Efficient Nearest Neighbor Search on Metric Time Series

While Deep-Learning approaches beat Nearest-Neighbor classifiers in an increasing number of areas, searching existing uncertain data remains an exclusive task for similarity search. Numerous specific solutions exist for different types of data and queries. This thesis aims at finding fast and general solutions for searching and indexing arbitrarily typed time series. A time series is considered a sequence of elements where the elements' order matters but not their actual time stamps. Since this thesis focuses on measuring distances between time series, the metric space is the most appropriate concept where the time series' elements come from. Hence, this thesis mainly considers metric time series as data type. Simple examples include time series in Euclidean vector spaces or graphs. For general similarity search solutions in time series, two primitive comparison semantics need to be distinguished, the first of which compares the time series' trajectories ignoring time warping. A ubiquitous example of such a distance function is the Dynamic Time Warping distance (DTW) developed in the area of speech recognition. The Dog Keeper distance (DK) is another time-warping distance that, opposed to DTW, is truly invariant under time warping and yields a metric space. After canonically extending DTW to accept multi-dimensional time series, this thesis contributes a new algorithm computing DK that outperforms DTW on time series in high-dimensional vector spaces by more than one order of magnitude. An analytical study of both distance functions reveals the reasons for the superiority of DK over DTW in high-dimensional spaces. The second comparison semantic compares time series in Euclidean vector spaces regardless of their position or orientation. This thesis proposes the Congruence distance that is the Euclidean distance minimized under all isometric transformations; thus, it is invariant under translation, rotation, and reflection of the time series and therefore disregards the position or orientation of the time series. A proof contributed in this thesis shows that there can be no efficient algorithm computing this distance function (unless P=NP). Therefore, this thesis contributes the Delta distance, a metric distance function serving as a lower bound for the Congruence distance. While the Delta distance has quadratic time complexity, the provided evaluation shows a speedup of more than two orders of magnitude against the Congruence distance. Furthermore, the Delta distance is shown to be tight on random time series, although the tightness can be arbitrarily bad in corner-case situations. Orthogonally to the previous mentioned comparison semantics, similarity search on time series consists of two different types of queries: whole sequence matching and subsequence search. Metric index structures (e. g., the M-Tree) only provide whole matching queries natively. This thesis contributes the concept of metric subset spaces and the SuperM-Tree for indexing metric subset spaces as a generic solution for subsequence search. Examples for metric subset spaces include subsequence search regarding the distance functions from the comparison semantics mentioned above. The provided evaluation shows that the SuperM-Tree outperforms a linear search by multiple orders of magnitude

Motor imagery ability in patients with traumatic brain injury

Oostra KM, Vereecke A, Jones K, Vanderstraeten G, Vingerhoets G. Motor imagery ability in patients with traumatic brain injury. Arch Phys Med Rehabil 2012;93:828-33. Objective: To assess motor imagery (MI) ability in patients with moderate to severe traumatic brain injury (TBI). Design: Prospective, cohort study. Setting: University hospital rehabilitation unit. Participants: Patients with traumatic brain injury (mean coma duration, 18d) undergoing rehabilitation (n=20) and healthy controls (n=17) matched for age and education level. Interventions: Not applicable. Main Outcome Measures: The vividness of MI was assessed using a revised version of the Movement Imagery Questionnaire-Revised second version (MIQ-RS); the temporal features were assessed using the time-dependent motor imagery (TDMI) screening test, the temporal congruence test, and a walking trajectory imagery test; and the accuracy of MI was assessed using a mental rotation test. Results: The MIQ-RS revealed a decrease of MI vividness in the TBI group. An increasing number of stepping movements was observed with increasing time periods in both groups during the TDMI screening test. The TBI group performed a significantly smaller number of imagery movements in the same movement time. The temporal congruence test revealed a significant correlation between imagery and actual stepping time in both groups. The walking trajectory test revealed an increase of the imagery and actual walking time with increasing path length in both groups, but the ratio of imaginary walking over actual walking time was significantly greater than 1 in the TBI group. Results of the hand mental rotation test indicated significant effects of rotation angles on imagery movement times in both groups, but rotation time was significantly slower in the TBI group. Conclusions: Our patients with TBI demonstrated a relatively preserved MI ability indicating that MI could be used to aid rehabilitation and subsequent functional recovery

Role of Horizons in Semiclassical Gravity: Entropy and the Area Spectrum

In any space-time, it is possible to have a family of observers who have access to only part of the space-time manifold, because of the existence of a horizon. We demand that \emph{physical theories in a given coordinate system must be formulated entirely in terms of variables that an observer using that coordinate system can access}. In the coordinate frame in which these observers are at rest, the horizon manifests itself as a (coordinate) singularity in the metric tensor. Regularization of this singularity removes the inaccessible region, and leads to the following consequences: (a) The non-trivial topological structure for the effective manifold allows one to obtain the standard results of quantum field theory in curved space-time. (b) In case of gravity, this principle requires that the effect of the unobserved degrees of freedom should reduce to a boundary contribution $A_{\rm boundary}$ to the gravitational action. When the boundary is a horizon, $A_{\rm boundary}$ reduces to a single, well-defined term proportional to the area of the horizon. Using the form of this boundary term, it is possible to obtain the full gravitational action in the semiclassical limit. (c) This boundary term must have a quantized spectrum with uniform spacing, $\Delta A_{boundary}=2\pi\hbar$, in the semiclassical limit. This, in turn, yields the following results for semiclassical gravity: (i) The area of any one-way membrane is quantized. (ii) The information hidden by a one-way membrane amounts to an entropy, which is always one-fourth of the area of the membrane in the leading order. (iii) In static space-times, the action for gravity can be given a purely thermodynamic interpretation and the Einstein equations have a formal similarity to laws of thermodynamics.Comment: Extends and presents the results of hep-th/0305165 in a broader context; clarifies some conceptual issues; 24 pages; revte