arima0(x, order = c(0, 0, 0), seasonal = list(order = c(0, 0, 0), period = NA), xreg = NULL, include.mean = TRUE, delta = 0.01, transform.pars = TRUE, fixed = NULL, init = NULL, method = c("ML", "CSS"), n.cond, optim.control = list())
"predict"(object, n.ahead = 1, newxreg, se.fit = TRUE, ...)
frequency(x)
).
This should be a list with components order
and
period
, but a specification of just a numeric vector of
length 3 will be turned into a suitable list with the specification
as the order
.x
.TRUE
for undifferenced series,
FALSE
for differenced ones (where a mean would not affect
the fit nor predictions).method = "CSS"
.NA
entries in
fixed
will be varied. transform.pars = TRUE
will be overridden (with a warning) if any ARMA parameters are
fixed.fixed
will be ignored.optim
.arima0
fit.xreg
to be used for
prediction. Must have at least n.ahead
rows.arima0
, a list of class "arima0"
with components:coef
.method = "ML"
fits.x
.optim
.predict.arima0
, a time series of predictions, or if
se.fit = TRUE
, a list with components pred
, the
predictions, and se
, the estimated standard errors. Both
components are time series.
delta
sets the
tolerance: at its default value the approximation is normally
negligible and the speed-up considerable. Exact computations can be
ensured by setting delta
to a negative value. If transform.pars
is true, the optimization is done using an
alternative parametrization which is a variation on that suggested by
Jones (1980) and ensures that the model is stationary. For an AR(p)
model the parametrization is via the inverse tanh of the partial
autocorrelations: the same procedure is applied (separately) to the
AR and seasonal AR terms. The MA terms are also constrained to be
invertible during optimization by the same transformation if
transform.pars
is true. Note that the MLE for MA terms does
sometimes occur for MA polynomials with unit roots: such models can be
fitted by using transform.pars = FALSE
and specifying a good
set of initial values (often obtainable from a fit with
transform.pars = TRUE
). Missing values are allowed, but any missing values
will force delta
to be ignored and full recursions used.
Note that missing values will be propagated by differencing, so the
procedure used in this function is not fully efficient in that case. Conditional sum-of-squares is provided mainly for expositional
purposes. This computes the sum of squares of the fitted innovations
from observation
n.cond
on, (where n.cond
is at least the maximum lag of
an AR term), treating all earlier innovations to be zero. Argument
n.cond
can be used to allow comparability between different
fits. The ‘part log-likelihood’ is the first term, half the
log of the estimated mean square. Missing values are allowed, but
will cause many of the innovations to be missing. When regressors are specified, they are orthogonalized prior to
fitting unless any of the coefficients is fixed. It can be helpful to
roughly scale the regressors to zero mean and unit variance.$$X_t = a_1X_{t-1} + \cdots + a_pX_{t-p} + e_t + b_1e_{t-1} + \dots + b_qe_{t-q}$$
and so the MA coefficients differ in sign from those of
S-PLUS. Further, if include.mean
is true, this formula
applies to $X-m$ rather than $X$. For ARIMA models with
differencing, the differenced series follows a zero-mean ARMA model.
The variance matrix of the estimates is found from the Hessian of the log-likelihood, and so may only be a rough guide, especially for fits close to the boundary of invertibility.
Optimization is done by optim
. It will work
best if the columns in xreg
are roughly scaled to zero mean
and unit variance, but does attempt to estimate suitable scalings.
Finite-history prediction is used. This is only statistically
efficient if the MA part of the fit is invertible, so
predict.arima0
will give a warning for non-invertible MA
models.
Gardner, G, Harvey, A. C. and Phillips, G. D. A. (1980) Algorithm AS154. An algorithm for exact maximum likelihood estimation of autoregressive-moving average models by means of Kalman filtering. Applied Statistics 29, 311--322.
Harvey, A. C. (1993) Time Series Models, 2nd Edition, Harvester Wheatsheaf, sections 3.3 and 4.4.
Harvey, A. C. and McKenzie, C. R. (1982) Algorithm AS182. An algorithm for finite sample prediction from ARIMA processes. Applied Statistics 31, 180--187.
Jones, R. H. (1980) Maximum likelihood fitting of ARMA models to time series with missing observations. Technometrics 22 389--395.
arima
, ar
, tsdiag
## Not run: arima0(lh, order = c(1,0,0))
arima0(lh, order = c(3,0,0))
arima0(lh, order = c(1,0,1))
predict(arima0(lh, order = c(3,0,0)), n.ahead = 12)
arima0(lh, order = c(3,0,0), method = "CSS")
# for a model with as few years as this, we want full ML
(fit <- arima0(USAccDeaths, order = c(0,1,1),
seasonal = list(order=c(0,1,1)), delta = -1))
predict(fit, n.ahead = 6)
arima0(LakeHuron, order = c(2,0,0), xreg = time(LakeHuron)-1920)
## Not run:
# ## presidents contains NAs
# ## graphs in example(acf) suggest order 1 or 3
# (fit1 <- arima0(presidents, c(1, 0, 0), delta = -1)) # avoid warning
# tsdiag(fit1)
# (fit3 <- arima0(presidents, c(3, 0, 0), delta = -1)) # smaller AIC
# tsdiag(fit3)## End(Not run)
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