stsm (version 1.7)

maxlik.td: Maximization of the Time Domain Likelihood Function

Description

Maximize the time domain log-likelihood function of a structural time series model by means of a scoring algorithm or a general purpose optimization algorithm available in optim.

Usage

maxlik.td.optim(m,  KF.version = eval(formals(KFKSDS::KalmanFilter)$KF.version), KF.args = list(), check.KF.args = TRUE, barrier = list(type = c("1", "2"), mu = 0), inf = 99999,  method = c("BFGS", "L-BFGS-B", "Nelder-Mead", "CG", "SANN", "AB-NM"), gr = c("numerical", "analytical"), optim.control = list())
maxlik.td.scoring(m, step = NULL,  KF.args = list(), check.KF.args = TRUE, ls = list(type = "optimize", tol = .Machine$double.eps^0.25, cap = 1), control = list(maxit = 100, tol = 0.001, trace = FALSE, silent = FALSE),  debug = FALSE)

Arguments

m

an object of class stsm.

KF.version

character indicating the implementation of the Kalman filter to be used.

KF.args

a list of parameters to be passed to the function chosen to run the Kalman filter.

check.KF.args

logical. If TRUE, the arguments passed through KF.args are checked for consistency with the interface chosen in KF.version: for maxlik.td.scoring in it is checked for consistency with interface KFKSDS.

barrier

a list defining a barrier term to penalize parameter values close to the bounds m@lower and m@upper.

inf

a numeric indicating the value to be returned if the value of the log-likelihood function happens to be NA or non-finite at some iteration of the optimization procedure.

method

character indicating the method to be used by optim.

gr

character indicating whether numerical or analytical derivatives should be used.

optim.control

a list of control parameters passed to optim.

step

if it is a numeric it stands for a fixed step size, otherwise an automatic procedure is used to choose the step size.

ls

control parameters for the line search procedure used to chose the step size in the scoring algorithm.

control

a list of control parameters for the scoring algorithm.

debug

logical. If TRUE, tracing information is printed for debugging purposes of the scoring algorithm.

Value

A list of class stsmFit. See the section ‘Value’ in maxlik.fd.

Details

The function maxlik.td.optim implements the common procedure of maximum likelihood, i.e., maximization of the time domain likelihood function by means of a numerical optimization algorithm (L-BFGS-B or other algorithms available in optim). The likelihood function is evaluated by means of the Kalman filter.

The function maxlik.td.scoring implements a scoring algorithm based on the analytical expression of the information matrix of the time domain likelihood function.

López-de-Lacalle (2013) discusses several options and details that are often omitted or ignored when maximizing the likelihood function of a structural time series models. The interface maxlik.td.optim allows the user to choose some options that may affect the results or convergence of the algorithm.

A novelty compared to other implementations such as StructTS is that the optimization procedure is enhanced by means of analytical derivatives. Another enhancement is that one of the parameters can be concentrated out of the likelihood function. The parameter to be concentrated is defined in the slot cpar of the input model m, see stsm. This option has not yet been implemented in the scoring algorithm.

For details about the options than can be passed through argument KF.args see the documentation of the same argument in function KalmanFilter in package KFKSDS.

For further information about the scoring algorithm see the following points in the details section of maxlik.fd.scoring: Bounds on parameters and barrier term, Control parameters for the scoring algorithm and Choice of the step size in the scoring algorithm.

If external regressors are included in the model m, starting values for their coefficients are obtained in a linear regression of the differenced series on the differenced regressors. The values in the slot pars are therefore overwritten and not used as initial values.

Note: ls$type = "wolfe" is not implemented for maxlik.td.scoring.

References

Durbin, J. and Koopman, S. J. (2001). Time Series Analysis by State Space Methods. Oxford University Press.

Harvey, A. C. (1989). Forecasting, Structural Time Series Models and the Kalman Filter. Cambridge University Press.

López-de-Lacalle, J. (2013). ‘101 Variations on a Maximum Likelihood Procedure for a Structural Time Series Model.’ Unpublished manuscript.

Nocedal, J. and Wright, J. W. (2006). Numerical Optimization. Springer-Verlag.

See Also

barrier.eval, mloglik.td, stsm, optim.

Examples

# replicate maximum likelihood procedure as implemented in 'stats::StructTS'
res0 <- stats::StructTS(log(AirPassengers), type = "BSM")
mairp <- stsm.model(model = "BSM", y = log(AirPassengers), 
  transPars = "StructTS")
res1 <- maxlik.td.optim(mairp, KF.version = "KFKSDS", 
  KF.args = list(P0cov = TRUE), method = "L-BFGS-B", gr = "numerical")
mairp1 <- set.pars(mairp, pmax(res1$par, .Machine$double.eps))
round(get.pars(mairp1), 6)
all.equal(get.pars(mairp1), res0$coef[c(4,1:3)], 
  tol = 1e-04, check.attributes = FALSE)

# smoothed components
ss1 <- char2numeric(mairp1, P0cov = TRUE)
kf1 <- KFKSDS::KF(mairp1@y, ss1)
ks1 <- KFKSDS::KS(mairp1@y, ss1, kf1)
plot(tsSmooth(res0)[,c(1,3)])
plot(ks1$ahat[,c(1,3)])

# the scoring algorithm reaches another local optimum where
# the seasonal component is more homoscedastic
mairp <- stsm.model(model = "BSM", y = log(AirPassengers), 
  transPars = NULL)
res2 <- maxlik.td.scoring(mairp, step = NULL, 
  ls = list(type = "optimize", tol = .Machine$double.eps^0.25, cap = 1),
  control = list(maxit = 100, tol = 0.001, trace = FALSE), debug = FALSE)
round(res2$par, 6)

mairp2 <- set.pars(mairp, res2$par)
ss2 <- char2numeric(mairp2, P0cov = FALSE)
kf2 <- KFKSDS::KF(mairp2@y, ss2)
ks2 <- KFKSDS::KS(mairp2@y, ss2, kf2)
plot(ks2$ahat[,c(1,3)])