Linear Gaussian Approximation for Exponential Family State Space Model
Function approxSMM performs a linear Gaussian approximation of an
exponential family state space model.
approxSSM( model, theta, maxiter = 50, tol = 1e-08, expected = FALSE, H_tol = 1e+15 )
model |
A non-Gaussian state space model object of class |
theta |
Initial values for conditional mode theta. |
maxiter |
The maximum number of iterations used in approximation. Default is 50. |
tol |
Tolerance parameter for convergence checks. |
expected |
Logical value defining the approximation of H_t in case of Gamma
and negative binomial distribution. Default is |
H_tol |
Tolerance parameter for check |
This function is rarely needed itself, it is mainly available for illustrative and debugging purposes. The underlying Fortran code is used by other functions of KFAS for non-Gaussian state space modelling.
The linear Gaussian approximating model is defined by
ytilde[t] = Z[t]α[t] + ε[t], ε[t] ~ N(0,Htilde[t]),
α[t+1] = T[t]α[t] + R[t]η[t], η[t] ~ N(0,Q[t]),
and α[1] ~ N(a[1],P[1]),
where ytilde and Htilde are chosen in a way that the linear Gaussian approximating model has the same conditional mode of θ=Zα given the observations y as the original non-Gaussian model. Models also have a same curvature at the mode.
The approximation of the exponential family state space model is based on iterative weighted least squares method, see McCullagh and Nelder (1983) p.31 and Durbin Koopman (2012) p. 243.
An object of class SSModel which contains the approximating Gaussian state space model
with following additional components:
thetahat |
Mode of p(θ|y). |
iterations |
Number of iterations used. |
difference |
Relative difference in the last step of approximation algorithm. |
McCullagh, P. and Nelder, J. A. (1983). Generalized linear models. Chapman and Hall.
Koopman, S.J. and Durbin, J. (2012). Time Series Analysis by State Space Methods. Second edition. Oxford University Press.
importanceSSM, SSModel,
KFS, KFAS.
# A Gamma example modified from ?glm (with log-link)
clotting <- data.frame(
u = c(5,10,15,20,30,40,60,80,100),
lot1 = c(118,58,42,35,27,25,21,19,18),
lot2 = c(69,35,26,21,18,16,13,12,12))
glmfit1 <- glm(lot1 ~ log(u), data = clotting, family = Gamma(link = "log"))
glmfit2 <- glm(lot2 ~ log(u), data = clotting, family = Gamma(link = "log"))
# Model lot1 and lot2 together (they are still assumed independent)
# Note that Gamma distribution is parameterized by 1/dispersion i.e. shape parameter
model <- SSModel(cbind(lot1, lot2) ~ log(u),
u = 1/c(summary(glmfit1)$dispersion, summary(glmfit2)$dispersion),
data = clotting, distribution = "gamma")
approxmodel <- approxSSM(model)
# Conditional modes of linear predictor:
approxmodel$thetahat
cbind(glmfit1$linear.predictor, glmfit2$linear.predictor)
KFS(approxmodel)
summary(glmfit1)
summary(glmfit2)
# approxSSM uses modified step-halving for more robust convergence than glm:
y <- rep (0:1, c(15, 10))
suppressWarnings(glm(formula = y ~ 1, family = binomial(link = "logit"), start = 2))
model <- SSModel(y~1, dist = "binomial")
KFS(model, theta = 2)
KFS(model, theta = 7)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.