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Confidence Tubes for Curves on SO(3) and Identification of Subject-Specific Gait Change after Kneeling

Fabian J. E. Telschow, Michael R. Pierrynowski, Stephan F. Huckemann

Abstract

In order to identify changes of gait patterns, e.g. due to prolonged occupational kneeling, which is believed to be major risk factor, among others, for the development of knee osteoarthritis, we develop confidence tubes for curves following a Gaussian perturbation model on SO(3). These are based on an application of the Gaussian kinematic formula to a process of Hotelling statistics and we approximate them by a computible version, for which we show convergence. Simulations endorse our method, which in application to gait curves from eight volunteers undergoing kneeling tasks, identifies phases of the gait cycle that have changed due to kneeling tasks. We find that after kneeling, deviation from normal gait is stronger, in particular for older aged male volunteers. Notably our method adjusts for different walking speeds and marker replacement at different visits.

Confidence Tubes for Curves on SO(3) and Identification of Subject-Specific Gait Change after Kneeling

Abstract

In order to identify changes of gait patterns, e.g. due to prolonged occupational kneeling, which is believed to be major risk factor, among others, for the development of knee osteoarthritis, we develop confidence tubes for curves following a Gaussian perturbation model on SO(3). These are based on an application of the Gaussian kinematic formula to a process of Hotelling statistics and we approximate them by a computible version, for which we show convergence. Simulations endorse our method, which in application to gait curves from eight volunteers undergoing kneeling tasks, identifies phases of the gait cycle that have changed due to kneeling tasks. We find that after kneeling, deviation from normal gait is stronger, in particular for older aged male volunteers. Notably our method adjusts for different walking speeds and marker replacement at different visits.

Paper Structure

This paper contains 23 sections, 6 theorems, 52 equations, 4 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Testing Gaussian Perturbation Models on Lie Groups Modulo Sample-Specific Spatio-Temporal Action
  3. Confidence Tubes on $G$
  4. GP Models and Approximating Confidence Tubes on $SO(3)$
  5. GP Models on $SO(3)$
  6. Approximating Confidence Tubes on $SO(3)$
  7. The Gaussian kinematic formula (GKF).
  8. Estimation of the quantile $\hat{h}_{\gamma,N,\alpha}$.
  9. Simulations of Covering Rates
  10. GP models used for simulation.
  11. Design of simulation of simultaneous confidence tubes (SCTs) for center curves of GP models.
  12. Results of simulation of SCT for center curves of GP models.
  13. Application: Assessing Kneeling Effects on Gait
  14. Study design.
  15. Dealing with the marker replacement effect.
  16. ...and 8 more sections

Key Result

Theorem 3.2

Let $\gamma_1,\ldots,\gamma_N$ be a sample of a random curve $\gamma \in \Gamma$ following a GP model around a center curve $\gamma_0$. Let $\hat{\gamma}_{N}$ be an estimator for $\gamma_0$ and assume $\hat{S}^{x,N}_t$ is non-singular for all $t\in[0,1]$. Then and hence this set forms a simultaneous $(1-\alpha)$-confidence tube for $\gamma_0$.

Figures (4)

  • Figure 1: Depicting standard naming convention for gait events with respect to the flexion-extension angle.
  • Figure 2: Depicting for Volunteer 2 all three Euler angles of sampled gait curves for each of two different sessions. PEM curves are fat and vertical lines indicate loci of non-overlapping simultaneous $0.05$-confidence tubes in $SO(3)$. The largely varying curves are flexion-extension angles, cf. Figure \ref{['gait-event:fig']}, the middle curves with least variation are abduction-adduction and the bottom ones are internal-external angles.
  • Figure 3: Depicting with notation from Figure \ref{['conf-tubes-2:fig']} for Volunteer 6 all three Euler angles of sampled gait curves for each of two different sessions with PEM curves and loci of non-overlapping simultaneous $0.05$-confidence tubes in $SO(3)$.
  • Figure 4: Depicting with notation from Figure \ref{['conf-tubes-2:fig']} for Volunteers 1, 3, 4 and 7 all three Euler angles of sampled gait curves for each of two different sessions with PEM curves and loci of non-overlapping simultaneous $0.05$-confidence tubes in $SO(3)$.

Theorems & Definitions (8)

  • Definition 2.1
  • Definition 3.1
  • Theorem 3.2
  • Theorem 4.1
  • Corollary 4.2
  • Theorem 4.3: Approximations for Concentrated Errors
  • Corollary 4.4: Asymptotically genuine Hotelling process
  • Theorem 4.5