cusp: Fit a Cusp Catatrophe Model to Data

In cusp: Cusp-Catastrophe Model Fitting Using Maximum Likelihood

Description

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.

Usage

cusp(formula, alpha, beta, data, weights, offset, ..., control =
    glm.control(), method = "cusp.fit", optim.method = "L-BFGS-B", model = TRUE,
    contrasts = NULL)

Arguments

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

Details

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) = Ψ \exp(α y + β y^2/2 - y^4/4),

where Ψ 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:

\code{y ~ model},

\code{alpha ~ model},

\code{beta ~ model},

in which model can be any valid formula specified in terms of variables that are present in the data.frame.

Value

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

Author(s)

Raoul Grasman

References

See cusp-package

See Also

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],

Examples

set.seed(123)
# 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)

Example output

Call:  cusp(formula = y ~ z, alpha = alpha ~ x1 + x2, beta = beta ~      x1 + x2, data = data) 

Coefficients:
a[(Intercept)]           a[x1]           a[x2]  b[(Intercept)]           b[x1]  
      -2.93474         6.15488        -0.16132        -2.23399         0.59292  
         b[x2]  w[(Intercept)]            w[z]  
       3.81285         0.07288         0.95201  

Degrees of Freedom: 149 Total (i.e. Null);  142 Residual
Null Deviance:	    192.7 
Delay Deviance:	 67.49 	AIC: 242.4 

Call:
cusp(formula = y ~ z, alpha = alpha ~ x1 + x2, beta = beta ~ 
    x1 + x2, data = data)

Deviance Residuals: 
     Min        1Q    Median        3Q       Max  
-2.47861  -0.42193  -0.01327   0.37423   1.64024  

Coefficients:
               Estimate Std. Error z value Pr(>|z|)    
a[(Intercept)] -2.93474    0.61408  -4.779 1.76e-06 ***
a[x1]           6.15488    0.95874   6.420 1.36e-10 ***
a[x2]          -0.16132    0.50064  -0.322   0.7473    
b[(Intercept)] -2.23399    0.89596  -2.493   0.0127 *  
b[x1]           0.59292    1.16289   0.510   0.6101    
b[x2]           3.81285    0.66732   5.714 1.11e-08 ***
w[(Intercept)]  0.07288    0.11371   0.641   0.5216    
w[z]            0.95201    0.05280  18.032  < 2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1


  Null deviance: 192.732  on 149  degrees of freedom
Linear deviance:  82.579  on 146  degrees of freedom
Logist deviance:      NA  on  NA  degrees of freedom
 Delay deviance:  67.486  on 142  degrees of freedom

             R.Squared    logLik npar      AIC     AICc      BIC
Linear model 0.6116778 -168.0745    4 344.1491 344.4250 356.1916
Cusp model   0.6521824 -113.1932    8 242.3865 243.4078 266.4716
---
Note: R.Squared for cusp model is Cobb's pseudo-R^2. This value
      can become negative.

	Chi-square test of linear vs. cusp model

X-squared = 109.8, df = 4, p-value = 0

Number of optimization iterations: 51

cusp documentation built on May 2, 2019, 6:51 p.m.