Recovery of gradients from sparsely observed functional data

Description

A nonparametric empirical Bayes method for recovering gradients (or growth velocities) from observations of smooth functions (e.g., growth curves) at isolated time points.

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

posterior covariance kernel

Description

Internal function. Finds the posterior covariance kernel on the fine grid tgrid (same for every subject); see Section 2 of the referenced article for further details.

Usage

	Khatf(tgrid, tobs, sigma, Sigmahat)

Arguments

cross validation error

Description

Computes the cross validation error resulting from the removal of the data at a given interior observation time as a function of the infinitessimal standard deviation sigma on a grid of k equispaced values in the interval [a, b].

Usage

	cv.growth(data, tobs, d, K, a, b, r) 

Arguments

Details

The data for the rth observation time (for a given 1 < r < n) are removed and the mean squared error of the reconstructed data at that time point computed over the grid for sigma.

Value

Author(s)

Sara Lopez-Pintado and Ian W. McKeague

Maintainer: Ian W. McKeague <im2131@columbia.edu>

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):
    ## WARNING: cv.growth is time-consuming. This example uses only part of the data.
    data(height_data);
    ht = height_data[1:100,];
    tobs=c(0,1/3,2/3,1,3,4,7);
    cvg=cv.growth(ht, tobs, 100, 21, 1, 5, 2);

    ## Plot of the cross validation error as a function of sigma:
    plot(cvg$sigmavec, cvg$CVer, xlab="Sigma", ylab="Cross validation error");

    ## Value of sigma that minimizes the cross validation error:
    sigmaopt=cvg$sigmavec[which(cvg$CVer==min(cvg$CVer))]; 	

## End(Not run)

converts height data into difference quotients

Description

Internal function. Calculates the crude one-sided difference quotients and the second-order difference quotients based on the height data and the observed time points.

Usage

	diffquo(data, tobs) 

Arguments

Value

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

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

Author(s)

Sara Lopez-Pintado and Ian W. McKeague

Maintainer: Ian W. McKeague <im2131@columbia.edu>

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)

height data

Description

The heights of 532 girls at birth, four, eight, and twelve months, and three, four, and seven years of age.

Usage

	data(height_data)

Format

A matrix with 532 observations on 7 variables (height at each age).

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

	data(height_data)

recovery of growth velocity for a new subject

Description

Computes the posterior mean and covariance kernel for a new subject having data at observation times newtobs different from tobs (apart from the first and the last). growth needs to be run first.

Usage

	new.growth(newdata, newtobs, sigma, d, muhatcurve, Khat, tgrid) 

Arguments

Value

Author(s)

Sara Lopez-Pintado and Ian W. McKeague

Maintainer: Ian W. McKeague <im2131@columbia.edu>

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 
##    (after first running growth to obtain the output g):
## suppose a new subject has 5 observation times (including 0 and 7)
	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);

	newtobs=c(0, 2, 3, 5, 7);
	newdata=t(as.vector(c(50,70,87,100,115)));
	ng=new.growth(newdata,newtobs,sigma,d,g$muhatcurve,g$Khat,g$tgrid);

## plot of the posterior mean growth velocity for the new subject:
	plot(g$tgrid,ng$muhatcurvenew,xlab="Age (years)",ylab="Growth
	velocity (cms/year)");

## End(Not run)

posterior mean

Description

Internal function. Finds the posterior mean on the fine grid tgrid for every subject.

Usage

	posteriordistribcurve(muhatmatrix, Sigmahat, sigma, tobs, d, YI)

Arguments

posterior mean and covariance at observation times

Description

Internal function. Finds the posterior mean and covariance at the observation times tobs, for every subject; see Section 2 of the referenced article for further details.

Usage

	posteriorobs(Sigma0inv, sigma, muprior, Xtilda, tobs, YI)

Arguments

prior mean

Description

Internal function. Finds the sample mean of the n-vector y for use in specifying the prior mean {\boldsymbol\mu}_0; see Section 2 of the referenced article for further details.

Usage

	priormeanobs(YI, tobs)

Arguments