Mean and Covariance Estimation for Functional Snippets

We consider estimation of mean and covariance functions of functional snippets, which are short segments of functions possibly observed irregularly on an individual specific subinterval that is much shorter than the entire study interval. Estimation of the covariance function for functional snippets is challenging since information for the far off-diagonal regions of the covariance structure is completely missing. We address this difficulty by decomposing the covariance function into a variance function component and a correlation function component. The variance function can be effectively estimated nonparametrically, while the correlation part is modeled parametrically, possibly with an increasing number of parameters, to handle the missing information in the far off-diagonal regions. Both theoretical analysis and numerical simulations suggest that this hybrid strategy is effective. In addition, we propose a new estimator for the variance of measurement errors and analyze its asymptotic properties. This estimator is required for the estimation of the variance function from noisy measurements.

Sparse functional linear discriminant analysis

Functional linear discriminant analysis provides a simple yet efficient method for classification, with the possibility of achieving perfect classification. Several methods have been proposed in the literature that mostly address the dimensionality of the problem. On the other hand, there is growing interest in interpretability of the analysis, which favours a simple and sparse solution. In this paper we propose a new approach that incorporates a type of sparsity that identifies nonzero subdomains in the functional setting, yielding a solution that is easier to interpret without compromising performance. Given the need to embed additional constraints in the solution, we reformulate functional linear discriminant analysis as a regularization problem with an appropriate penalty. Inspired by the success of $\ell_1$-type regularization at inducing zero coefficients for scalar variables, we develop a new regularization method for functional linear discriminant analysis that incorporates an $L^1$-type penalty, $\int |f|$, to induce zero regions. We demonstrate that our formulation has a well-defined solution that contains zero regions, achieving functional sparsity in the sense of domain selection. In addition, the misclassification probability of the regularized solution is shown to converge to the Bayes error if the data are Gaussian. Our method does not assume that the underlying function has zero regions in the domain, but it produces a sparse estimator that consistently estimates the true function whether or not the latter is sparse. Using both simulated and real data examples, we demonstrate this property of our method in finite samples through comparisons with existing methods.

Basis expansions for functional snippets

Estimation of mean and covariance functions is fundamental for functional data analysis. While this topic has been studied extensively in the literature, a key assumption is that there are enough data in the domain of interest to estimate both the mean and covariance functions. We investigate mean and covariance estimation for functional snippets in which observations from a subject are available only in an interval of length strictly, and often much, shorter than the length of the whole interval of interest. For such a sampling plan, no data is available for direct estimation of the off-diagonal region of the covariance function. We tackle this challenge via a basis representation of the covariance function. The proposed estimator enjoys a convergence rate that is adaptive to the smoothness of the underlying covariance function, and has superior finite-sample performance in simulation studies.