Partial Spline Models

The linear partial spline model assumes that (Wahba, 1990)

$$y_i= \beta_1 x_{1i} + \cdots + \beta_d x_{di} + f(t_i) + \epsilon_i , ~~~~i=1,\cdots,n,$$ (14)

where the first part is a linear model of covariates $x_1,\cdots,x_d$, and $f\in{\cal H}$ as in (). Note that an SS ANOVA model discussed in the next section can also be used for $f$ when $t$ is multivariate. Partial spline models provide a tool to model multiple covariates when the relationship is unknown for only a few variables. Note that some $x_k$'s may be functions of $t$. For example, $x=(t-t_0)_+^q$ allows a jump in the $q$th derivative at $t_0$.

Let $X$ be the design matrix of $x_1,\cdots,x_d$: $X^T = {\{ x_{ji} \}_{j=1}^d }_{i=1}^n$, and $S=(X~ T)$. If $S$ is of full column rank, the estimate of $f$ has the same representation as in (). Furthermore, coefficients $(\beta_1,\cdots,\beta_d,\mbox{\boldmath$d$}^T)^T$ and $\mbox{\boldmath$c$}$ are solutions to equations () with $T$ replaced by $S$.

The linear model for $x_1,\cdots,x_d$ in () can be easily specified by adding these covariates to the right hand side of formula. For example, supposing $d=3$ and a cubic spline for $f$, we can fit model () by

    ssr(y~x1+x2+x3+t, rk=cubic(t))