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Cosine Series Representation

Moo K. Chung

TL;DR

The paper develops a functional data analysis framework that represents biomedical signals as smooth functions in a Hilbert space using an explicit orthonormal cosine basis. It builds a unified pipeline—comprising least-squares projection in $L^2[0,1]$, coefficient-wise Wiener filtering, and Karhunen–Loève–based stochastic representations—enabling scalable smoothing, regularization, and hypothesis testing directly on basis coefficients. Key contributions include the explicit cosine-series formulation, efficient LSE implementations, and integration of probabilistic structure for denoising and inference, demonstrated in diffusion-tensor imaging tract modeling and EEG denoising. The work provides a transparent, interpretable, and computationally efficient FDA approach with practical MATLAB resources.

Abstract

We present a functional data analysis (FDA) framework based on explicit orthonormal basis expansion for modeling and denoising complex biomedical signals. Observed functional data are represented as smooth functions in a Hilbert space, and statistical inference is performed directly on their basis coefficients. This formulation provides a transparent and flexible approach to smoothing, regularization, and hypothesis testing. Applications to diffusion tensor imaging tract modeling and EEG denoising demonstrate the advantages of explicit basis representations for scalable and interpretable functional modeling.

Cosine Series Representation

TL;DR

The paper develops a functional data analysis framework that represents biomedical signals as smooth functions in a Hilbert space using an explicit orthonormal cosine basis. It builds a unified pipeline—comprising least-squares projection in , coefficient-wise Wiener filtering, and Karhunen–Loève–based stochastic representations—enabling scalable smoothing, regularization, and hypothesis testing directly on basis coefficients. Key contributions include the explicit cosine-series formulation, efficient LSE implementations, and integration of probabilistic structure for denoising and inference, demonstrated in diffusion-tensor imaging tract modeling and EEG denoising. The work provides a transparent, interpretable, and computationally efficient FDA approach with practical MATLAB resources.

Abstract

We present a functional data analysis (FDA) framework based on explicit orthonormal basis expansion for modeling and denoising complex biomedical signals. Observed functional data are represented as smooth functions in a Hilbert space, and statistical inference is performed directly on their basis coefficients. This formulation provides a transparent and flexible approach to smoothing, regularization, and hypothesis testing. Applications to diffusion tensor imaging tract modeling and EEG denoising demonstrate the advantages of explicit basis representations for scalable and interpretable functional modeling.

Paper Structure

This paper contains 7 sections, 1 theorem, 54 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Hilbert space theory in $[0,1]$
  3. Least squares estimation
  4. Wiener filter
  5. Karhunen--Loève Expansion
  6. Cosine basis
  7. Acknowledgment

Key Result

Theorem 1

The minimization of (eq:L2min) is given by where the $l$-th degree Fourier coefficient$\langle \zeta_i, \psi_l \rangle$ is given by the inner product

Figures (1)

  • Figure 1: A. fMRI time series at two different voxels at time point $t_1$. B. fMRI time series at voxel 1 (red) and 2 (black) shown for all 1200 time points. C. Normalized and scaled time series at voxel 1 and its cosine series representation with degree $k=59$. D. Normalized and scaled time series at voxel 2 and its cosine series representation with degree $k=59$. Such high-frequency denosising is often necessary for functional signals like fMRI and EEG.

Theorems & Definitions (1)

  • Theorem 1