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Wearable devices permit the continuous monitoring of biological processes,such as blood glucose metabolism, and behavior, such as sleep quality andphysical activity. The continuous monitoring often occurs in epochs of 60seconds over multiple days, resulting in high dimensional longitudinal curvesthat are best described and analyzed as functional data. From this perspective,the functional data are smooth, latent functions obtained at discrete timeintervals and prone to homoscedastic white noise. However, the assumption ofhomoscedastic errors might not be appropriate in this setting because thedevices collect the data serially. While researchers have previously addressedmeasurement error in scalar covariates prone to errors, less work has been doneon correcting measurement error in high dimensional longitudinal curves proneto heteroscedastic errors. We present two new methods for correctingmeasurement error in longitudinal functional curves prone to complexmeasurement error structures in multi-level generalized functional linearregression models. These methods are based on two-stage scalable regressioncalibration. We assume that the distribution of the scalar responses and thesurrogate measures prone to heteroscedastic errors both belong in theexponential family and that the measurement errors follow Gaussian processes.In simulations and sensitivity analyses, we established some finite sampleproperties of these methods. In our simulations, both regression calibrationmethods for correcting measurement error performed better than estimators basedon averaging the longitudinal functional data and using observations from asingle day. We also applied the methods to assess the relationship betweenphysical activity and type 2 diabetes in community dwelling adults in theUnited States who participated in the National Health and Nutrition ExaminationSurvey.

Scalable regression calibration approaches to correcting measurement  error in multi-level generalized functional linear regression models with  heteroscedastic measurement errors