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Multi-class classification of biomechanical data: A functional LDA approach based on multi-class penalized functional PLS

M Carmen Aguilera-Morillo, Ana M Aguilera

TL;DR

A functional linear discriminant analysis approach to classify a set of kinematic data (human movement curves of individuals performing different physical activities) is performed.

Abstract

A functional linear discriminant analysis approach to classify a set of kinematic data (human movement curves of individuals performing different physical activities) is performed. Kinematic data, usually collected in linear acceleration or angular rotation format, can be identified with functions in a continuous domain (time, percentage of gait cycle, etc.). Since kinematic curves are measured in the same sample of individuals performing different activities, they are a clear example of functional data with repeated measures. On the other hand, the sample curves are observed with noise. Then, a roughness penalty might be necessary in order to provide a smooth estimation of the discriminant functions, which would make them more interpretable. Moreover, because of the infinite dimension of functional data, a reduction dimension technique should be considered. To solve these problems, we propose a multi-class approach for penalized functional partial least squares (FPLS) regression. Then linear discriminant analysis (LDA) will be performed on the estimated FPLS components. This methodology is motivated by two case studies. The first study considers the linear acceleration recorded every two seconds in 30 subjects, related to three different activities (walking, climbing stairs and down stairs). The second study works with the triaxial angular rotation, for each joint, in 51 children when they completed a cycle walking under three conditions (walking, carrying a backpack and pulling a trolley). A simulation study is also developed for comparing the performance of the proposed functional LDA with respect to the corresponding multivariate and non-penalized approaches.

Multi-class classification of biomechanical data: A functional LDA approach based on multi-class penalized functional PLS

TL;DR

A functional linear discriminant analysis approach to classify a set of kinematic data (human movement curves of individuals performing different physical activities) is performed.

Abstract

A functional linear discriminant analysis approach to classify a set of kinematic data (human movement curves of individuals performing different physical activities) is performed. Kinematic data, usually collected in linear acceleration or angular rotation format, can be identified with functions in a continuous domain (time, percentage of gait cycle, etc.). Since kinematic curves are measured in the same sample of individuals performing different activities, they are a clear example of functional data with repeated measures. On the other hand, the sample curves are observed with noise. Then, a roughness penalty might be necessary in order to provide a smooth estimation of the discriminant functions, which would make them more interpretable. Moreover, because of the infinite dimension of functional data, a reduction dimension technique should be considered. To solve these problems, we propose a multi-class approach for penalized functional partial least squares (FPLS) regression. Then linear discriminant analysis (LDA) will be performed on the estimated FPLS components. This methodology is motivated by two case studies. The first study considers the linear acceleration recorded every two seconds in 30 subjects, related to three different activities (walking, climbing stairs and down stairs). The second study works with the triaxial angular rotation, for each joint, in 51 children when they completed a cycle walking under three conditions (walking, carrying a backpack and pulling a trolley). A simulation study is also developed for comparing the performance of the proposed functional LDA with respect to the corresponding multivariate and non-penalized approaches.
Paper Structure (11 sections, 28 equations, 11 figures, 2 tables)

This paper contains 11 sections, 28 equations, 11 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Multi-class classification of functional data
  3. Multi-class functional PLS regression
  4. Penalized multi-class FPLS regression
  5. The problem of repeated measures
  6. Functional LDA on a reduced set of functional PLS components
  7. Simulation study
  8. Case studies
  9. Human activity data
  10. Gait data
  11. Conclusions

Figures (11)

  • Figure 1: Simulation study. Sample paths related to classes 1, 2 and 3 (solid, dashed and dotted line, respectively) for two subjects A and B, black and grey lines, respectively.
  • Figure 2: Simulation study. Smooth and noisy sample paths (left and right panel, respectively) related to classes 1, 2 and 3 (solid line, dashed line and dotted line, respectively) for 30 subjects.
  • Figure 3: Simulation study. Box plots showing to the number of PLS components used in the LDA (bottom panel), the correct classification rates from the cross-validation on the training sample and from the classification of the test sample (top-left and top-right panels, respectively). Experiments run 500 times.
  • Figure 4: Human activity data. Raw data. Sample paths displayed separately by stimulus: walking, walking upstairs and walking downstairs, from left to right, respectively.
  • Figure 5: Human activity data. Raw data. Sample paths related to walking (solid line), walking upstairs (dashed line) and walking downstairs (dotted line) for two subjects A and B, left and right panel, respectively.
  • ...and 6 more figures