View source: R/TSfamilyARIMAX.R
ARIMAXff  R Documentation 
(p, d, q)
Autoregressive Integrated Moving
Average Model (ARIMA(p, d, q)) with covariatesMaximum likelihood estimation of the drift,
standard deviation or variance of the random noise,
and coefficients of an autoregressive integrated
moving average process of order(p, d, q)
with covariates by MLE using Fisher scoring.
No seasonal terms handled yet. No seasonal components
handled yet.
ARIMAXff(order = c(1, 1, 0),
zero = c("ARcoeff", "MAcoeff"),
diffCovs = TRUE,
xLag = 0,
include.current = FALSE,
type.EIM = c("exact", "approximate")[1],
var.arg = TRUE,
nodrift = FALSE,
noChecks = FALSE,
ldrift = "identitylink",
lsd = "loglink",
lvar = "loglink",
lARcoeff = "identitylink",
lMAcoeff = "identitylink",
idrift = NULL,
isd = NULL,
ivar = NULL,
iARcoeff = NULL,
iMAcoeff = NULL)
order 
Integer vector with three components,
( 
zero 
Integer or character–strings vector.
Name(s) or position(s) of the parameters/linear predictors
to be modeled as interceptonly. Details at

diffCovs 
Logical. The default is 
xLag 
Integer, non–negative. If 
include.current 
Logical. Same as

type.EIM 
The type of expected information matrix (EIM) of the ARMA process
to be utilized in Fisher scoring.

var.arg 
Logical. If 
nodrift 
Logical. 
noChecks 
Logical. If 
ldrift, lsd, lvar, lARcoeff, lMAcoeff 
Link functions applied to the intercept, the random noise standard deviation (or optionally, the variance), and the coefficients in the ARMA–type conditional–mean model. 
idrift, isd, ivar, iARcoeff, iMAcoeff 
Optional initial values for the intercept (drift), noise SD
(or variance), and ARMA coeffcients (a vector of length 
Let \boldsymbol{x}_t
be a (probably time–varying) vector of
suitable covariates. The ARIMAX model handled by ARIMAXff
is
\nabla^d Y_t = \mu^{\star} + \boldsymbol{\beta}^T
\nabla^d \boldsymbol{x}_t +
\theta_1 \nabla^d Y_{t  1} + \cdots +
\theta_p \nabla^d Y_{t  p} +
\phi_1 \varepsilon_{t  1} + \cdots +
\phi_q \varepsilon_{t  q} + \varepsilon,
with \nabla^d (\cdot)
the operator differencing a
series d
times. If diffCovs = TRUE
, this function
differencing the covariates d
times too.
Similarly, ARMAXff
manages
\nabla^d Y_t = \mu^{\star} + \boldsymbol{\beta}^T \boldsymbol{x}_t +
\theta_1 Y_{t  1} + \cdots + \theta_p Y_{t  p} +
\phi_1 \varepsilon_{t  1} + \cdots +
\phi_q \varepsilon_{t  q} + \varepsilon,
where
\varepsilon_{t  \Phi_{t  1}}
\sim N(0, \sigma_{\varepsilon_t  \Phi_{t  1}}^2).
Note, \sigma_{\varepsilon  \Phi_{t  1}}^2
is conditional on
\Phi_{t  1}
,
the information
of the joint process
\left(Y_{t  1}, \boldsymbol{x}_t \right)
,
at time t
, and hence may be
modelled in terms of \boldsymbol{x}_t
,
if required.
ARIMAXff()
and ARMAXff()
handle multiple responses, thus a
matrix can be used as the response.
Note, no seasonal terms handled. This feature is to be
incorporated shortly.
The default linear predictor is
\boldsymbol{\eta} = \left(
\mu, \log \sigma^{2}_{\varepsilon_{t  \Phi_{t  1}}},
\theta_1, \ldots, \theta_p,
\phi_1, \ldots, \phi_q
\right)^T.
Other links are also handled. See Links
.
Further choices for the random noise, besides Gaussian, will be implemented over time.
As with ARXff
and MAXff
,
choices for the EIMs are "exact"
and "approximate"
.
Covariates may be incorporated in the fit for any linear
predictor above. Hence, ARIMAXff
supports non–stationary
processes (\sigma_{\varepsilon_t  \Phi_{t  1}}^2
) may depend on
\boldsymbol{X}_t
.
Also, constraint matrices
on the linear predictors may be
entered through cm.ARMA
or
using the argument constraints
, from
vglm
.
Checks on stationarity and
invertibility on the esitmated process are performed by default.
Set noChecks = TRUE
to dismiss this step.
An object of class "vglmff"
(see vglmffclass
)
to be used by VGLM/VGAM modelling functions, e.g.,
vglm
or vgam
.
zero
can be a numeric or a character–strings
vector specifying the position(s) or the name(s) of
the parameter(s) modeled as intercept–only.
Numeric values can be set as
usual (See CommonVGAMffArguments
).
If names are entered, the parameter names in this
family function are:
c("drift.mean", "noiseVar"  "noiseSD", "ARcoeff", "MAcoeff")
.
Manually modify this if required. For simplicity, the second choice is recommended.
No seasonal components handled yet.
If no covariates, \boldsymbol{x}_t
,
are incorporated in the analysis,
then ARIMAXff
fits an ordinary ARIMA model.
Ditto with ARMAXff
.
If nodrift = TRUE
, then the 'drift' is removed from the
vector of parameters and is not estimated.
By default, an ARMA model of order–c(1, 0)
with
order–1 differences is fitted. When initial
values are entered (isd
, iARcoeff
, etc.),
they are recycled
according to the number of responses.
Also, the ARMA coefficients
are intercept–only (note, zero = c("ARcoeff",
"MAcoeff")
)
This may altered via zero
, or by
constraint matrices (See constraints
)
using cm.ARMA
.
Checks on stationarity and/or
invertibility can be manually via
checkTS.VGAMextra
.
Victor Miranda and T. W. Yee
Miranda, V. and Yee, T.W. (2018) Vector Generalized Linear Time Series Models. In preparation.
Porat, B., and Friedlander, B. (1986) Computation of the Exact Information Matrix of Gaussian Time Series with Stationary Random Components. IEEE Transactions on Acoustics, Speech and Signal Processing. ASSp34(1), 118–130.
ARXff
,
MAXff
,
checkTS.VGAMextra
,
cm.ARMA
,
CommonVGAMffArguments
,
constraints
,
vglm
.
set.seed(3)
nn < 90
theta < c(0.12, 0.17) # AR coefficients
phi < c(0.15, 0.20) # MA coefficients.
sdWNN < exp(1.0) # SDs
mu < c(1.25, 0.85) # Mean (not drift) of the process.
covX < runif(nn + 1) # A single covariate.
mux3 < mu[1] + covX
##
## Simulate ARMA processes. Here, the drift for 'tsd3' depends on covX.
##
tsdata < data.frame(TS1 = mu[1] + arima.sim(model = list(ar = theta, ma = phi,
order = c(2, 1, 2)), n = nn, sd = sdWNN ),
TS2 = mu[2] + arima.sim(model = list(ar = theta, ma = phi,
order = c(2, 1, 2)), n = nn, sd = exp(2 + covX)),
TS3 = mux3 + arima.sim(model = list(ar = theta, ma = phi,
order = c(2, 1, 2)), n = nn, sd = exp(2 + covX) ),
x2 = covX)
### EXAMPLE 1. Fitting a simple ARIMA(2, 1, 2) using vglm().
# Note that no covariates involved.
fit.ARIMA1 < vglm(TS1 ~ 1, ARIMAXff(order = c(2, 1, 2), var.arg = FALSE,
# OPTIONAL INITIAL VALUES
# idrift = c(1.5)*(1  sum(theta)),
# ivar = exp(4), isd = exp(2),
# iARcoeff = c(0.20, 0.3, 0.1),
# iMAcoeff = c(0.25, 0.35, 0.1),
type.EIM = "exact"),
data = tsdata, trace = TRUE, crit = "log")
Coef(fit.ARIMA1)
summary(fit.ARIMA1)
vcov(fit.ARIMA1, untransform = TRUE)
##
# Fitting same model using arima().
##
# COMPARE to EXAMPLE1
( fitArima < arima(tsdata$TS1, order = c(2, 1, 2)) )
### EXAMPLE 2. Here only the ARMA coefficients and drift are interceptonly.
# The random noise variance is not constant.
fit.ARIMA2 < vglm(TS2 ~ x2, ARIMAXff(order = c(2, 1, 2), var.arg = TRUE,
lARcoeff = "rhobitlink", lMAcoeff = "identitylink",
type.EIM = c("exact", "approximate")[1],
# NOTE THE ZERO ARGUMENT.
zero = c("drift.mean", "ARcoeff", "MAcoeff")),
data = tsdata, trace = TRUE)
coef(fit.ARIMA2, matrix = TRUE)
summary(fit.ARIMA2)
constraints(fit.ARIMA2)
### EXAMPLE 3. Here only ARMA coefficients are interceptonly.
# The random noise variance is not constant.
# Note that the "drift" and the "variance" are "generated" in
# terms of 'x2' above for TS3.
fit.ARIMA3 < vglm(TS3 ~ x2, ARIMAXff(order = c(1, 1, 2), var.arg = TRUE,
lARcoeff = "identitylink", lMAcoeff = "identitylink",
type.EIM = c("exact", "approximate")[1], nodrift = FALSE,
zero = c( "ARcoeff", "MAcoeff")), # NOTE THE ZERO ARGUMENT.
data = tsdata, trace = TRUE)
coef(fit.ARIMA3, matrix = TRUE)
summary(fit.ARIMA3)
constraints(fit.ARIMA3)
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