# smooth.Pspline: Fit a Polynomial Smoothing Spline of Arbitrary Order In pspline: Penalized Smoothing Splines

## Description

Returns an object of class `"smooth.Pspline"` which is a natural polynomial smooth of the input data of order fixed by the user.

## Usage

 ```1 2 3``` ```smooth.Pspline(x, y, w=rep(1, length(x)), norder=2, df=norder + 2, spar=0, method=1) sm.spline(x, y, w, cv=FALSE, ...) ```

## Arguments

 `x` values of the predictor variable. These must be strictly increasing, and there must be at least `2*norder + 1` of them. `sm.spline` provides a simplified interface, in which the `x` values can be unsorted, or a list with components `"x"` and `"y"` or a two-column matrix or a complex vector. `y` one or more sets of response variable values. If there is one response variable, `y` is an array of the same length as `x`; if more than one, then `y` is a matrix with `length(x)` rows and number of columns equal to the number of variables. `w` vector of positive weights for smoothing of the same length as `x`. If measurements at different values of `x` have different variances, `w` should be inversely proportional to the variances. The default is that all weights are one. `norder` the order of the spline. `norder = 2` gives the cubic smoothing spline, and more generally the smoothing function is a piecewise polynomial of degree `2*norder - 1`. If derivatives are to be computed from the smoothing using `predict.smooth.Pspline`, the order should be one or two more than the highest order of derivative. `df` a number which specifies the degrees of freedom = trace(S). Here S is the implicit smoothing matrix. `df` controls the amount of smoothing if `method = 2`. `spar` the usual smoothing parameter for smoothing splines, which is the coefficient of the integrated squared derivative of order `norder`. `spar` controls the amount of smoothing if `method = 1`. `cv` logical: should ordinary cross-validation be used (true) or generalized cross-validation. `method` the method for controlling the amount of smoothing. `method = 1` uses the value supplied for `spar`. `method = 2` adjusts `spar` so that the degrees of freedom is equal to `df`. `method = 3` adjusts `spar` so that the generalized cross-validation criterion is minimized. `method = 4` adjusts `spar` so that the ordinary cross-validation criterion is minimized. If `method = 3` or `method = 4`, `spar` defines the initial value for the minimization algorithm if positive; otherwise an internally generated value is used. `sm.spline` chooses this automatically based on the supplied values and that of `cv`. `...` additional arguments to be passed to `smooth.Pspline`.

## Details

The method produces results similar to function `smooth.spline`, but the smoothing function is a natural smoothing spline rather than a B-spline smooth, and as a consequence will differ slightly for `norder = 2` over the initial and final intervals.

The main extension is the possibility of setting the order of derivative to be penalized, so that derivatives of any order can be computed using the companion function `predict.smooth.Pspline`. The algorithm is of order N, meaning that the number of floating point operations is proportional to the number of values being smoothed. Note that the argument values must be strictly increasing, a condition that is not required by `smooth.spline`.

Note that the appropriate or minimized value of the smoothing parameter `spar` will depend heavily on the order; the larger the order, the smaller this parameter will tend to be.

## Value

an object of class `"smooth.Pspline"` is returned, consisting of the fitted smoothing spline evaluated at the supplied data, some fitting criteria and constants. This object contains the information necessary to evaluate the smoothing spline or one of its derivatives at arbitrary argument values using `predict.smooth.Pspline`. The components of the returned list are

 `norder` the order of the spline `x` values of the predictor variable `ysmth` a matrix with `length(x)` rows, each column of which contains the smoothed response variable values for the corresponding column of `y`. `lev` leverage values, which are the diagonal elements of the smoother matrix S. `gcv` generalized cross-validation criterion value `cv` ordinary cross-validation criterion value `df` a number which supplies the degrees of freedom = trace(S) rather than a smoothing parameter. `spar` the final smoothing parameter for smoothing splines. This is unchanged if `method = 1`, but adjusted otherwise. `call` the call that produced the fit.

## References

Heckman, N. and Ramsay, J. O. (1996) Spline smoothing with model based penalties. McGill University, unpublished manuscript.

`predict.smooth.Pspline`, `smooth.spline`

## Examples

 ```1 2 3 4 5 6 7``` ```data(cars) attach(cars) plot(speed, dist, main = "data(cars) & smoothing splines") cars.spl <- sm.spline(speed, dist) cars.spl lines(cars.spl, col = "blue") lines(sm.spline(speed, dist, df=10), lty=2, col = "red") ```

### Example output ```Call:
smooth.Pspline(x = ux, y = tmp[, 1], w = tmp[, 2], method = method)

Smoothing Parameter (Spar): 366.8429
Equivalent Degrees of Freedom (Df): 2.428851
GCV Criterion: 29.54554
CV  Criterion: 39.18787
```

pspline documentation built on May 2, 2019, 4:01 a.m.