Computational Topology Techniques for Characterizing Time-Series Data

Topological data analysis (TDA), while abstract, allows a characterization of time-series data obtained from nonlinear and complex dynamical systems. Though it is surprising that such an abstract measure of structure - counting pieces and holes - could be useful for real-world data, TDA lets us compare different systems, and even do membership testing or change-point detection. However, TDA is computationally expensive and involves a number of free parameters. This complexity can be obviated by coarse-graining, using a construct called the witness complex. The parametric dependence gives rise to the concept of persistent homology: how shape changes with scale. Its results allow us to distinguish time-series data from different systems - e.g., the same note played on different musical instruments.Comment: 12 pages, 6 Figures, 1 Table, The Sixteenth International Symposium on Intelligent Data Analysis (IDA 2017

Betti number signatures of homogeneous Poisson point processes

The Betti numbers are fundamental topological quantities that describe the k-dimensional connectivity of an object: B_0 is the number of connected components and B_k effectively counts the number of k-dimensional holes. Although they are appealing natural descriptors of shape, the higher-order Betti numbers are more difficult to compute than other measures and so have not previously been studied per se in the context of stochastic geometry or statistical physics. As a mathematically tractable model, we consider the expected Betti numbers per unit volume of Poisson-centred spheres with radius alpha. We present results from simulations and derive analytic expressions for the low intensity, small radius limits of Betti numbers in one, two, and three dimensions. The algorithms and analysis depend on alpha-shapes, a construction from computational geometry that deserves to be more widely known in the physics community.Comment: Submitted to PRE. 11 pages, 10 figure

Supersonic aircraft Patent

Design of supersonic aircraft with novel fixed, swept wing planfor

Sparse Nerves in Practice

Topological data analysis combines machine learning with methods from algebraic topology. Persistent homology, a method to characterize topological features occurring in data at multiple scales is of particular interest. A major obstacle to the wide-spread use of persistent homology is its computational complexity. In order to be able to calculate persistent homology of large datasets, a number of approximations can be applied in order to reduce its complexity. We propose algorithms for calculation of approximate sparse nerves for classes of Dowker dissimilarities including all finite Dowker dissimilarities and Dowker dissimilarities whose homology is Cech persistent homology. All other sparsification methods and software packages that we are aware of calculate persistent homology with either an additive or a multiplicative interleaving. In dowker_homology, we allow for any non-decreasing interleaving function $\alpha$. We analyze the computational complexity of the algorithms and present some benchmarks. For Euclidean data in dimensions larger than three, the sizes of simplicial complexes we create are in general smaller than the ones created by SimBa. Especially when calculating persistent homology in higher homology dimensions, the differences can become substantial

Tactile Interactions with a Humanoid Robot : Novel Play Scenario Implementations with Children with Autism

Acknowledgments: This work has been partially supported by the European Commission under contract number FP7-231500-ROBOSKIN. Open Access: This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.The work presented in this paper was part of our investigation in the ROBOSKIN project. The project has developed new robot capabilities based on the tactile feedback provided by novel robotic skin, with the aim to provide cognitive mechanisms to improve human-robot interaction capabilities. This article presents two novel tactile play scenarios developed for robot-assisted play for children with autism. The play scenarios were developed against specific educational and therapeutic objectives that were discussed with teachers and therapists. These objectives were classified with reference to the ICF-CY, the International Classification of Functioning – version for Children and Youth. The article presents a detailed description of the play scenarios, and case study examples of their implementation in HRI studies with children with autism and the humanoid robot KASPAR.Peer reviewedFinal Published versio

Robot-mediated interviews: : Do robots possess advantages over human interviewers when talking to children with special needs?

Wood L.J., Dautenhahn K., Lehmann H., Robins B., Rainer A., Syrdal D.S. (2013) 'Robot-Mediated Interviews: Do Robots Possess Advantages over Human Interviewers When Talking to Children with Special Needs?', In: Herrmann G., Pearson M.J., Lenz A., Bremner P., Spiers A., Leonards U. (eds) Social Robotics. ICSR 2013. Lecture Notes in Computer Science, vol 8239. Springer, Cham Available online at doi: 10.1007/978-3-319-02675-6-6 © Springer-Verlag Berlin Heidelberg 2013Children that have a disability are up to four times more likely to be a victim of abuse than typically developing children. However, the number of cases that result in prosecution is relatively low. One of the factors influencing this low prosecution rate is communication difficulties. Our previous research has shown that typically developing children respond to a robotic interviewer very similar compared to a human interviewer. In this paper we conduct a follow up study investigating the possibility of Robot-Mediated Interviews with children that have various special needs. In a case study we investigated how 5 children with special needs aged 9 to 11 responded to the humanoid robot KASPAR compared to a human in an interview scenario. The measures used in this study include duration analysis of responses, detailed analysis of transcribed data, questionnaire responses and data from engagement coding. The main questions in the interviews varied in difficulty and focused on the theme of animals and pets. The results from quantitative data analysis reveal that the children interacted with KASPAR in a very similar manner to how they interacted with the human interviewer, providing both interviewers with similar information and amounts of information regardless of question difficulty. However qualitative analysis suggests that some children may have been more engaged with the robotic interviewer

Parallel Mapper

The construction of Mapper has emerged in the last decade as a powerful and effective topological data analysis tool that approximates and generalizes other topological summaries, such as the Reeb graph, the contour tree, split, and joint trees. In this paper, we study the parallel analysis of the construction of Mapper. We give a provably correct parallel algorithm to execute Mapper on multiple processors and discuss the performance results that compare our approach to a reference sequential Mapper implementation. We report the performance experiments that demonstrate the efficiency of our method

Hypergraphic LP Relaxations for Steiner Trees

We investigate hypergraphic LP relaxations for the Steiner tree problem, primarily the partition LP relaxation introduced by Koenemann et al. [Math. Programming, 2009]. Specifically, we are interested in proving upper bounds on the integrality gap of this LP, and studying its relation to other linear relaxations. Our results are the following. Structural results: We extend the technique of uncrossing, usually applied to families of sets, to families of partitions. As a consequence we show that any basic feasible solution to the partition LP formulation has sparse support. Although the number of variables could be exponential, the number of positive variables is at most the number of terminals. Relations with other relaxations: We show the equivalence of the partition LP relaxation with other known hypergraphic relaxations. We also show that these hypergraphic relaxations are equivalent to the well studied bidirected cut relaxation, if the instance is quasibipartite. Integrality gap upper bounds: We show an upper bound of sqrt(3) ~ 1.729 on the integrality gap of these hypergraph relaxations in general graphs. In the special case of uniformly quasibipartite instances, we show an improved upper bound of 73/60 ~ 1.216. By our equivalence theorem, the latter result implies an improved upper bound for the bidirected cut relaxation as well.Comment: Revised full version; a shorter version will appear at IPCO 2010

Processes controlling atmospheric dispersion through city centres

We develop a process-based model for the dispersion of a passive scalar in the turbulent flow around the buildings of a city centre. The street network model is based on dividing the airspace of the streets and intersections into boxes, within which the turbulence renders the air well mixed. Mean flow advection through the network of street and intersection boxes then mediates further lateral dispersion. At the same time turbulent mixing in the vertical detrains scalar from the streets and intersections into the turbulent boundary layer above the buildings. When the geometry is regular, the street network model has an analytical solution that describes the variation in concentration in a near-field downwind of a single source, where the majority of scalar lies below roof level. The power of the analytical solution is that it demonstrates how the concentration is determined by only three parameters. The plume direction parameter describes the branching of scalar at the street intersections and hence determines the direction of the plume centreline, which may be very different from the above-roof wind direction. The transmission parameter determines the distance travelled before the majority of scalar is detrained into the atmospheric boundary layer above roof level and conventional atmospheric turbulence takes over as the dominant mixing process. Finally, a normalised source strength multiplies this pattern of concentration. This analytical solution converges to a Gaussian plume after a large number of intersections have been traversed, providing theoretical justification for previous studies that have developed empirical fits to Gaussian plume models. The analytical solution is shown to compare well with very high-resolution simulations and with wind tunnel experiments, although re-entrainment of scalar previously detrained into the boundary layer above roofs, which is not accounted for in the analytical solution, is shown to become an important process further downwind from the source