lspline {assist} R Documentation

## Calculate Reproducing Kernels for Some L-splines

### Description

Return a matrix evaluating reproducing kernels for some L-splines at observed points.

### Usage

```lspline(x,y=x, type="exp", ...)
```

### Arguments

 `x` a numeric vector on which reproducing kerenls are evaluated. `y` an optional vector, specifying the second argument of reproducing kernels. Default is `x`. `type` a string indicating the type of L-splines. Available options are "exp", "logit","sine", "sine1", and "linSinCos". Default is "exp". `...` other arguments needed.

### Details

Denote L as the differential oprator, H_0 as the null (kernel) space. The available kernels correspond to the following L:

• exp: L=rD+D^2, H_0=span\{1,exp(-rx)\}. r>0, default to be 1;

• logit: L=D-1/(1+e^t), H_0=span\{e^t/(1+e^t)\};

• sine0: L=D^2+(2π)^2, H_0=span\{sin(2π x),cos(2π x)\};

• sine1: L=D(D^2+(2π)^2), H_0=span\{1, sin(2π x),cos(2π x)\};

• linSinCos: L=D^4+D^2, H_0=spac\{1, x, sin(x), cos(x)\}.

### Value

a matrix with the numbers of row and column equal to the lengths of x and y respectively. The [i, j] element is the reproducing kernel evaluated at (x[i], y[j]).

### Author(s)

Chunlei Ke chunlei\_ke@pstat.ucsb.edu and Yuedong Wang yuedong@pstat.ucsb.edu

### References

Wahba, G. (1990). Spline Models for Observational Data. SIAM, Vol. 59.

Heckman, N and Ramsay, J. O. (2000). Penalised regression with model-based penalties. To appear in Canadian Journal of Statisitcs.

`ssr`
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