Observation of the Crab Nebula with the HAWC Gamma-Ray Observatory

The Crab Nebula is the brightest TeV gamma-ray source in the sky and has been used for the past 25 years as a reference source in TeV astronomy, for calibration and verification of new TeV instruments. The High Altitude Water Cherenkov Observatory (HAWC), completed in early 2015, has been used to observe the Crab Nebula at high significance across nearly the full spectrum of energies to which HAWC is sensitive. HAWC is unique for its wide field-of-view, nearly 2 sr at any instant, and its high-energy reach, up to 100 TeV. HAWC's sensitivity improves with the gamma-ray energy. Above $\sim$1 TeV the sensitivity is driven by the best background rejection and angular resolution ever achieved for a wide-field ground array. We present a time-integrated analysis of the Crab using 507 live days of HAWC data from 2014 November to 2016 June. The spectrum of the Crab is fit to a function of the form $\phi(E) = \phi_0 (E/E_{0})^{-\alpha -\beta\cdot{\rm{ln}}(E/E_{0})}$. The data is well-fit with values of $\alpha=2.63\pm0.03$, $\beta=0.15\pm0.03$, and log$_{10}(\phi_0~{\rm{cm}^2}~{\rm{s}}~{\rm{TeV}})=-12.60\pm0.02$ when $E_{0}$ is fixed at 7 TeV and the fit applies between 1 and 37 TeV. Study of the systematic errors in this HAWC measurement is discussed and estimated to be $\pm$50\% in the photon flux between 1 and 37 TeV. Confirmation of the Crab flux serves to establish the HAWC instrument's sensitivity for surveys of the sky. The HAWC survey will exceed sensitivity of current-generation observatories and open a new view of 2/3 of the sky above 10 TeV.Comment: Submitted 2017/01/06 to the Astrophysical Journa

Revealing networks from dynamics: an introduction

What can we learn from the collective dynamics of a complex network about its interaction topology? Taking the perspective from nonlinear dynamics, we briefly review recent progress on how to infer structural connectivity (direct interactions) from accessing the dynamics of the units. Potential applications range from interaction networks in physics, to chemical and metabolic reactions, protein and gene regulatory networks as well as neural circuits in biology and electric power grids or wireless sensor networks in engineering. Moreover, we briefly mention some standard ways of inferring effective or functional connectivity.Comment: Topical review, 48 pages, 7 figure

The Dantzig selector: Statistical estimation when pp is much larger than nn

In many important statistical applications, the number of variables or parameters $p$ is much larger than the number of observations $n$. Suppose then that we have observations $y=X\beta+z$, where $\beta\in\mathbf{R}^p$ is a parameter vector of interest, $X$ is a data matrix with possibly far fewer rows than columns, $n\ll p$, and the $z_i$'s are i.i.d. $N(0,\sigma^2)$. Is it possible to estimate $\beta$ reliably based on the noisy data $y$? To estimate $\beta$, we introduce a new estimator--we call it the Dantzig selector--which is a solution to the $\ell_1$-regularization problem \min_{\tilde{\b eta}\in\mathbf{R}^p}\|\tilde{\beta}\|_{\ell_1}\quad subject to\quad \|X^*r\|_{\ell_{\infty}}\leq(1+t^{-1})\sqrt{2\log p}\cdot\sigma, where $r$ is the residual vector $y-X\tilde{\beta}$ and $t$ is a positive scalar. We show that if $X$ obeys a uniform uncertainty principle (with unit-normed columns) and if the true parameter vector $\beta$ is sufficiently sparse (which here roughly guarantees that the model is identifiable), then with very large probability, $$\|\hat{\beta}-\beta\|_{\ell_2}^2\le C^2\cdot2\log p\cdot \Biggl(\sigma^2+\sum_i\min(\beta_i^2,\sigma^2)\Biggr).$$ Our results are nonasymptotic and we give values for the constant $C$. Even though $n$ may be much smaller than $p$, our estimator achieves a loss within a logarithmic factor of the ideal mean squared error one would achieve with an oracle which would supply perfect information about which coordinates are nonzero, and which were above the noise level. In multivariate regression and from a model selection viewpoint, our result says that it is possible nearly to select the best subset of variables by solving a very simple convex program, which, in fact, can easily be recast as a convenient linear program (LP).Comment: This paper discussed in: [arXiv:0803.3124], [arXiv:0803.3126], [arXiv:0803.3127], [arXiv:0803.3130], [arXiv:0803.3134], [arXiv:0803.3135]. Rejoinder in [arXiv:0803.3136]. Published in at http://dx.doi.org/10.1214/009053606000001523 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Improving the performance of translation wavelet transform using BMICA

Research has shown Wavelet Transform to be one of the best methods for denoising biosignals. Translation-Invariant form of this method has been found to be the best performance. In this paper however we utilize this method and merger with our newly created Independent Component Analysis method – BMICA. Different EEG signals are used to verify the method within the MATLAB environment. Results are then compared with those of the actual Translation-Invariant algorithm and evaluated using the performance measures Mean Square Error (MSE), Peak Signal to Noise Ratio (PSNR), Signal to Distortion Ratio (SDR), and Signal to Interference Ratio (SIR). Experiments revealed that the BMICA Translation-Invariant Wavelet Transform out performed in all four measures. This indicates that it performed superior to the basic Translation- Invariant Wavelet Transform algorithm producing cleaner EEG signals which can influence diagnosis as well as clinical studies of the brain

Variational Bayesian Inference of Line Spectra

In this paper, we address the fundamental problem of line spectral estimation in a Bayesian framework. We target model order and parameter estimation via variational inference in a probabilistic model in which the frequencies are continuous-valued, i.e., not restricted to a grid; and the coefficients are governed by a Bernoulli-Gaussian prior model turning model order selection into binary sequence detection. Unlike earlier works which retain only point estimates of the frequencies, we undertake a more complete Bayesian treatment by estimating the posterior probability density functions (pdfs) of the frequencies and computing expectations over them. Thus, we additionally capture and operate with the uncertainty of the frequency estimates. Aiming to maximize the model evidence, variational optimization provides analytic approximations of the posterior pdfs and also gives estimates of the additional parameters. We propose an accurate representation of the pdfs of the frequencies by mixtures of von Mises pdfs, which yields closed-form expectations. We define the algorithm VALSE in which the estimates of the pdfs and parameters are iteratively updated. VALSE is a gridless, convergent method, does not require parameter tuning, can easily include prior knowledge about the frequencies and provides approximate posterior pdfs based on which the uncertainty in line spectral estimation can be quantified. Simulation results show that accounting for the uncertainty of frequency estimates, rather than computing just point estimates, significantly improves the performance. The performance of VALSE is superior to that of state-of-the-art methods and closely approaches the Cram\'er-Rao bound computed for the true model order.Comment: 15 pages, 8 figures, accepted for publication in IEEE Transactions on Signal Processin

Learning Graphs from Linear Measurements: Fundamental Trade-offs and Applications

We consider a specific graph learning task: reconstructing a symmetric matrix that represents an underlying graph using linear measurements. We present a sparsity characterization for distributions of random graphs (that are allowed to contain high-degree nodes), based on which we study fundamental trade-offs between the number of measurements, the complexity of the graph class, and the probability of error. We first derive a necessary condition on the number of measurements. Then, by considering a three-stage recovery scheme, we give a sufficient condition for recovery. Furthermore, assuming the measurements are Gaussian IID, we prove upper and lower bounds on the (worst-case) sample complexity for both noisy and noiseless recovery. In the special cases of the uniform distribution on trees with n nodes and the Erdős-Rényi (n,p) class, the fundamental trade-offs are tight up to multiplicative factors with noiseless measurements. In addition, for practical applications, we design and implement a polynomial-time (in n ) algorithm based on the three-stage recovery scheme. Experiments show that the heuristic algorithm outperforms basis pursuit on star graphs. We apply the heuristic algorithm to learn admittance matrices in electric grids. Simulations for several canonical graph classes and IEEE power system test cases demonstrate the effectiveness and robustness of the proposed algorithm for parameter reconstruction