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Regularized Nonlinear Regression with Dependent Errors and its Application to a Biomechanical Model

Hojun You, Kyubaek Yoon, Wei-Ying Wu, Jongeun Choi, Chae Young Lim

Abstract

A biomechanical model often requires parameter estimation and selection in a known but complicated nonlinear function. Motivated by observing that data from a head-neck position tracking system, one of biomechanical models, show multiplicative time dependent errors, we develop a modified penalized weighted least squares estimator. The proposed method can be also applied to a model with non-zero mean time dependent additive errors. Asymptotic properties of the proposed estimator are investigated under mild conditions on a weight matrix and the error process. A simulation study demonstrates that the proposed estimation works well in both parameter estimation and selection with time dependent error. The analysis and comparison with an existing method for head-neck position tracking data show better performance of the proposed method in terms of the variance accounted for (VAF).

Regularized Nonlinear Regression with Dependent Errors and its Application to a Biomechanical Model

Abstract

A biomechanical model often requires parameter estimation and selection in a known but complicated nonlinear function. Motivated by observing that data from a head-neck position tracking system, one of biomechanical models, show multiplicative time dependent errors, we develop a modified penalized weighted least squares estimator. The proposed method can be also applied to a model with non-zero mean time dependent additive errors. Asymptotic properties of the proposed estimator are investigated under mild conditions on a weight matrix and the error process. A simulation study demonstrates that the proposed estimation works well in both parameter estimation and selection with time dependent error. The analysis and comparison with an existing method for head-neck position tracking data show better performance of the proposed method in terms of the variance accounted for (VAF).
Paper Structure (8 sections, 2 theorems, 11 equations, 3 figures, 9 tables)

This paper contains 8 sections, 2 theorems, 11 equations, 3 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Methods
  3. Modified Weighted Least Squares
  4. Notations and Assumptions
  5. Theoretical results
  6. Simulation Study
  7. Application to a head-neck position tracking system
  8. Conclusion

Key Result

Theorem 1

For any $\varepsilon>0$ and $b_n = (\lambda_\epsilon/n)^{1/2}+c_n$, under Assumptions 1-(1), (2), (3), (5) and assumption:penalty-(1),(2), there exists a positive constant $C$ such that for large enough $n$. Therefore, with probability tending to 1, there exists a local minimizer of $\bm Q_n (\bm \theta)$, say $\hat{\bm\theta}$, in the ball centered at $\bm\theta_0$ with the radius $b_n \bm v$. B

Figures (3)

  • Figure 1: The black curve represents the measured responses (the observations) from the subject No. 8 in the head-neck position tracking experiment. The red dashed curve represents the estimated responses (the fitted values) from the nonlinear regression model with additive errors introduced in yoon2022regularized.
  • Figure 3: Estimation (left) and prediction (right) results for the subject No.2. Measured responses (black line) and fitted values from the additive approach in yoon2022regularized (blue dashed line, $--$), PWLS (yellow dot-dashed line, $\cdot -\cdot$), and PMWLS (red dotted line, $\cdots$) are exhibited.
  • Figure 4: Same configurations as in Figure \ref{['fig:real_results_2']}, but with the subject No. 8.

Theorems & Definitions (6)

  • Definition 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 1
  • Theorem 2: Oracle property