This function fits a logistic curve model to data using maximum likelihood under the assumption of normal errors (i.e., nonlinear least squares). Both the response variable may be modeled by a linear combination of variables and design factors, as well as the normal/asymmetry factor alpha and bifurcation/splitting factor beta.
cusp.logist(formula, alpha, beta, data, ..., model = TRUE, x =
FALSE, y = TRUE)List with components
Objective function value at minimum
Coordinates of objective function minimum
Gradient of objective function at minimum.
Convergence code returned by optim
Number of iterations used by optim
A named vector of estimates of \(a_j, b_j\)'s
Estimates of \(\alpha_i\)'s and \(\beta_i\)'s.
Predicted values of \(y_i\)'s as determined from the linear.predictors
Residuals
Numerical rank of matrix of predictors for \(\alpha_i\)'s plus rank of matrix of predictors for \(\beta_i\)'s plus rank of matrix of predictors for the \(y_i\)'s.
Residual sum of squares.
Log of the likelihood at the minimum.
Akaike's information criterion
R Squared (proportion of explained variance)
Degrees of freedom for the residual
Degrees of freedom for the Null residual
Residual sum of squares
Hessian matrix of objective function at the minimum if hessian=TRUE.
Hessian matrix of log-likelihood function at the minimum (currently unavailable)
QR decomposition of the hessian matrix
Boolean indicating if optimization convergence is proper (based on exit code optim, gradient, and, if hessian=TRUE eigen values of the hessian).
weights (currently unused)
the matched call
If requested (the default), the matrix of response variables used.
If requested, the model matrix used.
The sum of squared deviations from the mean of the estimated \(y_i\)'s.
formulas for the response variable and the regression variables (see below)
data.frame containing \(n\) observations of all the variables named in the formulas
named arguments that are passed to nlm
logicals. If TRUE the corresponding components of the fit (the model frame, the model matrix, and the response are returned.
Raoul Grasman
A nonlinear regression is carried out of the model
$$y_i = \frac{1}{1+\exp(-\alpha_i/\beta_i^2)} + \epsilon_i$$
for \(i = 1, 2, \ldots, n\),
where
$$y_i = w_0 + w_1 Y_{i1} + \cdots + w_p Y_{ip}$$
$$\alpha_i = a_0 + a_1 X_{i1} + \cdots + a_p X_{ip}$$
$$\beta_i = b_0 + b_1 X_{i1} + \cdots + b_q X_{iq}$$
in which the \(a_j\)'s, and \(b_j\)'s, are estimated. The \(Y_{ij}\)'s are variables in the data set
and specified by formula; the \(X_{ij}\)'s are variables in the data set and are specified in alpha
and beta. Variables in alpha and beta need not be the same. The \(w_j\)'s are estimated implicitly
using concentrated likelihood methods, and are not returned explicitly.
Hartelman PAI (1997). Stochastic Catastrophe Theory. Amsterdam: University of Amsterdam, PhD thesis.
summary.cusp