# Fitting Smoothing Spline ANOVA Models with Correlated Data Fit smoothing spline ANOVA models with correlated Gaussian data. The symbolic model specification via `formula` follows the same rules as in `lm`. ```r ssanova9(formula, type=NULL, data=list(), subset, offset, na.action=na.omit, partial=NULL, method="v", alpha=1.4, varht=1, id.basis=NULL, nbasis=NULL, seed=NULL, cov, skip.iter=FALSE) para.arma(fit) ``` ## Arguments - `formula`: Symbolic description of the model to be fit. - `type`: List specifying the type of spline for each variable. See `mkterm` for details. - `data`: Optional data frame containing the variables in the model. - `subset`: Optional vector specifying a subset of observations to be used in the fitting process. - `offset`: Optional offset term with known parameter 1. - `na.action`: Function which indicates what should happen when the data contain NAs. - `partial`: Optional symbolic description of parametric terms in partial spline models. - `method`: Method for smoothing parameter selection. Supported are `method="v"` for V, `method="m"` for M, and `method="u"` for U; see the reference for definitions of U, V, and M. - `alpha`: Parameter modifying V or U; larger absolute values yield smoother fits. Ignored when `method="m"` are specified. - `varht`: External variance estimate needed for `method="u"`. Ignored when `method="v"` or `method="m"` are specified. - `id.basis`: Index designating selected "knots". - `nbasis`: Number of "knots" to be selected. Ignored when `id.basis` is supplied. - `seed`: Seed to be used for the random generation of "knots". Ignored when `id.basis` is supplied. - `cov`: Input for covariance functions. See `mkcov` for details. - `skip.iter`: Flag indicating whether to use initial values of theta and skip theta iteration. See notes on skipping theta iteration. - `fit`: `ssanova9` fit with ARMA error. ## Details The model specification via `formula` is intuitive. For example, `y~x1*x2` yields a model of the form $$ y = C + f_{1}(x1) + f_{2}(x2) + f_{12}(x1,x2) + e $$ with the terms denoted by `"1"`, `"x1"`, `"x2"`, and `"x1:x2"`. The model terms are sums of unpenalized and penalized terms. Attached to every penalized term there is a smoothing parameter, and the model complexity is largely determined by the number of smoothing parameters. A subset of the observations are selected as "knots." Unless specified via `id.basis` or `nbasis`, the number of "knots" $q$ is determined by $max(30,10n^{2/9})$, which is appropriate for the default cubic splines for numerical vectors. Using $q$ "knots," `ssanova` calculates an approximate solution to the penalized least squares problem using algorithms of the order $O(nq^{2})$, which for $q<

ssanova9 function