Gait speed as a powerful biomarker of mobility is mostly assessed in controlled environments, e.g. in the clinic. With wearable inertial sensors, gait speed can be estimated in an objective manner. However, most of the previous works have validated the gait speed estimation algorithms in clinical settings which can be different than the home assessments in which the patients demonstrate their actual performance. Moreover, to provide comfort for the users, devising an algorithm based on a single sensor setup is essential. To this end, the goal of this study was to develop and validate a new gait speed estimation method based on a machine learning approach to predict gait speed in both clinical and home assessments by a sensor on the lower back. Moreover, two methods were introduced to detect walking bouts during daily activities at home. We have validated the algorithms in 35 patients with multiple sclerosis as it often presents with mobility difficulties. Therefore, the robustness of the algorithm can be shown in an impaired or slow gait. Against silver standard multi-sensor references, we achieved a bias close to zero and a precision of 0.15 m/s for gait speed estimation. Furthermore, the proposed machine learning-based locomotion detection method had a median of 96.8% specificity, 93.0% sensitivity, 96.4% accuracy, and 78.6% F1-score in detecting walking bouts at home. The high performance of the proposed algorithm showed the feasibility of the unsupervised mobility assessment introduced in this study.