Optimal method estimates noisy functional data for linear regression
Optimal estimation for Functional Linear Regression with Noisy Discretized Data
Machine Learning
Summary
Many times, we want to predict a single number from data that changes over time, but this data is only measured with noise and on fixed time points. The authors study how to best reconstruct these noisy curves and then estimate how they influence the prediction, using a two-step process involving Fourier techniques and model selection. They prove that their method achieves the best possible accuracy under certain mathematical conditions if enough data points are available. They also demonstrate their approach on simulated examples and weather data.
What this means in practice
- •For meteorological analysts: Improve predictive accuracy of weather models when sensor data is noisy and discretely sampled.
- •For biomedical device developers: Enhance estimation of health signals recorded at discrete noisy points for better diagnostic models.
Authors
Sixtine Sphabmixay
Abstract
In this paper, we consider the scalar-on-function linear regression model under a realistic sampling scheme in which the functional covariates are observed on a regular grid and contaminated by additive noise. We propose a two-step estimation procedure: first, the underlying curves are reconstructed from the discrete noisy observations using a Fourier-based projection method; second, the slope function is estimated by a penalized least-squares criterion over finite-dimensional trigonometric spaces, with data-driven selection of the model dimension. We establish oracle-type inequalities for the prediction error, both with respect to the reconstructed curves and to the true latent curves. Under regularity assumptions on the slope function and polynomial decay of the eigenvalues of the covariate, we derive convergence rates for the prediction error and show that our estimator attains the minimax rate when the number of grid points is sufficiently large. Finally, the proposed method is illustrated on simulated data and on a real meteorological dataset.