fda (version 1.2.3)

smooth.monotone: Monotone Smoothing of Data

Description

When the discrete data that are observed reflect a smooth strictly increasing or strictly decreasing function, it is often desirable to smooth the data with a strictly monotone function, even though the data themselves may not be monotone due to observational error. An example is when data are collected on the size of a growing organism over time. This function computes such a smoothing function, but, unlike other smoothing functions, for only for one curve at a time. The smoothing function minimizes a weighted error sum of squares criterion. This minimization requires iteration, and therefore is more computationally intensive than normal smoothing.

The monotone smooth is beta[1]+beta[2]*integral(exp(Wfdobj)), where Wfdobj is a functional data object. Since exp(Wfdobj)>0, its integral is monotonically increasing.

Usage

smooth.monotone(x, y, WfdParobj, wt=rep(1,nobs),
                zmat=matrix(1,nobs,1), conv=.0001, iterlim=20,
                active=c(FALSE,rep(TRUE,ncvec-1)), dbglev=1)

Arguments

x

a vector of argument values.

y

a vector of data values. This function can only smooth one set of data at a time.

WfdParobj

a functional parameter object that provides an initial value for the coefficients defining function $W(t)$, and a roughness penalty on this function.

wt

a vector of weights to be used in the smoothing.

zmat

a design matrix or a matrix of covariate values that also define the smooth of the data.

conv

a convergence criterion.

iterlim

the maximum number of iterations allowed in the minimization of error sum of squares.

active

a logical vector specifying which coefficients defining $W(t)$ are estimated. Normally, the first coefficient is fixed.

dbglev

either 0, 1, or 2. This controls the amount information printed out on each iteration, with 0 implying no output, 1 intermediate output level, and 2 full output. If either level 1 or 2 is specified, it can be helpful to turn off the output buffering fea

Value

• a named list of length 5 containing:
• Wfdobj
  a functional data object defining function $W(x)$ that optimizes the fit to the data of the monotone function that it defines.
• beta
  the optimal regression coefficient values.
• Flist
  a named list containing three results for the final converged solution: (1) f: the optimal function value being minimized, (2) grad: the gradient vector at the optimal solution, and (3) norm: the norm of the gradient vector at the optimal solution.
• iternum
  the number of iterations.
• iterhist
  iternum+1 by 5 matrix containing the iteration history.

Details

The smoothing function $f(x)$ is determined by three objects that need to be estimated from the data:

• $W(x)$, a functional data object that is first exponentiated and then the result integrated. This is the heart of the monotone smooth. The closer $W(x)$ is to zero, the closer the monotone smooth becomes a straight line. The closer $W(x)$ becomes a constant, the more the monotone smoother becomes an exponential function. It is assumed that $W(0) = 0.$
• $b0$, an intercept term that determines the value of the smoothing function at $x = 0$.
• $b1$, a regression coefficient that determines the slope of the smoothing function at $x = 0$.

In addition, it is possible to have the intercept $b0$ depend in turn on the values of one or more covariates through the design matrix Zmat as follows: $b0 = Z c$. In this case, the single intercept coefficient is replaced by the regression coefficients in vector $c$ multipying the design matrix.

See Also

smooth.basis, smooth.pos, smooth.morph

Examples

#  Estimate the acceleration functions for growth curves
#  See the analyses of the growth data.
#  Set up the ages of height measurements for Berkeley data
age <- c( seq(1, 2, 0.25), seq(3, 8, 1), seq(8.5, 18, 0.5))
#  Range of observations
rng <- c(1,18)
#  First set up a basis for monotone smooth
#  We use b-spline basis functions of order 6
#  Knots are positioned at the ages of observation.
norder <- 6
nage   <- 31
nbasis <- nage + norder - 2
wbasis <- create.bspline.basis(rng, nbasis, norder, age)
#  starting values for coefficient
cvec0 <- matrix(0,nbasis,1)
Wfd0  <- fd(cvec0, wbasis)
#  set up functional parameter object
Lfdobj    <- 3          #  penalize curvature of acceleration
lambda    <- 10^(-0.5)  #  smoothing parameter
growfdPar <- fdPar(Wfd0, Lfdobj, lambda)
#  Set up wgt vector
wgt   <- rep(1,nage)
#  Smooth the data for the first girl
hgt1 = growth$hgtf[,1]
result <- smooth.monotone(age, hgt1, growfdPar, wgt)
#  Extract the functional data object and regression
#  coefficients
Wfd  <- result$Wfdobj
beta <- result$beta
#  Evaluate the fitted height curve over a fine mesh
agefine <- seq(1,18,len=101)
hgtfine <- beta[1] + beta[2]*eval.monfd(agefine, Wfd)
#  Plot the data and the curve
plot(age, hgt1, type="p")
lines(agefine, hgtfine)
#  Evaluate the acceleration curve
accfine <- beta[2]*eval.monfd(agefine, Wfd, 2)
#  Plot the acceleration curve
plot(agefine, accfine, type="l")
lines(c(1,18),c(0,0),lty=4)