the Michaelis-Menten equation. Biometrics 43:793-803.
Sepkoski, J.J. 1984. A kinetic model of Phanerozoic taxonomic diversity. Paleobiology 10:246-267.