asp2: Fit a semiparametric regression model with spatially adaptive...

In AdaptFitOS: Adaptive Semiparametric Additive Regression with Simultaneous Confidence Bands and Specification Tests

Fit a semiparametric regression model with spatially adaptive penalized splines

Description

Fits semiparametric additive regression models using the mixed model representation of penalized splines with spatially adaptive penalties. It includes the availability of simultaneous confidence bands (also for the derivatives of the smooth curves) and B-spline basis functions. Note that random effects, autocorrelations and interaction surfaces are not supported. Further, only Gaussian responses are supported. Also note that estimated curves are centered to have zero mean. See aspHetero for incorporation of heteroscedastic errors, scbM for some more details on the simulataneous confidence bands and summary.asp for computation of associated specification (lack-of-fit) tests.

Usage

asp2(form, spar.method = "REML", contrasts=NULL, 
     omit.missing = NULL, returnFit=FALSE, 
     niter = 20, niter.var = 50, tol=1e-6, tol.theta=1e-6, 
     control=NULL)

Arguments

form	a formula describing the model to be fitted. See aspFormula for further information. Note, that an intercept is always included, whether given in the formula or not.

spar.method	method for automatic smoothing parameter selection. May be "REML" (restricted maximum likelihood) or "ML" (maximum likelihood).
contrasts	an optional list. See the contrasts.arg of model.matrix.default.
omit.missing	a logical value indicating whether fields with missing values are to be omitted.
niter	a maximum number of iterations for the mean estimation, default is 20.
niter.var	a maximum number of iterations for the variance of random effects estimation, default is 50.
tol	tolerance for the convergence criterion. Default is 1e-6.
tol.theta	tolerance for the convergence criterion (smoothing parameter function routine). Default is 1e-6.
returnFit	a logical value indicating whether the fitted object should be returned when the maximum number of iterations is reached without convergence of the algorithm. Default is FALSE.
control	see lmeControl in the documentation to nlme.

Details

See Wiesenfarth et al (2012) for technical details and Wiesenfarth (2012, Chapter 5.1) for some more details on the use of the package (including a demonstration on how plots in Wiesenfarth et al are obtained).

Value

A list object of class asp containing the fitted model. The components are:

fitted	fitted values.
coef.mean	estimated mean coefficients.
design.matrices	design matrices both for knots und subknots.
x	x values.
knots	knots.
y.cov	estimated covariance matrix of the response.
random.var	estimated covariance matrix of the random effects.
subknots	subknots.
coef.random	estimated spline coefficients of the covariance matrix of the random effects.
var.random.var	estimated variance of the spline coefficients of the covariance matrix of the random effects.
fit	mimics fit object of lme().
info	information about the inputs.
aux	auxiliary information such as variability estimates.

References

Krivobokova, T., Crainiceanu, C.M. and Kauermann, G. (2008)
Fast Adaptive Penalized Splines. Journal of Computational and Graphical Statistics. 17(1) 1-20.

Ruppert, D., Wand, M.P. and Carroll, R.J. (2003)
Semiparametric Regression Cambridge University Press.
https://web.stat.tamu.edu/~carroll/semiregbook/

Wiesenfarth, M., Krivobokova, T., Klasen, S., Sperlich, S. (2012).
Direct Simultaneous Inference in Additive Models and its Application to Model Undernutrition. Journal of the American Statistical Association, 107(500): 1286-1296.

Wiesenfarth, M. (2012). Estimation and Inference in Special Nonparametric Models. Doctoral dissertation, Goettingen, Georg-August Universitaet, Diss., 2012. http://d-nb.info/104297182X/34

See Also

gam (in package ‘mgcv’), lme (in package ‘nlme’)

Examples

############
  ## scatterplot smoothing
    x <- 1:1000/1000
    mu <- exp(-400*(x-0.6)^2)+
    			5*exp(-500*(x-0.75)^2)/3+2*exp(-500*(x-0.9)^2)
    y <- mu+0.5*rnorm(1000)

  #fit with default knots
    y.fit <- asp2(y~f(x,adap=TRUE))
    plot(y.fit,residuals=TRUE,lwd=2,scb.lwd=2,scb.lty="dashed")
    # with shaded confidence region. 
    # Use scb.lty="blank" to plot the shades only.
      plot(y.fit,residuals=TRUE,shade=TRUE,scb.lty="blank")

 ## Not run: 
  ## Model with heteroscedastic errors
    attach(mcycle)

    y=accel
    kn1 <- default.knots(times,20)
    # fit model with constant residual variance
      fit= asp2(accel~f(times,basis="os",degree=3,knots=kn1,adap=FALSE),
  								niter = 20, niter.var = 200)

    # fit model with varying residual variance
      fith=aspHetero(fit,times,tol=1e-8)
    op <- par(mfrow = c(1,3))
    plot(fit);plot(fith)
    #sigma() returns the fitted varying residual variance
    plot(sort(times),sigma(fith)[order(times)],type="l")
    par(op)

 ## additive models
    x1 <- 1:300/300
    x2 <- runif(300)
    mu1 <- exp(-400*(x1-0.6)^2)+
    			 5*exp(-500*(x1-0.75)^2)/3+2*exp(-500*(x1-0.9)^2)
    mu2 <- sin(2*pi*x2)
    y2 <- mu1+mu2+0.3*rnorm(300)

    y2.fit <- asp2(y2~f(x1,adap=TRUE)+f(x2,adap=TRUE))
    # switch off adaptive fitting for the first function
      y21.fit <- asp2(y2~f(x1,adap=FALSE)+f(x2,adap=TRUE))
    op <- par(mfrow = c(2, 2))
    plot(y2.fit)
    plot(y21.fit)
    par(op)


  ## scatterplot smoothing with specified knots and subknots
    x <- 1:400/400
    mu <- sqrt(x*(1-x))*sin((2*pi*(1+2^((9-4*6)/5)))/(x+2^((9-4*6)/5)))
    y <- mu+0.2*rnorm(400)

    kn <- default.knots(x,80)
    kn.var <- default.knots(kn,20)

    y.fit <- asp2(y~f(x,knots=kn))
    y.fit2 <- asp2(y~f(x,knots=kn,var.knots=kn.var,adap=TRUE))
    op <- par(mfrow = c(1, 2))
    plot(y.fit)
    plot(y.fit2)
    par(op)

##################
#more examples
  beta=function(l,m,x) 
  		return(gamma(l+m)*(gamma(l)*gamma(m))^(-1)*x^(l-1)*(1-x)^(m-1))
  f1 = function(x) return((0.6*beta(30,17,x)+0.4*beta(3,11,x))*1/0.958)
  f2 = function(z) return((sin(2*pi*(z-0.5))^2)*1/.3535)
  f3 = function(z) 
  		return((exp(-400*(z-0.6)^2)+
  				5/3*exp(-500*(z-0.75)^2)+2*exp(-500*(z-0.9)^2))*1/0.549)

  set.seed(1)
  N <- 500
  x1 = runif(N,0,1)
  x2 = runif(N,0,1)
  x3 = runif(N,0,1)


  kn1 <- default.knots(x1,40)
  kn2 <- default.knots(x2,40)
  kn3 <- default.knots(x3,40)
  kn.var3 <- default.knots(kn3,5)

  y <- f1(x1)+f2(x2)+f3(x3)+0.3*rnorm(N)

  # semiparametric model
    fit1=  asp2(y~x1+f(x2,basis="os",degree=3,knots=kn2,adap=FALSE)
                    +f(x3,basis="os",degree=3,
                    	   knots=kn3,var.knots=kn.var3,adap=FALSE),
                    niter = 20, niter.var = 200)
    summary(fit1)
    plot(fit1,pages=1)


  # all effects flexible
  # fit model with all smoothing parameters constant
    fit2a= asp2(y~f(x1,basis="os",degree=3,knots=kn1,adap=FALSE)
                  +f(x2,basis="os",degree=3,knots=kn2,adap=FALSE)
                  +f(x3,basis="os",degree=3,knots=kn3,adap=FALSE),
                  niter = 20, niter.var = 200)
    plot(fit2a,pages=1)

  # fit model with last smoothing parameter adaptive
    fit2b= asp2(y~f(x1,basis="os",degree=3,knots=kn1,adap=FALSE)
                  +f(x2,basis="os",degree=3,knots=kn2,adap=FALSE)
                  +f(x3,basis="os",degree=3,knots=kn3,adap=TRUE,
                     var.knots=kn.var3,var.basis="os",var.degree=3),
                 niter = 20, niter.var = 200)

  # plot smoothing parameter function for covariate x3.
  # Note that in the case of B-splines additional knots are added, 
  # see references.
    plot(seq(0,1,length.out=42), fit2b$y.cov/fit2b$random.var[85:126],
                ylab=expression(lambda(x3)),xlab="x3",type="l",lwd=3)

  # compute 95% simultaneous confidence bands.
  # You could skip this and use "fit2b" indstead of "scb2b" later on, however,
  # if N is large, computing the SCBs various times can take some time
  # if you don't need fitted values and bounds for all covariate points
  # (can be computationally intensive due to large matrix dimensions),
  # set calc.stdev=F such that these are not computed.
    scb2b<- scbM(fit2b,calc.stdev=FALSE)
    plot(scb2b,pages=1)

  # plot only f(x2).
    plot(scb2b,select=2,mfrow=c(1,1),lwd=3,ylab="f(x2)",xlab="x2")
  # plot.scbm (and plot.asp) returns fitted values and confidence limits,
  # if you only need the returned object set plot=FALSE
    pscb=plot(scb2b,plot=FALSE)
  # add pointwise confidence intervals to the plot
    polygon(c(pscb$grid.x[[2]], rev(pscb$grid.x[[2]])),
            c(pscb$fitted[[2]]+1.96*pscb$Stdev[[2]],
              rev(pscb$fitted[[2]]-1.96*pscb$Stdev[[2]])),
            col = grey(0.85), border = NA)
    lines(pscb$grid.x[[2]],pscb$lcb[[2]],lty="dotted",lwd=3)
    lines(pscb$grid.x[[2]],pscb$fitted[[2]],lwd=3)
    lines(pscb$grid.x[[2]],pscb$ucb[[2]],lty="dotted",lwd=3)

  # plot first derivative of f(x1). 
  # Useful to check statistical significance of certain features (such 
  # as bumps) in a curve.
    scb2bdrv<- scbM(fit2b,drv=1,calc.stdev=FALSE)
    plot(scb2bdrv,select=1)
    #the following would give the same result
    #x11();plot(fit2b,select=1,drv=1)
  # different style
    plot(scb2bdrv,select=1,scb.lty="blank", 
    				shade=TRUE,shade.col="steelblue")

## End(Not run)

AdaptFitOS documentation built on July 21, 2022, 5:10 p.m.