bentcableAR (version 0.3.0)

cable.ar.p.iter: Bent-Cable Regression for Independent or AR Data, With Exception

Description

This function is the main engine for bentcable.ar. It performs bent-cable (including broken-stick) regression to AR(p) time-series data or independent data (time-series or otherwise). However, it cannot fit broken sticks to independent data (see stick.ar.0).

Usage

cable.ar.p.iter(init, y.vect, t.vect = NULL, n = NA, tol,
	method0 = "css", method1 = "yw", stick = FALSE)

Arguments

init

A numeric vector of initial values, in the form of c(b0,b1,b2,tau,gamma,phi.1,...,phi.p) when stick=FALSE, and c(b0,b1,b2,tau,phi.1,...,phi.p) when stick=TRUE. phi values correspond to AR

y.vect

A numeric vector of response data.

t.vect

A numeric vector of design points, which MUST be equidistant with unit increments if AR(p) is assumed. They need not be equidistant for independent data. Specifying t.vect=NULL is equivalent to specifying the default time points

n

Length of response vector (optional).

tol

Tolerance for determining convergence.

method0, method1

The fitting method when p>0. "css" stands for conditional sum-of-squares and corresponds to conditional maximum likelihood. "yw" stands for Yule-Walker, and "mle" for (full) maximum likelihoo

stick

A logical value; if TRUE, a broken-stick regression is performed.

Value

• fit
  An nls object, returned if independent data are assumed. It is the maximum likelihood bent-cable fit.
• estimate
  A numeric vector, returned if AR(p>0) is assumed. It is the estimated value of ($\theta,\phi$).
• ar.p.fit
  Returned if AR(p>0) is assumed. If "css" is used, converges, and yields a $\phi$ estimate that corresponds to stationarity, then $ar.p.fit is an optim object containing the CML fit. If "yw" or "mle" is used and converges, then $ar.p.fit is an ar object containing the CML-ML(-MM) fit.
• y, t, n, p, stick
  As supplied by the user; always returned.
• method
  A character string, returned if AR(p>0) is assumed. It indicates the method that yielded the returned fit.

Details

The bent cable has the form $f(t) = b_0 + b_1 t + b_2 q(t)$, where $q(t)$ is the basic bent cable $$q(t)=\frac{(t-\tau+\gamma)^2}{4\gamma} I{|t-\tau|\le\gamma} + (t-\tau) I{t>\tau+\gamma}$$ for $\gamma\ge 0$. For independent data (time series or otherwise), bent-cable regression by maximum likelihood is performed via nonlinear least-squares estimation of $\theta=(b_0,b_1,b_2,\tau,\gamma)$ through the built-in Rfunction nls. For AR(p) data, conditional maximum likelihood (CML) estimation of $(\theta,\phi)$ (conditioned on the first p data points) is performed through the built-in Rfunction optim with the "BFGS" algorithm, where $\phi=(\phi_1,\ldots,\phi_p)$ are the AR coefficients. In either case, the estimation relies on the user-supplied initial values in init. A Gaussian model is assumed, so that CML estimation is equivalent to minimizing the conditional sum-of-squares error, specified as "css" by default for method0. However, "css" sometimes fails to converge, or the resulting $\phi$ estimate sometimes corresponds to non-stationarity. In this case, the alternative estimation approach specified for method1 is attempted. "mle" specifies the CML-ML hybrid algorithm, and "yw" the CML-ML-MM hybrid algorithm (MM stands for method of moments; see References.) Both "yw" and "mle" guarantee stationarity, but often take much longer than "css" to converge.

The bent-cable likelihood / deviance often has multiple peaks. Thus, the user should be aware of different local maxima on which the optimization algorithm can converge despite initial values for $\theta$ that are very similar. The user is advised to combine several exploratory analyses as well as model diagnoses before settling on a best fit. See Details on the bentcable.ar help page for a detailed description.

References

See the bentcableAR package references.

See Also

stick.ar.0, fullcable.t, bentcable.dev.plot, nls, optim, ar.

Examples

data(stagnant)
data(sockeye)

# 'stagnant': independent data cable fit
fit0 <- cable.ar.p.iter( c(.6,-.4,-.7,0,.5),
	stagnant$loght, stagnant$logflow )    # 'nls' fit
	# compare to this:
	# bentcable.ar( stagnant$loght, t.vect=stagnant$logflow,
	#	init.cable=c(.6,-.4,-.7,0,.5) )

fit0$fit   # 'fit0' SSE=0.005


# 'sockeye': AR(2) cable fit
fit1 <- cable.ar.p.iter( c(13,.1,-.5,11,4,.5,-.5),
	sockeye$logReturns, tol=1e-4 )    # "css" successful
	# compare to this:
	# fit1 <- bentcable.ar( sockeye$logReturns, 
	#	init.cable=c(13,.1,-.5,11,4), p=2 )

fit1$ar.p.fit$value     # 'fit1' SSE=4.9


# 'sockeye': AR(2) cable fit
fit2 <- cable.ar.p.iter( c(10,0,0,5,.1,.5,-.5), sockeye$logReturns, 
	tol=1e-4 )    # "css" unsuccessful, switched to "yw"
	# compare to this:
	# fit2 <- bentcable.ar(sockeye$logReturns, 
	#	init.cable=c(10,0,0,5,.1), p=2 )

cable.ar.p.iter( fit2$est, sockeye$logReturns, 
	tol=1e-4 )   # 'fit2' SSE=13.8 (from first line of screen output)


# 'sockeye': AR(4) stick fit
cable.ar.p.iter( c(13,.1,-.5,11,.5,-.5,.5,-.5),
	sockeye$logReturns, tol=1e-4, stick=TRUE )
	# compare to this:
	# bentcable.ar( sockeye$logReturns,
	#	init.cable=c(13,.1,-.5,11), p=4, stick=TRUE )