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Similarity distance measure between two trajectories is an essential tool tounderstand patterns in motion, for example, in Human-Robot Interaction orImitation Learning. The problem has been faced in many fields, from SignalProcessing, Probabilistic Theory field, Topology field or Statisticsfield.Anyway, up to now, none of the trajectory similarity measurements metricsare invariant to all possible linear transformation of the trajectories(rotation, scaling, reflection, shear mapping or squeeze mapping). Also not allof them are robust in front of noisy signals or fast enough for real-timetrajectory classification. To overcome this limitation this paper proposes asimilarity distance metric that will remain invariant in front of any possiblelinear transformation.Based on Pearson Correlation Coefficient and theCoefficient of Determination, our similarity metric, the Generalized MultipleCorrelation Coefficient (GMCC) is presented like the natural extension of theMultiple Correlation Coefficient. The motivation of this paper is two fold.First, to introduce a new correlation metric that presents the best propertiesto compute similarities between trajectories invariant to lineartransformations and compare it with some state of the art similaritydistances.Second, to present a natural way of integrating the similarity metricin an Imitation Learning scenario for clustering robot trajectories.

Generalized Multiple Correlation Coefficient as a Similarity Measurements between Trajectories