# Different number of time-points in functional data analysis

I have a data set that I wish to analyze using functional data analysis methods. Data consists of repeated measures of some characteristic on a number of inviduals. I have the time of the measurements and these differ between individuals. So far, I see no problem in analyzing this, e.g. by using ready-to-use ${\tt R}$ packages. However, the number of measurements per individual differs as well for my data. How can I handle this? Can anyone point me in the direction of literature on this set-up or any packages that can handle it?

What I want to do with data to start with is basically something like functional linear model (functional response and scalar predictors).

I'm aware that I could smooth the data using a very light smoothing procedure (using many basis functions and no or a very small penalty parameter) and use these smoothed curves for my analysis. But is this the best way to approach the problem? It doesn't take into account that some curves are determined with more precision than are others. An additional problem to this approach is that some of the individuals only have measurements in a subinterval (say, [200, 1000]) of the full interval of the majority of the curves (say, [0, 1000]). Thus, the smooth would be very poorly estimated in those regions. Naturally, I could restrict attention to the interval [200, 1000] but this seems like a waste of data.

To sum up my question, what would you do in this case? Any thoughts and hints will be much appreciated.

The Matlab package PACE can deal with data with different observed time points (irregular data), which is exactly suitable for your analysis. In your example, as long as all the data pooled together are dense in [0, 1000], the FPCA function in the package can estimate an individual curve even if its measurements are on [200, 1000].

The method used by FPCA to deal with irregular data is Principal components analysis by Conditional Expectation (PACE, see Yao, Mueller, and Wang 2005). An individual curve can be represented by its functional principal components (FPC, Wikipedia has an entry) in the eigenbasis. The method aims to estimate the FPCs even if individual data are observed sparsely, i.e. with only a few observations. The estimated individual curves can then be recovered from the FPCs.

One choice for fitting a functional linear model with functional response and scalar predictors is to obtain the FPCs of the response first, and then reduce the problem to a linear regression model with multivariate response. You can consider using the FPCreg function in PACE, although it require a functional predictor (you can make a scalar predictor a constant function).

If you prefer manipulating the data in R while using Matlab for FPCA you can try the R package R.matlab, which let you pass around some R and Matlab objects.

Though this is now an old questions, to update the answer given by CrossD, the same analysis can be performed in R using the fdapace package. This is basically the R implementation of PACE and contains functions FPCA and FPCreg just as in the Matlab package.

An alternative approach is described by Ivanescu, Staicu, Scheipl, & Greven (2015) and implemented in the pffr function of the refund R package. Their approach uses spline basis expansions (as opposed to the FPCA basis used in the PACE approach) of the functional coefficients and covariates. The model parameters are then estimated in a mixed effects, penalized regression framework where the penalization helps to prevent overfitting the data. A sparsely- and irregularly-observed functional response is allowed in their framework as discussed in section 3.2 of their paper. The authors argue that this spline-based approach is preferable to the PCA-based approaches largely because the pffr approach is less dependent on the dimension of the spline basis and parameter estimation (thanks to the penalization).

Note that while the paper and accompanying pffr function are named after function-on-function regression, the function-on-scalar model described in the question fits into this framework as well.