cusp {cusp} | R Documentation |
This function fits a cusp catatrophe model to data using the maximum likelihood method of Cobb. Both the state variable may be modelled by a linear combination of variables and design factors, as well as the normal/asymmetry factor alpha
and bifurction/splitting factor beta
.
cusp(formula, alpha, beta, data, weights, offset, ..., control = glm.control(), method = "cusp.fit", optim.method = "L-BFGS-B", model = TRUE, contrasts = NULL)
formula |
formula that models the canonical state variable |
alpha |
formula that models the canonical normal/asymmetry factor |
beta |
formula that models the canonical bifurcation/splitting factor |
data |
data.frame that contains all the variables named in the formulas |
weights |
vector of weights by which each data point is weighted (experimental) |
offset |
vector of offsets for the data (experimental) |
... |
named arguments that are passed to optim |
control |
glm.control object, currently unused |
method |
string, currently unused |
optim.method |
string passed to optim to choose the optimization algorithm |
model |
should the model matrix be returned? |
contrasts |
matrix of contrasts , experimental |
cusp
fits a cusp catastrophe model to data. Cobb's definition for the canonical form of the stochastic cusp catastrophe is the stochastic differential equation
dY(t) = (α + β Y(t_ - Y(t)^3)dt + dW(t).
The stationary distribution of the ‘behavioral’, or ‘state’ variable Y, given the control parameters α (‘asymmetry’ or ‘normal’ factor) and β (‘bifurcation’ or ‘splitting’ factor) is
f(y) = Psi exp(α y + β y^2/2 - y^4/4),
where Psi is a normalizing constant.
The behavioral variable and the asymmetry and bifurcation factors are usually not directly related to the dependent and independent variables in the data set. These are therefore used to predict the state variable and control parameters:
y[i] = w[0] + w[1] * Y[i,1] + cdots + w[p] * Y[i,p],
α[i] = a[0] + a[1] * X[i,1] + cdots + a[p] * X[i,p],
β[i] = b[0] + b[1] * X[i,1] + cdots + b[p] * X[i,p],
in which the a[j]'s, b[j]'s, and w[j]'s are estimated by means of maximum likelihood. Here, the Y[i,j]'s and X[i,j]'s are variables constructed from variables in the data set. Variables predicting the α's and β's need not be the same.
The state variable and control parameters can be modelled by specifying a model formula
:
y ~ model
,
alpha ~ model
,
beta ~ model
,
in which model
can be any valid formula
specified in terms of variables that are present in the data.frame
.
List with components
coefficients |
Estimated coefficients |
rank |
rank of Hessian matrix |
qr |
qr decomposition of the Hessian matrix |
linear.predictors |
two column matrix containing the α[i]'s and β[i]'s for each case |
deviance |
sum of squared errors using Delay convention |
aic |
AIC |
null.deviance |
variance of canonical state variable |
iter |
number of optimization iterations |
weights |
weights provided through weights argument |
df.residual |
residual degrees of freedom |
df.null |
degrees of freedom of constant model for state variable |
y |
predicted values of state variable |
converged |
convergence status given by optim |
par |
parameter estimates for qr standardized data |
Hessian |
Hessian matrix of negative log likelihood function at minimum |
hessian.untransformed |
Hessian matrix of negative log likelihood for qr standardized data |
code |
optim convergence indicator |
model |
list with model design matrices |
call |
function call that created the object |
formula |
list with the formulas |
OK |
logical. TRUE if Hessian matrix is positive definite at the minimum obtained |
data |
original data.frame |
Raoul Grasman
See cusp-package
summary.cusp
for summaries and model assessment.
The generic functions coef
, effects
, residuals
, fitted
, vcov
.
predict
for estimated values of the control parameters α[i] and β[i],
# example with regressors x1 = runif(150) x2 = runif(150) z = Vectorize(rcusp)(1, 4*x1-2, 4*x2-1) data <- data.frame(x1, x2, z) fit <- cusp(y ~ z, alpha ~ x1+x2, beta ~ x1+x2, data) print(fit) summary(fit) ## Not run: plot(fit) cusp3d(fit) ## End(Not run) # useful use of OK ## Not run: while(!fit$OK) fit <- cusp(y ~ z, alpha ~ x1+x2, beta ~ x1+x2, data, start=rnorm(fit$par)) # use different starting values ## End(Not run)