Sorted L-One Penalized Estimator (SLOPE) is a relatively new convex optimization procedure for selecting predictors in high dimensional regression analyses. SLOPE extends LASSO by replacing the L1 penalty norm with a Sorted L1 norm, based on the non-increasing sequence of tuning parameters. This allows SLOPE to adapt to unknown sparsity and achieve an asymptotic minimax convergency rate under a wide range of high dimensional generalized linear models. Additionally, in the case when the design matrix is orthogonal, SLOPE with the sequence of tuning parameters λ BH corresponding to the sequence of decaying thresholds for the Benjamini-Hochberg multiple testing correction provably controls the False Discovery Rate (FDR) in the multiple regression model. In this article we provide new asymptotic results on the properties of SLOPE when the elements of the design matrix are iid random variables from the Gaussian distribution. Specifically, we provide conditions under which the asymptotic FDR of SLOPE based on the sequence λ BH converges to zero and the power converges to 1. We illustrate our theoretical asymptotic results with an extensive simulation study. We also provide precise formulas describing FDR of SLOPE under different loss functions, which sets the stage for future investigation on the model selection properties of SLOPE and its extensions.

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Summary Human health is strongly associated with person's lifestyle and levels of physical activity. Therefore, characterization of daily human activity is an important task. Accelerometers have been used to obtain precise measurements of body acceleration. Wearable accelerometers collect data as a three‐dimensional time series with frequencies up to 100 Hz. Using such accelerometry signal, we are able to classify different types of physical activity. In our work, we present a novel procedure for physical activity classification based on the raw accelerometry signal. Our proposal is based on the spherical representation of the data. We classify four activity types: resting, upper body activities (sitting), upper body activities (standing), and lower body activities. The classifier is constructed using decision trees with extracted features consisting of spherical coordinates summary statistics, moving averages of the radius and the angles, radius variance, and spherical variance. The classification accuracy of our method has been tested on data collected on a sample of 47 elderly individuals who performed a series of activities in laboratory settings. The achieved classification accuracy is over 90% when the subject‐specific data are used and 84% when the group data are used. Main contributor to the classification accuracy is the angular part of the collected signal, especially spherical variance. To the best of our knowledge, spherical variance has never been previously used in the analysis of the raw accelerometry data. Its major advantage over other angular measures is its invariance to the accelerometer location shifts.

Human health is strongly associated with person's lifestyle and levels of physical activity. Therefore, characterization of daily human activity is an important task. Accelerometers have been used to obtain precise measurements of body acceleration. Wearable accelerometers collect data as a three-dimensional time series with frequencies up to 100Hz. Using such accelerometry signal, we are able to classify different types of physical activity.In our work, we present a novel procedure for physical activity classification based on the raw accelerometry signal. Our proposal is based on the spherical representation of the data. We classify four activity types: resting, upper body activities (sitting), upper body activities (standing) and lower body activities. The classifier is constructed using decision trees with extracted features consisting of spherical coordinates summary statistics, moving averages of the radius and the angles, radius variance and spherical variance.The classification accuracy of our method has been tested on data collected on a sample of 47 elderly individuals who performed a series of activities in 1 laboratory settings. The achieved classification accuracy is over 90% when the subject-specific data are used and 84% when the group data are used. Main contributor to the classification accuracy is the angular part of the collected signal, especially spherical variance. To the best of our knowledge, spherical variance has never been previously used in the analysis of the raw accelerometry data. Its major advantage over other angular measures is its invariance to the accelerometer location shifts.