# ssm_mlg: General multivariate linear Gaussian state space models In bssm: Bayesian Inference of Non-Linear and Non-Gaussian State Space Models

## Description

Construct an object of class ssm_mlg by directly defining the corresponding terms of the model.

## Usage

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ssm_mlg( y, Z, H, T, R, a1, P1, init_theta = numeric(0), D, C, state_names, update_fn = default_update_fn, prior_fn = default_prior_fn )

## Arguments

 y Observations as multivariate time series or matrix with dimensions n x p. Z System matrix Z of the observation equation as p x m matrix or p x m x n array. H Lower triangular matrix H of the observation. Either a scalar or a vector of length n. T System matrix T of the state equation. Either a m x m matrix or a m x m x n array. R Lower triangular matrix R the state equation. Either a m x k matrix or a m x k x n array. 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. init_theta Initial values for the unknown hyperparameters theta. D Intercept terms for observation equation, given as a p x n matrix. C Intercept terms for state equation, given as m x n matrix. state_names Names for the states. update_fn Function which returns list of updated model components given input vector theta. This function should take only one vector argument which is used to create list with elements named as Z, H 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. prior_fn Function which returns log of prior density given input vector theta.

## Details

The general multivariate linear-Gaussian model is defined using the following observational and state equations:

y_t = D_t + Z_t α_t + H_t ε_t, (\textrm{observation equation})

α_{t+1} = C_t + T_t α_t + R_t η_t, (\textrm{transition equation})

where ε_t \sim N(0, I_p), η_t \sim N(0, I_k) and α_1 \sim N(a_1, P_1) independently of each other. Here p is the number of time series and 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, H 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_mlg, 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

## Value

Object of class ssm_mlg.

## Examples

 1 2 3 4 data("GlobalTemp", package = "KFAS") model_temp <- ssm_mlg(GlobalTemp, H = matrix(c(0.15,0.05,0, 0.05), 2, 2), R = 0.05, Z = matrix(1, 2, 1), T = 1, P1 = 10) ts.plot(cbind(model_temp$y, smoother(model_temp)$alphahat),col=1:3)

bssm documentation built on July 10, 2021, 9:07 a.m.