#### Funded by the US National Science Foundation, grant DMS 0604396, June 2006 – May 2010.

### Project Summary

The analysis of samples of curves (a.k.a. Functional Data Analysis) is a field of growing importance in Statistics. Samples of curves arise, for instance, in longitudinal studies where a random process is observed on groups of individuals. In many cases, some observed trajectories are atypical compared with the rest of the sample, which may lead to invalid statistical inference. The main goal of this project is to develop outlying-resistant methods that can provide valid statistical inference even in presence of a significant proportion of outlying curves. In particular, outlier-resistant estimators for the mean and the variance components will be developed.

### Publications

- Gervini, D. (2012). Functional robust regression for longitudinal data.
*ArXiv*1211.7332. - Gervini, D. (2012). Outlier detection and trimmed estimation for general functional data.
*Statistica Sinica***22**1639-1660.- Available supplements: Technical Report, EEM movie, and LogEEM movie.

- Gervini, D. (2010). The functional singular value decomposition for bivariate stochastic processes.
*ArXiv*1211.7336.- Technical Report available.
- A note note on the publication of this paper.

- Gervini, D. (2009). Detecting and handling outlying trajectories in irregularly sampled functional datasets.
*The Annals of Applied Statistics***3**1758-1775.- Technical Report available.

- Gervini, D. (2008). Robust functional estimation using the median and spherical principal components.
*Biometrika***95**587-600- Technical Supplement available.

### Computer programs

#### GMt robust functional regression estimators

The Matlab functions GMt and boot_GMt compute the functional regression estimators of Gervini (2012, *ArXiv*) and their bootstrap versions for inference. They must be used in conjunction with the reduced-rank t-model programs given below.

#### Trimmed functional estimators

The Matlab functions fte and fte2 compute the trimmed mean and covariance estimators introduced in Gervini (2012, *Statistica Sinica*). The function fte.m handles univariate vector-valued functional data (that is, one-to-p functions, including of course the most common case p = 1). The function fte2.m is for bivariate real-valued data (that is, 2-to-1 surfaces, like EEM matrices).

#### Reduced-rank t models for sparse data

The Matlab functions in the .zip file FtMod_Matlab.zip compute reduced-rank t and Normal model estimators for the mean and principal components of curves observed on sparse, irregular grids (of course, they also work for regular grids). For details, see Gervini (2009).

An example of data analysis using these functions is given here.

#### Spatial median and spherical PCs

The following Matlab functions compute the spatial median and the spherical principal components (see Gervini 2008).

Spatial median: SpMed.m, Spherical PCs: SpPC.m, external function used: gsj.m

For quick comparison, RawPC.m computes the raw principal components.

An example of data analysis using these functions is given here.

Last updated: 10 Jan 2018, 14:00 hs