  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.
