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Linear convergence of the subspace constrained mean shift algorithm: from Euclidean to directional data.

ABSTRACT: This paper studies the linear convergence of the subspace constrained mean shift (SCMS) algorithm, a well-known algorithm for identifying a density ridge defined by a kernel density estimator. By arguing that the SCMS algorithm is a special variant of a subspace constrained gradient ascent (SCGA) algorithm with an adaptive step size, we derive the linear convergence of such SCGA algorithm. While the existing research focuses mainly on density ridges in the Euclidean space, we generalize density ridges and the SCMS algorithm to directional data. In particular, we establish the stability theorem of density ridges with directional data and prove the linear convergence of our proposed directional SCMS algorithm.

SUBMITTER: Zhang Y

PROVIDER: S-EPMC9893762 | biostudies-literature | 2023 Mar

REPOSITORIES: biostudies-literature

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Linear convergence of the subspace constrained mean shift algorithm: from Euclidean to directional data.

Zhang Yikun Y Chen Yen-Chi YC

Information and inference : a journal of the IMA 20220409 1

This paper studies the linear convergence of the subspace constrained mean shift (SCMS) algorithm, a well-known algorithm for identifying a density ridge defined by a kernel density estimator. By arguing that the SCMS algorithm is a special variant of a subspace constrained gradient ascent (SCGA) algorithm with an adaptive step size, we derive the linear convergence of such SCGA algorithm. While the existing research focuses mainly on density ridges in the Euclidean space, we generalize density ...[more]

PMID: 36761435

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Project description:Objectives(1) To develop and internally-validate Euclidean Norm Minus One (ENMO) and Mean Amplitude Deviation (MAD) thresholds for separating sedentary behaviours from common light-intensity physical activities using raw acceleration data collected from both hip- and wrist-worn tri-axial accelerometers; and (2) to compare and evaluate the performances between the ENMO and MAD metrics.MethodsThirty-three adults [mean age (standard deviation (SD)) = 27.4 (5.9) years; mean BMI (SD) = 23.9 (3.7) kg/m2; 20 females (60.6%)] wore four accelerometers; an ActiGraph GT3X+ and a GENEActiv on the right hip; and an ActiGraph GT3X+ and a GENEActiv on the non-dominant wrist. Under laboratory-conditions, participants performed 16 different activities (11 sedentary behaviours and 5 light-intensity physical activities) for 5 minutes each. ENMO and MAD were computed from the raw acceleration data, and logistic regression and receiver-operating-characteristic (ROC) analyses were implemented to derive thresholds for activity discrimination. Areas under ROC curves (AUROC) were calculated to summarise performances and thresholds were assessed via executing leave-one-out-cross-validations.ResultsFor both hip and wrist monitor placements, in comparison to the ActiGraph GT3X+ monitors, the ENMO and MAD values derived from the GENEActiv devices were observed to be slightly higher, particularly for the lower-intensity activities. Monitor-specific hip and wrist ENMO and MAD thresholds showed excellent ability for separating sedentary behaviours from motion-based light-intensity physical activities (in general, AUROCs >0.95), with validation indicating robustness. However, poor classification was experienced when attempting to isolate standing still from sedentary behaviours (in general, AUROCs <0.65). The ENMO and MAD metrics tended to perform similarly across activities and accelerometer brands.ConclusionsResearchers can utilise these robust monitor-specific hip and wrist ENMO and MAD thresholds, in order to accurately separate sedentary behaviours from common motion-based light-intensity physical activities. However, caution should be taken if isolating sedentary behaviours from standing is of particular interest.

Dataset Information

Linear convergence of the subspace constrained mean shift algorithm: from Euclidean to directional data.

Publications

Linear convergence of the subspace constrained mean shift algorithm: from Euclidean to directional data.

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