arimaSSM: Create a State Space Representation of ARIMA Model

In jrnold/KFAS: Kalman Filter and Smoother for Exponential Family State Space Models.

Description

Function arimaSSM creates a state space representation of ARIMA model.

Usage

  arimaSSM(y, arima, H = NULL, Q = NULL, u = NULL,
    distribution = c("Gaussian", "Poisson", "Binomial"),
    transform = c("none", "ldl", "augment"),
    tolF = .Machine$double.eps^0.5,
    tol0 = .Machine$double.eps^0.5)

Arguments

arima	A list or a list of lists with components ar, ma and d, giving the autoregression and moving average coefficients, and the degree of differencing for each series. If arima is a single list, it is assumed that all p series have same ARIMA structure. Otherwise first sublist gives the ARIMA structure of the first series etc.
H	A p*p covariance matrix (or p*p*n array in of time-varying case) of the disturbance terms ε[t] of the observation equation. Default gives p*p zero matrix ie. ordinary ARIMA model without additional noise. Omitted in case of non-Gaussian distributions. Augment the state vector if you to add want additional noise.
Q	A p*p covariance matrix of the disturbance terms η[t] of the system equation. Default is p*p identity matrix ie. ordinary ARIMA model with disturbance terms having unit variance.
y	A time series object of class ts, or a object that can be coerced to such.
u	Only used with non-Gaussian distribution. See details.
distribution	Specify the distribution of the observations. Default is "Gaussian".
transform	The functions of KFAS require diagonal covariance matrix H[t]. If H[t] is not diagonal, model can be transformed using one of the two options. Option "ldl" performs LDL decomposition for covariance matrix H[t], and multiplies the observation equation with the L[t]^{-1}, so ε[t] ~ N(0,D[t]). Option "augment" adds ε[t] to the state vector, when Q[t] becomes block diagonal with blocks Q[t] and H[t]. In case of univariate series, option "ldl" only changes the H_type argument of the model to "Diagonal". Default is "none" which does no transformation but checks if H is diagonal. If not, H_type is set to "Untransformed".
tolF	Tolerance parameter for Finf. Smallest value not counted for zero.
tol0	Tolerance parameter for LDL decomposition, determines which diagonal values are counted as zero.

Details

The linear Gaussian state space model is given by

y[t] = Z[t]α[t] + ε[t], (observation equation)

α[t+1] = T[t]α[t] + R[t]η[t], (transition equation)

where ε[t] ~ N(0,H[t]), η[t] ~ N(0,Q[t]) and α[1] ~ N(a[1],P[1]) independently of each other. In case of non-Gaussian observations, the observation equation is of form p(y[t]|θ[t]) = p(y[t]|Z[t]α[t]), with p(y[t]|θ[t]) being one of the following:

If observations are Poisson distributed, parameter of Poisson distribution is u[t]λ[t] and θ[t]=log(λ[t]).

If observations are from binomial distribution, u is a vector specifying number the of trials at times 1,…,n, and θ[t] = log(π[t]/(1-π[t])), where π[t] is the probability of success at time t.

For non-Gaussian models u[t]=1 as a default. For Gaussian models, parameter is omitted.

Only univariate observations are supported when observation equation is non-Gaussian.

See Also

regSSM for state space representation of a regression model, structSSM for structural time series model, and SSModel for custom SSModel object.

jrnold/KFAS documentation built on May 19, 2019, 11:55 p.m.