growth {growthrate}

recovery of growth velocities

Description

Computes the mean function and covariance kernel (over a fine grid of equispaced time points) of the posterior growth velocity for each subject, based on growth data (e.g., heights) at fixed observation times.

Usage

	growth(data, tobs, sigma, d)

Arguments

data

Input matrix of size N (subjects) times n (observation times). Each column contains the heights (of all subjects) at a given observation time, each row contains the heights (at the observation times) for a given subject.

tobs

Row vector of n observation times (in increasing order, same for each subject).

sigma

A positive scalar representing the infinitessimal standard deviation of the tied-down Brownian motion in the prior. Can be selected by cross-validation.

d

Number of time points on the fine grid.

Details

The Bayesian reconstruction implemented here uses a prior growth velocity model that is specified by a general multivariate normal distribution at the n fixed observation times, and a tied-down Brownian motion (having infinitessimal standard deviation specified by sigma) between the observation times.

The prior mean and prior precision matrix at the observation times are estimated using the data on N subjects. Clime (constrained L1 minimization) provides the estimate of the prior precision matrix, with the clime constraint parameter lambda selected by 5-fold cross validation using the likelihood loss function.

Value

muhatcurve

Posterior means of the growth velocities (for each subject) on the fine grid tgrid. An N (subjects) times d matrix.

Khat

Posterior covariance kernel of the growth velocities on the fine grid tgrid. A d times d matrix (the same for every subject).

tgrid

The fine grid of d equispaced-time points (between the first and last observation times in tobs) at which the posterior means and covariances are computed.

Author(s)

Sara Lopez-Pintado and Ian W. McKeague

Maintainer: Ian W. McKeague <[e-mail]>

References

Lopez-Pintado, S. and McKeague, I. W. (2013). Recovering gradients from sparsely observed functional data. Biometrics 69, 396-404 (2013). http://www.columbia.edu/~im2131/ps/growthrate-package-reference.pdf

Examples

## Not run: 
## example using the height data provided in the package
## there are 7 observation times (age in years):
	data(height_data);
	tobs=c(0,1/3,2/3,1,3,4,7);
	d=200;
	sigma=1;
	g=growth(height_data,tobs,sigma,d);

## Plot of the posterior mean and credible interval for a specific individual
	indiv=1;
## posterior standard deviation (same for all subjects):
	postsd=sqrt(diag(g$Khat));
	plot(g$tgrid,g$muhatcurve[indiv,],type='l',
			xlab="Age (years)",ylab="Growth velocity (cms/year)");
	lines(g$tgrid,g$muhatcurve[indiv,]);
	lines(g$tgrid,g$muhatcurve[indiv,]+2*postsd,lty=2);
	lines(g$tgrid,g$muhatcurve[indiv,]-2*postsd,lty=2);

## Plot of a draw from the posterior growth velocity for a specific individual:
	draw=rmvnorm(n=1, mean=g$muhatcurve[indiv,], sigma=g$Khat, method="chol");
	plot(g$tgrid,draw,type='l',xlab="Age (years)",ylab="Growth
	velocity (cms/year)");

## End(Not run)