

   arima0 {ts}                                  R Documentation

   AARRIIMMAA MMooddeelllliinngg ooff TTiimmee SSeerriieess ---- PPrreelliimmiinnaarryy VVeerrssiioonn

   DDeessccrriippttiioonn::

        Fit an ARIMA model to a univariate time series by exact
        maximum likelihood, and forecast from the fitted model.

   UUssaaggee::

        arima0(x, order = c(0, 0, 0),
               seasonal = list(order = c(0, 0, 0), period = NA),
               xreg = NULL, include.mean, na.action = na.fail,
               delta = 0.01, transform.pars = 2)
        predict(arima0.obj, n.ahead = 1, newxreg, se.fit = TRUE)
        arima0.diag(fit, gof.lag = 10)

   AArrgguummeennttss::

          x: a univariate time series

      order: A specification of the non-seasonal part of the
             ARIMA model: the three components are (p, d, q),
             the AR order, the degree of differencing and the
             MA order.

   seasonal: A specification of the seasonal part of the ARIMA
             model, plus the period (which defaults to `fre-
             quency(x)').

       xreg: Optionally, a vector or matrix of external regres-
             sors, which must have the same number of rows as
             `x'.

   include.mean: Should the ARIMA model include a mean term?
             The default is `TRUE' for undifferenced series,
             `FALSE' for differenced ones (where a mean would
             not affect the fit nor predictions).

   na.action: Function to be applied to remove missing values.

      delta: A value to indicate at which point `fast recur-
             sions' should be used. See the Details section.

   transform.pars: If greater than 0, the ARMA parameters are
             transformed to ensure that they remain in the
             region of invertibility. If equal to 2, the opti-
             mization is rerun on the original scale to find
             the Hessian.

   arima0.obj, fit: The result of an `arima0' fit.

    newxreg: New values of `xreg' to be used for prediction.
             Must have at least `n.ahead' rows.

    n.ahead: The number of steps ahead for which prediction is
             required.

     se.fit: Logical: should standard errors of prediction be
             returned?

    gof.lag: Number of lags to be used in goodness-of-fit test.

   DDeettaaiillss::

        Different definitions of ARIMA models have different
        signs for the AR and/or MA coefficients. The definition
        here has

        `X[t] = a[1]X[t-1] + ... + a[p]X[t-p] + e[t] + b[1]e[t-1] + ... + b[q]e[t-q]'

        and so the MA coefficients differ in sign from those of
        S-PLUS. Further, if `include.mean' is true, this for-
        maula applies to X-m rather than X.

        The exact likelihood is computed via a state-space rep-
        resentation of the ARMA process, and the innovations
        and their variance found by a Kalman filter using the
        Fortran code of Gardener et al.  (1980).  This has the
        option to switch to `fast recursions' (assume an effec-
        tively infinite past) if the innovations variance is
        close enough to its asymptotic bound. The argument
        `delta' sets the tolerance: at its default value the
        approximation is normally negligible and the speed-up
        considerable. Exact computations can be ensured by set-
        ting `delta' to a negative value.

        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 (currently) done by `nlm'. It will work
        best if the columns in `xreg' are roughly scaled to
        zero mean and unit variance.

        Finite-history prediction is used. This is only statis-
        tically efficient if the MA part of the fit is invert-
        ible, so `predict.arima0' will give a warning for non-
        invertible MA models.

   VVaalluuee::

        For `arima0', a list of class `"arima0"' with compo-
        nents:

       coef: a vector of AR, MA and regression coefficients,

     sigma2: the MLE of the innovations variance.

   var.coef: the estimated variance matrix of the coefficients
             `coef'. If `transform.pars = 1', only the portion
             corresponding to the untransformed parameters is
             returned.

     loglik: the maximized log-likelihood (of the differenced
             data).

       arma: A compact form of the specification, as a vector
             giving the number of AR, MA, seasonal AR and sea-
             sonal MA coefficients, plus the period and the
             number of non-seasonal and seasonal differences.

        aic: the AIC value corresponding to the log-likelihood.

      resid: the residuals.

       call: the matched call.

     series: the name of the series `x'.

   convergence: the `code' returned by `nlm'.

             For `predict.arima0', a time series of predic-
             tions, or if `se.fit = TRUE', a list with compo-
             nents `pred', the predictions, and `se', the esti-
             mated standard errors. Both components are time
             series.

   NNoottee::

        This is a preliminary version, and will be replaced in
        due course.

        The standard errors of prediction exclude the uncer-
        tainty in the estimation of the ARMA model and the
        regression coefficients.

        The results are likely to be different from S-PLUS's
        `arima.mle', which computes a conditional likelihood
        and does not include a mean in the model. Further, the
        convention used by `arima.mle' reverses the signs of
        the MA coefficients.

   AAuutthhoorr((ss))::

        B.D. Ripley

   RReeffeerreenncceess::

        Brockwell, P. J. and Davis, R. A. (1996) Introduction
        to Time Series and Forecasting. Springer, New York.
        Sections 3.3 and 8.3.

        Gardener, 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 Statis-
        tics 29, 311-322.

        Harvey, A. C. (1993) Time Series Models, 2nd Edition,
        Harvester Wheatsheaf, section 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.

   SSeeee AAllssoo::

        `ar'

   EExxaammpplleess::

        data(lh)
        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)

        data(USAccDeaths)
        fit <- arima0(USAccDeaths, order=c(0,1,1), seasonal=list(order=c(0,1,1)))
        fit
        predict(fit, n.ahead=6)

        data(LakeHuron)
        arima0(LakeHuron, order=c(2,0,0), xreg=1:98)

