General univariate non-Gaussian state space model
Construct an object of class ssm_ung by directly defining the corresponding terms of
the model.
ssm_ung( y, Z, T, R, a1, P1, distribution, phi = 1, u = 1, init_theta = numeric(0), D, C, state_names, update_fn = default_update_fn, prior_fn = default_prior_fn )
y |
Observations as time series (or vector) of length n. |
Z |
System matrix Z of the observation equation. Either a vector of length m, a m x n matrix, or object which can be coerced to such. |
T |
System matrix T of the state equation. Either a m x m matrix or a m x m x n array, or object which can be coerced to such. |
R |
Lower triangular matrix R the state equation. Either a m x k matrix or a m x k x n array, or object which can be coerced to such. |
a1 |
Prior mean for the initial state as a vector of length m. |
P1 |
Prior covariance matrix for the initial state as m x m matrix. |
distribution |
Distribution of the observed time series. Possible choices are
|
phi |
Additional parameter relating to the non-Gaussian distribution. For negative binomial distribution this is the dispersion term, for gamma distribution this is the shape parameter, and for other distributions this is ignored. |
u |
Constant parameter vector for non-Gaussian models. For Poisson, gamma, and negative binomial distribution, this corresponds to the offset term. For binomial, this is the number of trials. |
init_theta |
Initial values for the unknown hyperparameters theta. |
D |
Intercept terms D_t for the observations equation, given as a scalar or vector of length n. |
C |
Intercept terms C_t for the state equation, given as a m times 1 or m times n matrix. |
state_names |
Names for the states. |
update_fn |
Function which returns list of updated model components given input vector theta. See details. |
prior_fn |
Function which returns log of prior density given input vector theta. |
The general univariate non-Gaussian model is defined using the following observational and state equations:
p(y_t | D_t + Z_t α_t), (\textrm{observation equation})
α_{t+1} = C_t + T_t α_t + R_t η_t, (\textrm{transition equation})
where η_t \sim N(0, I_k) and α_1 \sim N(a_1, P_1) independently of each other, and p(y_t | .) is either Poisson, binomial, gamma, or negative binomial distribution. Here k is the number of disturbance terms which can be less than m, the number of states.
The update_fn function should take only one
vector argument which is used to create list with elements named as
Z, phi T, R, a1, P1, D, and C,
where each element matches the dimensions of the original model.
If any of these components is missing, it is assumed to be constant wrt. theta.
Note that while you can input say R as m x k matrix for ssm_ung,
update_fn should return R as m x k x 1 in this case.
It might be useful to first construct the model without updating function and then check
the expected structure of the model components from the output.
Object of class ssm_ung.
data("drownings", package = "bssm")
model <- ssm_ung(drownings[, "deaths"], Z = 1, T = 1, R = 0.2,
a1 = 0, P1 = 10, distribution = "poisson", u = drownings[, "population"])
# approximate results based on Gaussian approximation
out <- smoother(model)
ts.plot(cbind(model$y / model$u, exp(out$alphahat)), col = 1:2)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.