stsm-sgf: Spectral Generating Function of Common Structural Time Series...
In stsm.class: Class and Methods for Structural Time Series Models

Description Usage Arguments Details Value References See Also Examples

Evaluate the spectral generating function of of common structural models: local level model, local trend model and basic structural model.

1	stsm.sgf(x, gradient = FALSE, hessian = FALSE, deriv.transPars = FALSE)

`x`	object of class `stsm`.
`gradient`	logical. If `TRUE`, the gradient is returned.
`hessian`	logical. If `TRUE`, `hessian` the gradient is returned.
`deriv.transPars`	logical. If `TRUE`, the gradient and the Hessian are scaled by the gradient of the function that transforms the parameters. Ignored if `x@transPars` is null.

The stationary form of the local level model is (Δ is the differencing operator):

Δ y[t] = v[t] + Δ e[t]

and its spectral generating function at each frequency λ[j] = 2π j/T for j=0,...,T-1 is:

g(λ[j]) = σ^2_2 + 2(1 - cos λ[j]) σ^2_1

The stationary form of the local trend model for a time series of frequency S is:

Δ^2 y[t] = Δ v[t] + w[t-1] + Δ^2 e[t]

and its spectral generating function is:

g(λ[j]) = 2(1 - cos λ[j]) σ^2_2 + σ^2_3 + 4(1 - cos λ[j]) σ^2_1

The stationary form of the basic structural model for a time series of frequency p is:

Δ Δ^p y[t] = Δ^p v[t] + S(L) w[t-1] + Δ^2 s[t] + Δ Δ^p e[t]

and its spectral generating function is:

g(λ[j]) = g_v(λ[j]) σ^2_2 + g_w(λ[j]) σ^2_3 + g_s(λ[j]) σ^2_4 + g_e(λ[j]) σ^2_1

with

g_v(λ[j]) = 2(1 - cos(λ[j] p))

g_w(λ[j]) = (1 - cos(λ[j] p))/(1 - cos(λ[j]))

g_s(λ[j]) = 4 (1 - cos(λ[j]))^2

g_e(λ[j]) = 4 (1 - cos(λ[j])) (1 - cos(λ[j] p))

A list containing the following results:

`sgf`	spectral generating function of the BSM model at each frequency λ[j] for j=0,…,T-1.
`gradient`	first order derivatives of the spectral generating function with respect of the parameters of the model.
`hessian`	second order derivatives of the spsectral generating function with respect of the parameters of the model.
`constants`	the terms g_v(λ[j]), g_w(λ[j]), g_s(λ[j]) and g_e(λ[j]) that do not depend on the variance parameters.

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

set.sgfc, stsm-class, stsm.model.

# spectral generating function of the local level plus seasonal model
m <- stsm.model(model = "llm+seas", y = JohnsonJohnson, 
  pars = c("var1" = 2, "var2" = 15), nopars = c("var3" = 30))
res <- stsm.sgf(m)
res$sgf
plot(res$sgf)
res$constants
# the element 'constants' contains the constant variables
# related to each component regardless of whether the 
# variances related to them are in the slot 'pars' or 'nopars'
names(get.pars(m))
colnames(res$constants)

# compare analytical and numerical derivatives
# identical values
m <- stsm.model(model = "llm+seas", y = JohnsonJohnson, 
  pars = c("var1" = 2, "var2" = 15, "var3" = 30))
res <- stsm.sgf(m, gradient = TRUE)

fcn <- function(x, model = m) {
  m <- set.pars(model, x)
  res <- stsm.sgf(m)
  sum(res$sgf)
}

a1 <- numDeriv::grad(func = fcn, x = get.pars(m))
a2 <- colSums(res$grad)
all.equal(a1, a2, check.attributes = FALSE)

# analytical derivatives are evaluated faster than numerically
system.time(a1 <- numDeriv::grad(func = fcn, x = get.pars(m)))
system.time(a2 <- colSums(stsm.sgf(m, gradient = TRUE)$grad))