stsm (version 1.7)

maxlik.em: Maximization of the Time Domain Likelihood Function via the Expectation-Maximization Algorithm

Description

Maximize the time domain log-likelihood function of a structural time series model by means of the Expectation-Maximization algorithm.

Usage

maxlik.em(model, type = c("standard", "modified", "mix"), 
  tol = 0.001, maxiter = 300, kfconv = c(0, 10, 1),  
  ur.maxiter = 1000, r.interval = c(0.001, var(model@y)*0.75, 20),
  mod.steps = seq(3, maxiter, 10), 
  parallel = FALSE, num.cores = NULL)

Arguments

model

an object of class stsm.

type

a character choosing the implementation of the algorithm, the original (standard), a modified version or a mixture of both.

tol

the convergence tolerance.

maxiter

the maximum number of iterations.

kfconv

a length-three vector with control parameters used to determine whether the Kalman filter has converged at a given iteration.

ur.maxiter

the maximum number of iterations used in the root finding procedure. Ignored with type = "standard".

r.interval

a three-length vector to control the ends of the interval in the root finding procedure.

mod.steps

a vector indicating the iterations at which the modified EM algorithm is to be run. Only for type = "mix".

parallel

logical. If TRUE the process is run in parallel by means of function mclapply in package parallel.

num.cores

an optional numeric. The number of processes executed in parallel if parallel is TRUE. By default it is set equal to the number of CPU cores.

Value

• A list containing the elements:
• Mpars
  a matrix with the values of the parameters stored by row for each iteration of the procedure.
• pars
  parameter values at the local optimum or point where the algorithm stopped.
• iter
  number of iterations until convergence or stopping.
• For type = "modified" or type = "mix" the element calls.v1 is also returned. It reports, for each parameter, the number of calls that were done to the traditional version due to failure to convergence of the root finding procedure.

Details

This function is based on the discussion given in López-de-Lacalle (2013) about the Expectation-Maximization (EM) algorithm in the context of structural time series models The previous reference includes an Rpackage called stsm.em but since it uses some non-standard options it has not been distributed. A version of the most relevant procedures is provided in the function maxlik.em.

The traditional design of the EM algorithm in the context of structural time series models is run with type = "standard". This approach is described for instance in Durbin and Koopman (2001) Section 7.3.4. A modified version introduced in López-de-Lacalle (2013) can be run using type = "modified". A mixture of both approaches is also possible by setting type = "mix". In that case, the modified version is run at those iterations indicated in mod.steps and the traditional version is run in the remaining iterations. As we do not know beforehand the number of iterations required for convergence, mod.steps should be defined considering up to maxiter possible iterations.

In pure variance models, the Kalman filter may converge to a steady state. The parameters in kfconv control how convergence of the filter is determined. It is considered that convergence is reached when the following is observed: the change in the variance of the prediction error over the last kfconv[2] consecutive iterations of the filter is below the tolerance kfconv[1]. The iteration at which the the Kalman smoother has converged is the iteration where the Kalman filter converged multiplied by the factor kfconv[3]. If provided, kfconv[3] should be equal or greater than unity.

The argument r.interval is a three-length vector. The first two elements are the initial lower and upper ends of the interval where the variance parameters are searched. The third element is the first iteration of the EM algorithm after which the initial searching interval is narrowed. A relatively width initial interval is recommended. As the algorithm makes progress, the interval is automatically narrowed according to the values and path followed in the first r.interval[3] and subsequent iterations.

References

Durbin, J. and Koopman, S.J. (2001). Section 7.3.4. Time Series Analysis by State Space Methods. Oxford University Press.

Harvey, A. C. (1989). Section 4.2.4. Forecasting, Structural Time Series Models and the Kalman Filter. Cambridge University Press.

Koopman, S.J. and Shephard, N. (1992) Exact Score for Time Series Models in State Space Form. Biometrika 79(4), pp. 823-826.

Koopman, S.J. (1993). Disturbance Smoother for State Space Models. Biometrika 80(1), pp. 117-126.

López-de-Lacalle, J. (2013). Why Does the Expectation-Maximization Algorithm Converge Slowly in Pure Variance Structural Time Series Models? Unpublished manuscript.

Shumway, R.H. and Stoffer, D.S. (1982) An Approach to Time Series Smoothing and Forecasting Using the EM Algorithm. Journal of Time Series Analysis 3(4), pp. 253-264.

See Also

maxlik.td.

Examples

# fit a local level plus seasonal model to a simulated sample series
# using the three versions of the EM algorithm
# the same solution is found by all versions (up to a tolerance)
# the modified version converges in fewer iterations, yet it involves 
# more computations
data("llmseas")
m <- stsm.model(model = "llm+seas", y = llmseas, 
  pars = c(var1 = 1, var2 = 1, var3 = 1))

# original version
res1 <- maxlik.em(m, type = "standard", 
  tol = 0.001, maxiter = 350, kfconv = c(0.01, 10, 1))
res1$pars
res1$iter

# modified version
res2 <- maxlik.em(m, type = "modified", 
  tol = 0.001, maxiter = 250, kfconv = c(0.01, 10, 1),  
  ur.maxiter = 1000, r.interval = c(0.001, var(m@y)*0.75, 20))
res2$pars
res2$iter
res2$calls.v1

# mixture, the modified version is run every 10 iterations
# starting in the third one
res3 <- maxlik.em(m, type = "mix", 
  tol = 0.001, maxiter = 250, kfconv = c(0.01, 10, 1),  
  ur.maxiter = 1000, r.interval = c(0.001, var(m@y)*0.75, 20),
  mod.steps = seq(3, 200, 10))
res3$pars
res3$iter
res3$calls.v1