Differential Calculus Notes on Wrapped Exponential Distribution
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C 2013 Johannes Traa
c 2013 Johannes Traa MULTICHANNEL SOURCE SEPARATION AND TRACKING WITH PHASE DIFFERENCES BY RANDOM SAMPLE CONSENSUS BY JOHANNES TRAA THESIS Submitted in partial fulfillment of the requirements for the degree of Master of Science in Electrical and Computer Engineering in the Graduate College of the University of Illinois at Urbana-Champaign, 2013 Urbana, Illinois Adviser: Assistant Professor Paris Smaragdis ABSTRACT Blind audio source separation (BASS) is a fascinating problem that has been tackled from many different angles. The use case of interest in this thesis is that of multiple moving and simultaneously-active speakers in a reverberant room. This is a common situation, for example, in social gatherings. We human beings have the remarkable ability to focus attention on a particular speaker while effectively ignoring the rest. This is referred to as the \cocktail party effect” and has been the holy grail of source separation for many decades. Replicating this feat in real-time with a machine is the goal of BASS. Single-channel methods attempt to identify the individual speakers from a single recording. However, with the advent of hand-held consumer electronics, techniques based on microphone array processing are becoming increasingly popular. Multichannel methods record a sound field from various locations to incorporate spatial information. If the speakers move over time, we need an algorithm capable of tracking their positions in the room. For compact arrays with 1-10 cm of separation between the microphones, this can be accomplished by applying a temporal filter on estimates of the directions-of-arrival (DOA) of the speakers. In this thesis, we review recent work on BSS with inter-channel phase difference (IPD) features and provide extensions to the case of moving speakers. -
Wrapped Log Kumaraswamy Distribution and Its Applications
International Journal of Mathematics and Computer Research ISSN: 2320-7167 Volume 06 Issue 10 October-2018, Page no. - 1924-1930 Index Copernicus ICV: 57.55 DOI: 10.31142/ijmcr/v6i10.01 Wrapped Log Kumaraswamy Distribution and its Applications K.K. Jose1, Jisha Varghese2 1,2Department of Statistics, St.Thomas College, Palai,Arunapuram Mahatma Gandhi University, Kottayam, Kerala- 686 574, India ARTICLE INFO ABSTRACT Published Online: A new circular distribution called Wrapped Log Kumaraswamy Distribution (WLKD) is introduced 10 October 2018 in this paper. We obtain explicit form for the probability density function and derive expressions for distribution function, characteristic function and trigonometric moments. Method of maximum likelihood estimation is used for estimation of parameters. The proposed model is also applied to a Corresponding Author: real data set on repair times and it is established that the WLKD is better than log Kumaraswamy K.K. Jose distribution for modeling the data. KEYWORDS: Log Kumaraswamy-Geometric distribution, Wrapped Log Kumaraswamy distribution, Trigonometric moments. 1. Introduction a test, atmospheric temperatures, hydrological data, etc. Also, Kumaraswamy (1980) introduced a two-parameter this distribution could be appropriate in situations where distribution over the support [0, 1], called Kumaraswamy scientists use probability distributions which have infinite distribution (KD), for double bounded random processes for lower and (or) upper bounds to fit data, where as in reality the hydrological applications. The Kumaraswamy distribution is bounds are finite. Gupta and Kirmani (1988) discussed the very similar to the Beta distribution, but has the important connection between non-homogeneous Poisson process advantage of an invertible closed form cumulative (NHPP) and record values. -
Recent Advances in Directional Statistics
Recent advances in directional statistics Arthur Pewsey1;3 and Eduardo García-Portugués2 Abstract Mainstream statistical methodology is generally applicable to data observed in Euclidean space. There are, however, numerous contexts of considerable scientific interest in which the natural supports for the data under consideration are Riemannian manifolds like the unit circle, torus, sphere and their extensions. Typically, such data can be represented using one or more directions, and directional statistics is the branch of statistics that deals with their analysis. In this paper we provide a review of the many recent developments in the field since the publication of Mardia and Jupp (1999), still the most comprehensive text on directional statistics. Many of those developments have been stimulated by interesting applications in fields as diverse as astronomy, medicine, genetics, neurology, aeronautics, acoustics, image analysis, text mining, environmetrics, and machine learning. We begin by considering developments for the exploratory analysis of directional data before progressing to distributional models, general approaches to inference, hypothesis testing, regression, nonparametric curve estimation, methods for dimension reduction, classification and clustering, and the modelling of time series, spatial and spatio- temporal data. An overview of currently available software for analysing directional data is also provided, and potential future developments discussed. Keywords: Classification; Clustering; Dimension reduction; Distributional -
Chapter 2 Parameter Estimation for Wrapped Distributions
ABSTRACT RAVINDRAN, PALANIKUMAR. Bayesian Analysis of Circular Data Using Wrapped Dis- tributions. (Under the direction of Associate Professor Sujit K. Ghosh). Circular data arise in a number of different areas such as geological, meteorolog- ical, biological and industrial sciences. We cannot use standard statistical techniques to model circular data, due to the circular geometry of the sample space. One of the com- mon methods used to analyze such data is the wrapping approach. Using the wrapping approach, we assume that, by wrapping a probability distribution from the real line onto the circle, we obtain the probability distribution for circular data. This approach creates a vast class of probability distributions that are flexible to account for different features of circular data. However, the likelihood-based inference for such distributions can be very complicated and computationally intensive. The EM algorithm used to compute the MLE is feasible, but is computationally unsatisfactory. Instead, we use Markov Chain Monte Carlo (MCMC) methods with a data augmentation step, to overcome such computational difficul- ties. Given a probability distribution on the circle, we assume that the original distribution was distributed on the real line, and then wrapped onto the circle. If we can unwrap the distribution off the circle and obtain a distribution on the real line, then the standard sta- tistical techniques for data on the real line can be used. Our proposed methods are flexible and computationally efficient to fit a wide class of wrapped distributions. Furthermore, we can easily compute the usual summary statistics. We present extensive simulation studies to validate the performance of our method. -
A General Approach for Obtaining Wrapped Circular Distributions Via Mixtures
UC Santa Barbara UC Santa Barbara Previously Published Works Title A General Approach for obtaining wrapped circular distributions via mixtures Permalink https://escholarship.org/uc/item/5r7521z5 Journal Sankhya, 79 Author Jammalamadaka, Sreenivasa Rao Publication Date 2017 Peer reviewed eScholarship.org Powered by the California Digital Library University of California 1 23 Your article is protected by copyright and all rights are held exclusively by Indian Statistical Institute. This e-offprint is for personal use only and shall not be self- archived in electronic repositories. If you wish to self-archive your article, please use the accepted manuscript version for posting on your own website. You may further deposit the accepted manuscript version in any repository, provided it is only made publicly available 12 months after official publication or later and provided acknowledgement is given to the original source of publication and a link is inserted to the published article on Springer's website. The link must be accompanied by the following text: "The final publication is available at link.springer.com”. 1 23 Author's personal copy Sankhy¯a:TheIndianJournalofStatistics 2017, Volume 79-A, Part 1, pp. 133-157 c 2017, Indian Statistical Institute ! AGeneralApproachforObtainingWrappedCircular Distributions via Mixtures S. Rao Jammalamadaka University of California, Santa Barbara, USA Tomasz J. Kozubowski University of Nevada, Reno, USA Abstract We show that the operations of mixing and wrapping linear distributions around a unit circle commute, and can produce a wide variety of circular models. In particular, we show that many wrapped circular models studied in the literature can be obtained as scale mixtures of just the wrapped Gaussian and the wrapped exponential distributions, and inherit many properties from these two basic models. -
Bayesian Orientation Estimation and Local Surface Informativeness for Active Object Pose Estimation Sebastian Riedel
Drucksachenkategorie Drucksachenkategorie Bayesian Orientation Estimation and Local Surface Informativeness for Active Object Pose Estimation Sebastian Riedel DEPARTMENT OF INFORMATICS TECHNISCHE UNIVERSITAT¨ MUNCHEN¨ Master’s Thesis in Informatics Bayesian Orientation Estimation and Local Surface Informativeness for Active Object Pose Estimation Bayessche Rotationsschatzung¨ und lokale Oberflachenbedeutsamkeit¨ fur¨ aktive Posenschatzung¨ von Objekten Author: Sebastian Riedel Supervisor: Prof. Dr.-Ing. Darius Burschka Advisor: Dipl.-Ing. Simon Kriegel Dr.-Inf. Zoltan-Csaba Marton Date: November 15, 2014 I confirm that this master’s thesis is my own work and I have documented all sources and material used. Munich, November 15, 2014 Sebastian Riedel Acknowledgments The successful completion of this thesis would not have been possible without the help- ful suggestions, the critical review and the fruitful discussions with my advisors Simon Kriegel and Zoltan-Csaba Marton, and my supervisor Prof. Darius Burschka. In addition, I want to thank Manuel Brucker for helping me with the camera calibration necessary for the acquisition of real test data. I am very thankful for what I have learned throughout this work and enjoyed working within this team and environment very much. This thesis is dedicated to my family, first and foremost my parents Elfriede and Kurt, who supported me in the best way I can imagine. Furthermore, I would like to thank Irene and Eberhard, dear friends of my mother, who supported me financially throughout my whole studies. vii Abstract This thesis considers the problem of active multi-view pose estimation of known objects from 3d range data and therein two main aspects: 1) the fusion of orientation measure- ments in order to sequentially estimate an objects rotation from multiple views and 2) the determination of informative object parts and viewing directions in order to facilitate plan- ning of view sequences which lead to accurate and fast converging orientation estimates. -
Bayesian Methods of Earthquake Focal Mechanism Estimation and Their Application to New Zealand Seismicity Data ’
Final Report to the Earthquake Commission on Project No. UNI/536: ‘Bayesian methods of earthquake focal mechanism estimation and their application to New Zealand seismicity data ’ David Walsh, Richard Arnold, John Townend. June 17, 2008 1 Layman’s abstract We investigate a new probabilistic method of estimating earthquake focal mech- anisms — which describe how a fault is aligned and the direction it slips dur- ing an earthquake — taking into account observational uncertainties. Robust methods of estimating focal mechanisms are required for assessing the tectonic characteristics of a region and as inputs to the problem of estimating tectonic stress. We make use of Bayes’ rule, a probabilistic theorem that relates data to hypotheses, to formulate a posterior probability distribution of the focal mech- anism parameters, which we can use to explore the probability of any focal mechanism given the observed data. We then attempt to summarise succinctly this probability distribution by the use of certain known probability distribu- tions for directional data. The advantages of our approach are that it (1) models the data generation process and incorporates observational errors, particularly those arising from imperfectly known earthquake locations; (2) allows explo- ration of all focal mechanism possibilities; (3) leads to natural estimates of focal mechanism parameters; (4) allows the inclusion of any prior information about the focal mechanism parameters; and (5) that the resulting posterior PDF can be well approximated by generalised statistical distributions. We demonstrate our methods using earthquake data from New Zealand. We first consider the case in which the seismic velocity of the region of interest (described by a veloc- ity model) is presumed to be precisely known, with application to seismic data from the Raukumara Peninsula, New Zealand. -
UNIVERSITY of CALIFORNIA Los Angeles Models for Spatial Point
UNIVERSITY OF CALIFORNIA Los Angeles Models for Spatial Point Processes on the Sphere With Application to Planetary Science A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Statistics by Meihui Xie 2018 c Copyright by Meihui Xie 2018 ABSTRACT OF THE DISSERTATION Models for Spatial Point Processes on the Sphere With Application to Planetary Science by Meihui Xie Doctor of Philosophy in Statistics University of California, Los Angeles, 2018 Professor Mark Stephen Handcock, Chair A spatial point process is a random pattern of points on a space A ⊆ Rd. Typically A will be a d-dimensional box. Point processes on a plane have been well-studied. However, not much work has been done when it comes to modeling points on Sd−1 ⊂ Rd. There is some work in recent years focusing on extending exploratory tools on Rd to Sd−1, such as the widely used Ripley's K function. In this dissertation, we propose a more general framework for modeling point processes on S2. The work is motivated by the need for generative models to understand the mechanisms behind the observed crater distribution on Venus. We start from a background introduction on Venusian craters. Then after an exploratory look at the data, we propose a suite of Exponential Family models, motivated by the Von Mises-Fisher distribution and its gener- alization. The model framework covers both Poisson-type models and more sophisticated interaction models. It also easily extends to modeling marked point process. For Poisson- type models, we develop likelihood-based inference and an MCMC algorithm to implement it, which is called MCMC-MLE. -
Manhattan World Inference in the Space of Surface Normals
IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 40, NO. 1, JANUARY 2018 235 The Manhattan Frame Model—Manhattan World Inference in the Space of Surface Normals Julian Straub , Member, IEEE, Oren Freifeld , Member, IEEE, Guy Rosman, Member, IEEE, John J. Leonard, Fellow, IEEE, and John W. Fisher III, Member, IEEE Abstract—Objects and structures within man-made environments typically exhibit a high degree of organization in the form of orthogonal and parallel planes. Traditional approaches utilize these regularities via the restrictive, and rather local, Manhattan World (MW) assumption which posits that every plane is perpendicular to one of the axes of a single coordinate system. The aforementioned regularities are especially evident in the surface normal distribution of a scene where they manifest as orthogonally-coupled clusters. This motivates the introduction of the Manhattan-Frame (MF) model which captures the notion of an MW in the surface normals space, the unit sphere, and two probabilistic MF models over this space. First, for a single MF we propose novel real-time MAP inference algorithms, evaluate their performance and their use in drift-free rotation estimation. Second, to capture the complexity of real-world scenes at a global scale, we extend the MF model to a probabilistic mixture of Manhattan Frames (MMF). For MMF inference we propose a simple MAP inference algorithm and an adaptive Markov-Chain Monte-Carlo sampling algorithm with Metropolis-Hastings split/merge moves that let us infer the unknown number of mixture components. We demonstrate the versatility of the MMF model and inference algorithm across several scales of man-made environments. -
On Wrapped Binomial Model Characteristics
Mathematics and Statistics 2(7): 231-234, 2014 http://www.hrpub.org DOI: 10.13189/ms.2014.020701 On Wrapped Binomial Model Characteristics S.V.S.Girija1,*, A.V. Dattatreya Rao2, G.V.L.N. Srihari 3 1Mathematics, Hindu College, Guntur, India 2Acharya Nagarjuna University, Guntur, India 3Mathematics, Aurora Engineering College, Bhongir *Corresponding Author: [email protected] Copyright © 2014 Horizon Research Publishing All rights reserved. Abstract In this paper, a new discrete circular model, the latter case, the probability density function f (θ ) the Wrapped Binomial model is constructed by applying the method of wrapping a discrete linear model. The exists and has the following basic properties. characteristic function of the Wrapped Binomial Distribution f (θ ) ≥ 0 is also derived and the population characteristics are studied. 2π Keywords Characteristic Function, Circular Models, ∫ fd(θθ) =1 Trigonometric Moments, Wrapped Models 0 ff(θ) =( θπ + 2 k) , for any integer k, That is , f (θ ) is periodic with period 2.π 1. Introduction 2.1. Wrapped Discrete Circular Random Variables Circular or directional data arise in many diverse fields, such as Biology, Geology, Physics, Meteorology, and If X is a discrete random variable on the set of integers, + psychology, Medicine, Image Processing, Political Science, then reduction modulo 2π mm( ∈ ) wraps the integers Economics and Astronomy (Mardia and Jupp (2000). on to the group of mth roots of unity which is a sub group of Wrapped circular models (Girija (2010)), stereographic unit circle. circular models (Phani (2013)) and Offset circular models i.e. Φ=2ππxm( mod 2 ) (Radhika (2014)) provide a rich and very useful class of models for circular as well as l-axial data. -
A New Unified Approach for the Simulation of a Wide Class of Directional Distributions
This is a repository copy of A New Unified Approach for the Simulation of a Wide Class of Directional Distributions. White Rose Research Online URL for this paper: http://eprints.whiterose.ac.uk/123206/ Version: Accepted Version Article: Kent, JT orcid.org/0000-0002-1861-8349, Ganeiber, AM and Mardia, KV orcid.org/0000-0003-0090-6235 (2018) A New Unified Approach for the Simulation of a Wide Class of Directional Distributions. Journal of Computational and Graphical Statistics, 27 (2). pp. 291-301. ISSN 1061-8600 https://doi.org/10.1080/10618600.2017.1390468 © 2018 American Statistical Association, Institute of Mathematical Statistics, and Interface Foundation of North America. This is an author produced version of a paper published in Journal of Computational and Graphical Statistics. Uploaded in accordance with the publisher's self-archiving policy. Reuse Items deposited in White Rose Research Online are protected by copyright, with all rights reserved unless indicated otherwise. They may be downloaded and/or printed for private study, or other acts as permitted by national copyright laws. The publisher or other rights holders may allow further reproduction and re-use of the full text version. This is indicated by the licence information on the White Rose Research Online record for the item. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing [email protected] including the URL of the record and the reason for the withdrawal request. [email protected] https://eprints.whiterose.ac.uk/ A new unified approach for the simulation of a wide class of directional distributions John T. -
Unscented Orientation Estimation Based on the Bingham Distribution
1 Unscented Orientation Estimation true limit distribution might be the wrapped normal distribution which Based on the Bingham Distribution arises by wrapping the density of a normal distribution around an interval of length 2π. Thus, it differs in its shape from the classical Igor Gilitschenski∗, Gerhard Kurzy, Simon J. Julierz, Gaussian distribution. and Uwe D. Hanebecky ∗ Autonomous Systems Laboratory (ASL) A. Orientation Estimation using Directional Statistics Institute of Robotics and Intelligent Systems Swiss Federal Institute of Technology (ETH) Zurich, Switzerland There has been a lot of work on orientation estimation but almost [email protected] all methods are based on the assumption that the uncertainty can be adequately represented by a Gaussian distribution. Thus, they are yIntelligent Sensor-Actuator-Systems Laboratory (ISAS) usually using the extended Kalman filter (EKF) or the unscented Institute for Anthropomatics and Robotics Kalman filter (UKF) [2]. Although three parameters are sufficient to Karlsruhe Institute of Technology (KIT), Germany represent orientation, they often suffer from an ambiguity problem [email protected], [email protected] known as “gimbal lock”. Therefore, many applications use quaternions [3], [4]. These represent uncertainty as a point on the surface of z Virtual Environments and Computer Graphics Group a four dimensional hypersphere. Moreover, current approaches use Department of Computer Science nonlinear projection [5] or other ad hoc approaches to push the state University College London (UCL), United Kingdom estimate back onto the surface of the hypersphere of unit quaternions. [email protected] For properly representing uncertain orientations, we need to use an antipodally symmetric probability distribution defined on this Abstract—In this work, we develop a recursive filter to estimate orientation in 3D, represented by quaternions, using directional distribu- hypersphere reflecting the fact that the unit quaternions q and −q tions.