Generalized projection-based shape outlier detection in functional data

Music City Center 

Shape outliers, or abnormally shaped functional data, are difficult to detect when masked by surrounding functions. Detection attempts range from visualization tricks to quantification techniques. They typically summarize high-dimensional shape information into a finite set of indices using tools like statistical depths or functional principal component analysis (FPCA). However, existing approaches overlook the varying importance of the derived indices. To address this, we propose the Generalized Trimmed Functional Score (GTFS), an outlyingness index that automatically reweighs the extracted indices. It is computed as the weighted sum of eigenscores, the projection of the curves onto FPCA eigenfunctions. The weighing plan we designed leverages the extreme value distribution of the squared eigenscore maxima to adaptively select only the eigenfunctions helpful for detection. We also introduce the specialized centering scheme that makes the index magnitude-invariant by un-masking the shape outliers. The thresholding rule based on the asymptotic distribution of GTFS, with which we control the false-positive rate is also provided. Theoretical studies explore the statistical power and some asymptotic properties. Finally, we validate the practicality via extensive simulations and a real-world application using the smartphone human activity signal data.

Functional data analysis

Shape outlier

Outlier detection

Generalized extreme value distribution

Functional principal component analysis

Reweighting 

Section on Statistical Learning and Data Science