# Monotone Smoothing of Data 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. ```r smooth.monotone(argvals, y, WfdParobj, wtvec=rep(1,n), zmat=NULL, conv=.0001, iterlim=50, active=rep(TRUE,nbasis), dbglev=1) ``` ## Arguments - `argvals`: Argument value array of length N, where N is the number of observed curve values for each curve. It is assumed that that these argument values are common to all observed curves. If this is not the case, you will need to run this function inside one or more loops, smoothing each curve separately. - `y`: a vector of data values. This function can only smooth one set of data at a time. Function value array (the values to be fit). If the functional data are univariate, this array will be an N by NCURVE matrix, where N is the number of observed curve values for each curve and NCURVE is the number of curves observed. If the functional data are multivariate, this array will be an N by NCURVE by NVAR matrix, where NVAR the number of functions observed per case. For example, for the gait data, NVAR = 2, since we observe knee and hip angles. - `WfdParobj`: A functional parameter or fdPar object. This object contains the specifications for the functional data object to be estimated by smoothing the data. See comment lines in function fdPar for details. The functional data object WFD in WFDPAROBJ is used to initialize the optimization process. Its coefficient array contains the starting values for the iterative minimization of mean squared error. - `wtvec`: 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 feature of S-PLUS. ## Details The smoothing function f(argvals) is determined by three objects that need to be estimated from the data: * W(argvals), a functional data object that is first exponentiated and then the result integrated. This is the heart of the monotone smooth. The closer W(argvals) is to zero, the closer the monotone smooth becomes a straight line. The closer W(argvals) 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 argvals = 0. * b1, a regression coefficient that determines the slope of the smoothing function at argvals = 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 multiplying the design matrix. ## Returns an object of class `monfd`, which is a list with the following 5 components: - **Wfdobj**: a functional data object defining function W(argvals) that optimizes the fit to the data of the monotone function that it defines. - **beta**: The regression coefficients b_0 and b_1 for each smoothed curve. If the curves are univariate and ... ZMAT is NULL, BETA is 2 by NCURVE. ... ZMAT has P columns, BETA is P+1 by NCURVE. If the curves are multivariate and ... ZMAT is NULL, BETA is 2 by NCURVE by NVAR. ... ZMAT has P columns, BETA is P+1 by NCURVE by NVAR. - **yhatfd**: A functional data object for the monotone curves that smooth the data. This object is constructed using the basis for WFDOBJ, and this basis may well be too simple to accommodate the curvature in the monotone function that Wfdobjnes. It may be necessary to discard this object and use a richer basis externally to smooth the values defined by beta[1] + beta[2]*eval.monfd(evalarg, Wfdobj). - **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. - **y2cMap**: For each estimated curve (and variable if functions are multivariate, this is an N by NBASIS matrix containing a linear mapping from data to coefficients that can be used for computing point-wise confidence intervals. If NCURVE = NVAR = 1, a matrix is returned. Otherwise an NCURVE by NVAR list is returned, with each slot containing this mapping. - **argvals**: input `argvals`, possibly modified / clarified by `argcheck`. - **y**: input argument `y`, possibly modified / clarified by `ycheck`. ## References Ramsay, James O., Hooker, Giles, and Graves, Spencer (2009), **Functional data analysis with R and Matlab**, Springer, New York. Ramsay, James O., and Silverman, Bernard W. (2005), **Functional Data Analysis, 2nd ed.**, Springer, New York. Ramsay, James O., and Silverman, Bernard W. (2002), **Applied Functional Data Analysis**, Springer, New York. ## See Also `smooth.basis`, `smooth.pos`, `smooth.morph` ## Examples ```r oldpar <- par(no.readonly=TRUE) # 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 <- length(age) 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$hgtf[,1] # conv=0.1 to reduce the compute time, # required to reduce the test time on CRAN # delete the test on CRAN because it takes too long if (!CRAN()) { result <- smooth.monotone(age, hgt1, growfdPar, wgt, conv=0.1) # Extract the functional data object and regression # coefficients Wfd <- result$Wfdobj beta <- result$beta # Evaluate the fitted height curve over a fine mesh agefine <- seq(1,18,len=73) 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) } par(oldpar) ```

smooth.monotone function