Identifying periodically expressed genes across different processes (e.g. the cell and metabolic cycles, circadian rhythms, etc) is a central problem in computational biology. Biological time series may contain (multiple) unknown signal shapes of systemic relevance, imperfections like noise, damping, and trending, or limited sampling density. While there exist methods for detecting periodicity, their design biases (e.g. toward a specific signal shape) can limit their applicability in one or more of these situations.
We introduce the first unified theory for target tracking using Multiple Hypothesis Tracking, Topological Data Analysis, and machine learning. Our string of innovations are 1) robust topological features are used to encode behavioral information, 2) statistical models are fitted to distributions over these topological features, and 3) the target type classification methods of Wigren and Bar Shalom et al. are employed to exploit the resulting likelihoods for topological features inside of the tracking procedure. To demonstrate the efficacy of our approach, we test our procedure on synthetic vehicular data generated by the Simulation of Urban Mobility package.
Table: Saccharomyces cerevisiae gene expression dynamics used in this study. The wild-type gene expression data of 28 genes over approximately two cell cycles were obtained from previous work [2]. Expression values from each profile were smoothed from 13 experimental time points to 40 time points by fitting a cubic spline (the function splinefun in R package stats). A Rescon Ltd. tool was used to rank the splined profiles by 50 % of the peak expression value. Genes are ordered from earliest (top) to latest (bottom) half-maximal expression and presented in the heat map in Fig. 1. (XLSX 51 kb)
We introduce a method called multi-scale local shape analysis, or MLSA, for extracting features that describe the local structure of points within a dataset. The method uses both geometric and topological features at multiple levels of granularity to capture diverse types of local information for subsequent machine learning algorithms operating on the dataset. Using synthetic and real dataset examples, we demonstrate significant performance improvement of classification algorithms constructed for these datasets with correspondingly augmented features.
This paper presents a processing pipeline for fusing `raw' and / or feature-level multi-sensor data - upstream fusion - and initial results from this pipeline using imagery, radar, and radio frequency (RF) signals data to determine which tracked object, among several, hosts an emitter of interest. Correctly making this determination requires fusing data across these modalities. Our approach performs better than standard fusion approaches that make detection / characterization decisions for each modality individually and then try to fuse those decisions - downstream (or post-decision) fusion. Our approach (1) fully exploits the inter-modality dependencies and phenomenologies inherent in different sensing modes, (2) automatically discovers compressive hierarchical representations that integrate structural and statistical characteristics to enhance target / event discriminability, and (3) completely obviates the need to specify features, manifolds, or model scope a priori. This approach comprises a unique synthesis of Deep Learning (DL), topological analysis over probability measure (TAPM), and hierarchical Bayesian non-parametric (HBNP) recognition models. Deep Generative Networks (DGNs - a deep generative statistical form of DL) create probability measures that provide a basis for calculating homologies (topological summaries over the probability measures). The statistics of the resulting persistence diagrams are inputs to HBNP methods that learn to discriminate between target types and distinguish emitting targets from non-emitting targets, for example. HBNP learning obviates batch-mode off-line learning. This approach overcomes the inadequacy of pre-defined features as a means for creating efficient, discriminating, low-dimensional representations from high-dimensional multi-modality sensor data collected under difficult, dynamic sensing conditions. The invariant properties in the resulting compact representations afford multiple compressive sensing benefits, including concise information sharing and enhanced performance. Machine learning makes adaptivity a central feature of our approach. Adaptivity is critical because it enables flexible processing that automatically accommodates a broad range of challenges that non-adaptive, standard fusion approaches would typically require manual intervention to begin to address. These include (a) interest in unknown or unanticipated targets, (b) desire to be rapidly able to fuse between different combinations of sensor modalities, and (c) potential need to transfer information between platforms that host different sensors. This paper presents results that demonstrate our approach enables accurate, real-time target detection, tracking, and recognition of known and unknown moving or stationary targets or events and their activities evolving over space and time.
Introduction to Maple Plotting Functions of 1 Variable Co-ordinates in 3-Space, Dot Products and Vector Products Parametric Lines and Planes Parametric Curves Finding Parametrizations Functions of Two Variables Level Curves and Contour Lines Functions of Three Variables and Level Surfaces Parametric Surfaces Cylindrical Co-ordinates Spherical Co-ordinates Partial Derivatives and the Chain Rule Gradients, Directional Derivatives and Vector Fields Tangent Planes Maxima, Minimums and Saddle Points The Method of Lagrange Multipliers Double Integrals Applications of Double Integrals Double Integrals in Polar Coordinates Triple Integrals in Rectangular Co-ordinates Triple Integrals in Cylindrical Co-ordinates Triple Integrals in Spherical Co-ordinates Type I and II Line Integrals Conservative Vector Fields Green's Theorem Surface Integrals Stokes' Theorem and the Divergence Theorem Maple Commands Answers to Drill Exercises.
Let ${mathcal M}_g^n$ be the moduli space of n-pointed Riemann surfaces of genus g. Denote by ${\bar {\mathcal M}}_g^n$ the Deligne-Mumford compactification of ${mathcal M}_g^n$. In the present paper, we calculate the orbifold and the ordinary Euler characteristic of ${\bar {\mathcal M}}_g^n$ for any g and n such that n>2-2g.
We introduce a method called multi-scale local shape analysis, or MLSA, for extracting features that describe the local structure of points within a dataset. The method uses both geometric and topological features at multiple levels of granularity to capture diverse types of local information for subsequent machine learning algorithms operating on the dataset. Using synthetic and real dataset examples, we demonstrate significant performance improvement of classification algorithms constructed for these datasets with correspondingly augmented features.
