Recovering gradients from sparsely observed functional data.

Area of Science:

• Statistics
• Functional Data Analysis
• Inverse Problems

Background:

• Estimating gradients of functional data from sparse observations is a complex ill-posed inverse problem.
• Existing methods may struggle with data sparsity and accuracy in estimating underlying growth velocities.

Purpose of the Study:

• To develop a robust Bayesian inversion approach for recovering gradients of sparsely observed functional data.
• To provide computationally tractable estimates of growth velocities even with limited data points.

Main Methods:

• A Bayesian inversion framework modeling the gradient via tied-down Brownian motion between observation times.
• Utilizing quadratic splines for explicit posterior mean and covariance kernel representations.
• Employing nonparametric empirical Bayes for hyperparameter specification and constrained ℓ₁ minimization for prior precision matrix estimation.
• Cross-validation for selecting the Brownian motion prior's infinitesimal variance.

Main Results:

• The proposed method yields explicit and computationally tractable representations of posterior means and covariance kernels for growth velocities.
• Demonstrated effectiveness on both simulated and real-world datasets, showcasing accurate gradient recovery.
• Successful estimation of prior precision matrix via constrained ℓ₁ minimization.

Conclusions:

• The developed Bayesian approach effectively addresses the ill-posed inverse problem of gradient recovery in sparse functional data.
• The method offers a computationally efficient and accurate solution for estimating growth velocities.
• Empirical Bayes and spline-based representations enhance the practical applicability of the model.