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The sample mean is one of the most fundamental concepts in statistics with far-reaching implications for data mining and pattern recognition. Household load profiles are compared to the aggregated levels more intermittent and a specific error measure based on local permutations has been proposed to cope with this when comparing profiles. We formally describe a distance based on this error, the local permutation invariant (LPI) distance, and introduce the sample mean problem in the LPI space. An existing exact solution has exponential complexity and is only tractable for very few profiles. We propose three subgradient-based approximation algorithms and compare them empirically on 100 households of the CER dataset. We find that stochastic subgradient descent can approximate the mean best, while the majorize-minimize mean is a good compromise for applications as no hyperparameter-tuning is needed. We show how the algorithms can be used in forecasting and clustering to achieve more appropriate results than by using the arithmetic mean.
Increasing user participation or changing behavior are key goals when applying gamification. Existing studies in domains such as education, health, and enterprise show that gamification can have a positive impact on meeting these goals. However, there is still a lack of detailed insights into how certain game design elements affect user behavior and motivation. To gain further insight, this paper presents a user study in the field with 20, 000 participants of a mobile e-commerce application over a one-month time period to analyze the impact of gamification in the e-commerce domain and to compare the effectiveness of tangible versus intangible rewards. Results show that gamification has a positive impact in the e-commerce domain. The study also reveals that tangible rewards increase the user activity substantially more than intangible rewards. We further show how tangible rewards affect certain user types and provide a first discussion on the lastingness of these rewards.