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

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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

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# 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)

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