# Cubic smoothing spline regression

### Description

Fits a cubic smoothing spline to univariate data. The amount of smoothness can be specified or estimated from the data by GCV. <!–brief description–>

### Usage

1 2 3 4 5 6 7 8 9 10 | ```
sreg(x, y, lambda = NA, df = NA, offset = 0,
weights = rep(1, length(x)), cost = 1,
nstep.cv = 80, tol=1e-5,find.diagA = TRUE, trmin = 2.01,
trmax = NA, lammin = NA,
lammax = NA, verbose = FALSE,
do.cv = TRUE, method = "GCV", rmse = NA,
na.rm = TRUE)
## S3 method for class 'sreg'
predict(object, x, derivative = 0, model = 1,...)
``` |

### Arguments

`x` |
Vector of x value |

`y` |
Vector of y values |

`lambda` |
Single smoothing parameter or a vector of values . If omitted smoothing parameter estimated by GCV. NOTE: lam here is equivalent to the value lambda*N in Tps/Krig where N is the number of unique observations. See example below. |

`object` |
An sreg object. |

`derivative` |
Order of deriviatve to evaluate. Must be 0,1, or 2. |

`df` |
Amount of smoothing in term of effective degrees of freedom for the spline |

`offset` |
an offset added to the term cost*degrees of freedom in the denominator of the GCV function. (This would be used for adjusting the df from fitting other models such as in back-fitting additive models.) |

`model` |
Specifies which model parameters to use. |

`weights` |
A vector that is proportional to the reciprocal variances of the errors. |

`cost` |
Cost value to be used in the GCV criterion. |

`nstep.cv ` |
Number of grid points of smoothing parameter for GCV grid search. |

`tol` |
Tolerance for convergence in minimizing the GCV or other criteria to estimate the smoothing parameter. |

`find.diagA` |
If TRUE calculates the diagonal elements of the smoothing matrix. The effective number of degrees of freedom is the sum of these diagonal elements. Default is true. This requires more stores if a grid of smoothing parameters is passed. ( See returned values below.) |

`trmin` |
Sets the minimum of the smoothing parameter range for the GCV grid search in terms of effective degrees of freedom. |

`trmax` |
Sets the maximum of the smoothing parameter range for the GCV grid search in terms of effective degrees of freedom. If NA the range is set to .99 of number of unique locations. |

`lammin` |
Same function as trmin but in the lambda scale. |

`lammax` |
Same function as trmax but in the lambda scale. |

`verbose` |
Print out all sorts of debugging info. Default is falseof course! |

`do.cv` |
Evaluate the spline at the GCV minimum. Default is true. |

`method` |
A character string giving the method for determining the smoothing parameter. Choices are "GCV", "GCV.one", "GCV.model", "pure error", "RMSE". Default is "GCV". |

`rmse` |
Value of the root mean square error to match by varying lambda. |

`na.rm` |
If TRUE NA's are removed from y before analysis. |

`...` |
Other optional arguments to pass to the predict function. |

### Details

MODEL: The assumed model is Y.k=f(x.k) +e.k where e.k should be approximately normal and independent errors with variances sigma**2/w.k

ESTIMATE: A smoothing spline is a locally weighted average of the y's based on the relative locations of the x values. Formally the estimate is the curve that minimizes the criterion:

(1/n) sum(k=1,n) w.k( Y.k - f( X.k))**2 + lambda R(f)

where R(f) is the integral of the squared second derivative of f over the range of the X values. Because of the inclusion of the (1/n) in the sum of squares the lambda parameter in sreg corresponds to the a value of lambda*n in the Tps function and in the Krig function.

The solution to this minimization is a piecewise cubic polynomial with the join points at the unique set of X values. The polynomial segments are constructed so that the entire curve has continuous first and second derivatives and the second and third derivatives are zero at the boundaries. The smoothing has the range [0,infinity]. Lambda equal to zero gives a cubic spline interpolation of the data. As lambda diverges to infinity ( e.g lambda =1e20) the estimate will converge to the straight line estimated by least squares.

The values of the estimated function at the data points can be expressed in the matrix form:

predicted values= A(lambda)Y

where A is an nXn symmetric matrix that does NOT depend on Y. The diagonal elements are the leverage values for the estimate and the sum of these (trace(A(lambda)) can be interpreted as the effective number of parameters that are used to define the spline function. IF there are replicate points the A matrix is the result of finding group averages and applying a weighted spline to the means. The A matrix is also used to find "Bayesian" confidence intervals for the estimate, see the example below.

CROSS-VALIDATION:The GCV criterion with no replicate points for a fixed value of lambda is

(1/n)(Residual sum of squares)/((1-(tr(A)-offset)*cost + offset)/n)**2,

Usually offset =0 and cost =1. Variations on GCV with replicate points are described in the documentation help file for Krig. With an appropriate choice for the smoothing parameter, the estimate of sigma**2 is found by (Residual sum of squares)/tr(A).

COMPUTATIONS: The computations for 1-d splines exploit the banded structure of the matrices needed to solve for the spline coefficients. Banded structure also makes it possible to get the diagonal elements of A quickly. This approach is different from the algorithms in Tps and tremendously more efficient for larger numbers of unique x values ( say > 200). The advantage of Tps is getting "Bayesian" standard errors at predictions different from the observed x values. This function is similar to the S-Plus smooth.spline. The main advantages are more information and control over the choice of lambda and also the FORTRAN source code is available (css.f).

See also the function `splint`

which is designed to be a bare bones
but fast smoothing spline.

### Value

Returns a list of class sreg. Some of the returned components are

`call` |
Call to the function |

`yM` |
Vector of dependent variables. If replicated data is given these are the replicate group means. |

`xM` |
Unique x values matching the y's. |

`weights` |
Proportional to reciprocal variance of each data point. |

`weightsM` |
Proportional to reciprocal pooled variance of each replicated mean data value (xM). |

`x` |
Original x data. |

`y` |
Original y data. |

`method` |
Method used to find the smoothing parameter. |

`pure.ss` |
Pure error sum of squares from replicate groups. |

`shat.pure.error` |
Estimate of sigma from replicate groups. |

`shat.GCV` |
Estimate of sigma using estimated lambda from GCV minimization |

`trace` |
Effective degrees of freedom for the spline estimate(s) |

`gcv.grid` |
Values of trace, GCV, shat. etc. for a grid of smoothing parameters. If lambda ( or df) is specified those values are used. |

`lambda.est` |
Summary of various estimates of the smoothing parameter |

`lambda` |
If lambda is specified the passed vector, if missing the estimated value. |

`residuals` |
Residuals from spline(s). If lambda or df is specified the residuals from these values. If lambda and df are omitted then the spline having estimated lambda. This will be a matrix with as many columns as the values of lambda. |

`fitted.values` |
Matrix of fitted values. See notes on residuals. |

`predicted` |
A list with components x and y. x is the unique values of xraw in sorted order. y is a matrix of the spline estimates at these values. |

`eff.df` |
Same as trace. |

`diagA` |
Matrix containing diagonal elements of the smoothing matrix. Number of columns is the number of lambda values. WARNING: If there is replicated data the diagonal elements are those for the smoothing the group means at the unique x locations. |

### See Also

Krig, Tps, splint

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 | ```
# fit a GCV spline to
# control group of rats.
fit<- sreg(rat.diet$t,rat.diet$con)
summary( fit)
set.panel(2,2)
plot(fit) # four diagnostic plots of fit
set.panel()
predict( fit) # predicted values at data points
xg<- seq(0,110,,50)
sm<-predict( fit, xg) # spline fit at 50 equally spaced points
der.sm<- predict( fit, xg, deriv=1) # derivative of spline fit
set.panel( 2,1)
plot( fit$x, fit$y) # the data
lines( xg, sm) # the spline
plot( xg,der.sm, type="l") # plot of estimated derivative
set.panel() # reset panel to 1 plot
# the same fit using the thin plate spline numerical algorithms
# sreg does not scale the obs so instruct Tps not to sacel either
# this will make lambda comparable within factor of n.
fit.tps<-Tps( rat.diet$t,rat.diet$con, scale="unscaled")
summary( fit.tps)
# compare sreg and Tps results to show the adjustment to lambda.
predict( fit)-> look
predict( fit.tps, lambda=fit$lambda*fit$N)-> look2
test.for.zero( look, look2) # silence means it checks to 1e-8
# finding approximate standard errors at observations
SE<- fit$shat.GCV*sqrt(fit$diagA)
# compare to predictSE( fit.tps) differences are due to
# slightly different lambda values and using shat.MLE instad of shat.GCV
#
# 95% pointwise prediction intervals
Zvalue<- qnorm(.0975)
upper<- fit$fitted.values + Zvalue* SE
lower<- fit$fitted.values - Zvalue* SE
#
# conservative, simultaneous Bonferroni bounds
#
ZBvalue<- qnorm(1- .025/fit$N)
upperB<- fit$fitted.values + ZBvalue* SE
lowerB<- fit$fitted.values - ZBvalue* SE
#
# take a look
plot( fit$x, fit$y)
lines( fit$predicted, lwd=2)
matlines( fit$x,
cbind( lower, upper, lowerB, upperB), type="l", col=c( 2,2,4,4), lty=1)
title( "95 pct pointwise and simultaneous intervals")
# or try the more visually honest:
plot( fit$x, fit$y)
lines( fit$predicted, lwd=2)
segments( fit$x, lowerB, fit$x, upperB, col=4)
segments( fit$x, lower, fit$x, upper, col=2, lwd=2)
title( "95 pct pointwise and simultaneous intervals")
set.panel( 1,1)
``` |