A one sided formula of the form ~ t, or ~ t | g, specifying a time covariate t and, optionally, a grouping factor g. A covariate for this correlation structure must be integer valued. When a grouping factor is present in formula, the correlation structure is assumed to apply only to observations within the same grouping level; observations with different grouping levels are assumed to be uncorrelated. Defaults to ~ 1, which corresponds to using the order of the observations in the data as a covariate, and no groups.

A non-negative integer specifying the moving average (MA) order of the ARMA structure. Default is 1.

A flag indicating whether ARMA effects should be estimated by means of residual covariance matrices. This is currently only possible for stationary ARMA effects of order 1. If the model family does not have natural residuals, latent residuals are added automatically. If FALSE (the default) a regression formulation is used that is considerably faster and allows for ARMA effects of order higher than 1 but is only available for gaussian models and some of its generalizations.