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### Some Special Spline Models

Example 1. Polynomial Spline. For the polynomial spline of order on , the model space is

 (6)

Let . Define an inner product on as

Then we have
 (7)

where and 's are Bernoulli polynomials (Craven and Wahba, 1979).

Let . Define inner product on as

Then (Wahba, 1990)
 (8)

Table lists statements inside ssr for four simple polynomial splines. We assume variables , and have been calculated based on Bernoulli polynomials. All expressions of the reproducing kernels in this table are available in our library. Note that the domain under construction () is restricted to while the domain under construction () is an arbitrary interval . Thus one needs to transform a variable into before using rk functions in the third column. The rk functions in the fourth column assume the domain for any fixed . One can calculate the reproducing kernel on by a translation, for example .

 m splines under construction () under construction () 1 linear y~1, rk=linear(t) y~1, rk=linear2(t) 2 cubic y~k1, rk=cubic(t) y~t, rk=cubic2(t) 3 quintic y~k1+k2, rk=quintic(t) y~t+t**2, rk=quintic2(t) 4 septic y~k1+k2+k3, rk=septic(t) y~t+t**2+t**3, rk=septic2(t)

Example 2. Stein Estimate. A James-Stein shrinkage estimate can be regarded as the solution to () with , and (Gu, 2002).

For shrinkage toward a constant,

 (9)

For shrinkage toward zero,

 (10)

The corresponding rk expressions for shrinkage toward a constant and shrinkage toward zero are shrink1(t) and shrink0(t) respectively. We use these shrinkage methods to model discrete covariates in smoothing spline ANOVA models as will be discussed later.

Example 3. Periodic Spline. For the -th order periodic spline on (Wahba, 1990),

The function periodic in our library calculates evaluated at specified points. The order is specified by the argument order with default order=2. For example, one may fit a cubic periodic spline () by
    ssr(y~1, rk=periodic(t))


Example 4. Thin plate spline (TPS) (Wahba, 1990). For a TPS of order on with ,

 (11)

where

A pseudo reproducing kernel (conditional positive definite rather than positive definite) for is , where is the Euclidean distance, and

The function tp.psuedo in our library calculates this pseudo kernel. The argument order of this function specifies with default order=2. For example, for and , one may fit the TSP model by
    ssr(y~t1+t2, rk=tp.pseudo(list(t1,t2)))


The true kernel discussed in Gu and Wahba (1993a) is calculated by the function tp. It takes longer to compute the true kernel and is only necessary for calculating posterior variances.

Example 5. Spline on the sphere is an extension of both the periodic spline defined on the unit circle and the TPS on . Let the domain be , where is the unit sphere. Any point on can be represented as , where ( ) is the longitude and ( ) is the latitude. Define

where is the surface Laplacian on the unit sphere

The model space , where and

is a RKHS with RK , where

is the rk of , is the angle between and , and 's are the Legendre polynomials. . is in the form of an infinite series which is inconvenient to compute. Closed form expressions are only available for and . Wahba (1981) proposed replacing by a topologically equivalent norm under which closed form of rk's can be derived. See Wahba (1981), Wahba (1982), Wahba (1990) and Wahba and Luo (1996) for more details. The function sphere in our library calculates under the norm for . The argument order of this function specifies with default order=2. For example, for , one may fit a spline on the sphere by
    ssr(y~t1+t2, rk=sphere(cbind(t1,t2),order=3))

where and are longitude and latitude respectively.

Example 6. -spline. The penalty term, , is usually used to penalize the roughness of the function . However, sometimes it is advantageous to use other forms of penalty. For example, prior information may be incorporated or even estimated by a penalty to the departure of the non-parametric function from a specific parametric model (Wahba, 1990; Heckman and Ramsay, 2000). Let be a linear differential operator , where denotes the th derivative operator and the 's are continuous real-valued weight functions. The spline estimate with the penalty is called an -spline. See Heckman and Ramsay (2000) and Gu (2002) for more details about the -spline. The lspline function in our library calculates reproducing kernels for the following four -spline models.

(a) Suppose that . If prior knowledge suggests that is close to a linear combination of and , one may use , and . Then

The statement for fitting such a model is

    ssr(y~sin(2*pi*t)+cos(2*pi*t)-1, rk=lspline(t,type="sine0"))


If we want to include the constant in the model space, then , ,

The statement for fitting such a model is
    ssr(y~sin(2*pi*t)+cos(2*pi*t), rk=lspline(t,type="sine1"))


(b) Suppose that . If prior knowledge suggests that is close to a linear combination of and , one may use , and . Then

 (12)

The statement for fitting such a model is
    ssr(y~exp(-t), rk=lspline(t,type="exp"))


(c) Suppose that . If prior knowledge suggests that is close to the logistic function , one may use , and . Then

The statement for fitting such a model is
    ssr(y~I(exp(t)/(1+exp(t)))-1, rk=lspline(t,type="logit"))


(d) Suppose that . If prior knowledge suggests that is close to a linear combination of , , and , one may use , and . Then

The statement is
    ssr(y~t+cos(t)+sin(t), rk=lspline(t,type="linSinCos"))


Next: The Smoothing Parameter Up: General Smoothing Spline Regression Previous: The ssr Function
Yuedong Wang 2004-05-19