In a nutshell, stay-points are locations that a person has stopped for some amount of time. Previous work depends mainly on stay-point identification methods using experimentally fine tuned threshold values. These behave well on their experimental datasets but may exhibit reduced performance on other datasets. In this work, we demonstrate the potential of a geometry-based method for stay-point extraction. This is accomplished by transforming the user's trajectory path to a two-dimensional discrete time series curve that in turn transforms the stay-points to the local minima of the first derivative of this curve. To demonstrate the soundness of the proposed method, we evaluated it on raw, noisy trajectory data acquired over the period of 28 different days using four different techniques. The results demonstrate, among others, that given a good trajectory tracking technique, we can identify correctly 86 to 98 of the stay-points.
@cite_4 propose a gradient-based visit extractor algorithm. This algorithm works as follows: insert all points into a buffer until the user has moved more than some threshold (computed via the gradient) or the time difference between the first point and the last point in the buffer is greater than another time threshold; then if there is some duration between the first and last points in the buffer, then a stay-point is identified. The threshold for the gradient, that controls the distance the user moved, depends on two parameters, which are experimentally tuned per dataset.
@cite_4
2779280457 2199718034 2895571900 2762725970
We tackle the problem of extracting stay regions from a geospatial trajectory where a user has stayed longer than a certain time threshold. There are four major difficulties with this problem: (1) stay regions are not only point-type ones such as at a bus-stop but large and arbitrary-shaped ones such as at a shopping mall; (2) trajectories contain spatial outliers; (3) there are missing points in trajectories; and (4) trajectories should be analyzed in an online mode. Previous algorithms cannot overcome these difficulties simultaneously. Density-based batch algorithms have advantages over the previous algorithms in discovering of arbitrary-shaped clusters from spatial data containing outliers; however, they do not consider temporal durations and thus have not been used for extracting stay regions. We extended a density-based algorithm so that it would work in a duration-based manner online and have robustness to missing points in stay regions while keeping its advantages. Experiments on real trajectories of 13 users conducting their daily activities for three weeks demonstrated that our algorithm statistically significantly outperformed five state-of-the-art algorithms in terms of F1 score and works well without trajectory preprocessing consisting of filtering, interpolating, and smoothing. Harnessing the latent knowledge present in geospatial trajectories allows for the potential to revolutionise our understanding of behaviour. This paper discusses one component of such analysis, namely the extraction of significant locations. Specifically, we: (i) present the Gradient-based Visit Extractor (GVE) algorithm capable of extracting periods of low mobility from geospatial data, while maintaining resilience to noise, and addressing the drawbacks of existing techniques, (ii) provide a comprehensive analysis of the properties of these visits and consequent locations, extracted through clustering, and (iii) demonstrate the applicability of GVE to the problem of visit extraction with respect to representative use-cases. We consider the problem of finding an approximate second-order stationary point of a constrained non-convex optimization problem. We first show that, unlike the unconstrained scenario, the vanilla projected gradient descent algorithm may converge to a strict saddle point even when there is only a single linear constraint. We then provide a hardness result by showing that checking ( , )-second order stationarity is NP-hard even in the presence of linear constraints. Despite our hardness result, we identify instances of the problem for which checking second order stationarity can be done efficiently. For such instances, we propose a dynamic second order Frank--Wolfe algorithm which converges to ( , )-second order stationary points in O ( ^ -2 , ^ -3 ) iterations. The proposed algorithm can be used in general constrained non-convex optimization as long as the constrained quadratic sub-problem can be solved efficiently. In this paper, we study the problem of sampling from a given probability density function that is known to be smooth and strongly log-concave. We analyze several methods of approximate sampling based on discretizations of the (highly overdamped) Langevin diffusion and establish guarantees on its error measured in the Wasserstein-2 distance. Our guarantees improve or extend the state-of-the-art results in three directions. First, we provide an upper bound on the error of the first-order Langevin Monte Carlo (LMC) algorithm with optimized varying step-size. This result has the advantage of being horizon free (we do not need to know in advance the target precision) and to improve by a logarithmic factor the corresponding result for the constant step-size. Second, we study the case where accurate evaluations of the gradient of the log-density are unavailable, but one can have access to approximations of the aforementioned gradient. In such a situation, we consider both deterministic and stochastic approximations of the gradient and provide an upper bound on the sampling error of the first-order LMC that quantifies the impact of the gradient evaluation inaccuracies. Third, we establish upper bounds for two versions of the second-order LMC, which leverage the Hessian of the log-density. We nonasymptotic guarantees on the sampling error of these second-order LMCs. These guarantees reveal that the second-order LMC algorithms improve on the first-order LMC in ill-conditioned settings.